5.3: A Population Proportion
During an election year, we see articles in the newspaper that state confidence intervals in terms of proportions or percentages. For example, a poll for a particular candidate running for president might show that the candidate has 40% of the vote within three percentage points (if the sample is large enough). Often, election polls are calculated with 95% confidence, so, the pollsters would be 95% confident that the true proportion of voters who favored the candidate would be between 0.37 and 0.43: (0.40 – 0.03,0.40 + 0.03).
Investors in the stock market are interested in the true proportion of stocks that go up and down each week. Businesses that sell personal computers are interested in the proportion of households in the United States that own personal computers. Confidence intervals can be calculated for the true proportion of stocks that go up or down each week and for the true proportion of households in the United States that own personal computers.
The procedure to find the confidence interval, the sample size, the error bound, and the confidence level for a proportion is similar to that for the population mean, but the formulas are different. How do you know you are dealing with a proportion problem? First, the underlying distribution is a binomial distribution. (There is no mention of a mean or average.) If \(X\) is a binomial random variable, then
\[X \sim B(n, p)\]
where \(n\) is the number of trials and \(p\) is the probability of a success.
To form a proportion, take \(X\), the random variable for the number of successes and divide it by \(n\), the number of trials (or the sample size). The random variable \(P′ \) (read "P prime") is that proportion,
\[P' = \dfrac{X}{n}\]
(Sometimes the random variable is denoted as \(\hat{P}\), read "P hat".)
When \(n\) is large and \(p\) is not close to zero or one, we can use the normal distribution to approximate the binomial.
\[X \sim N(np, \sqrt{npq})\]
If we divide the random variable, the mean, and the standard deviation by \(n\), we get a normal distribution of proportions with \(P′ \), called the estimated proportion, as the random variable. (Recall that a proportion as the number of successes divided by \(n\).)
\[\dfrac{X}{n} = P' - N\left(\dfrac{np}{n}, \dfrac{\sqrt{npq}}{n}\right)\]
Using algebra to simplify:
\[\dfrac{\sqrt{npq}}{n} = \sqrt{\dfrac{pq}{n}}\]
P′ follows a normal distribution for proportions:
\[\dfrac{X}{n} = P' - N\left(\dfrac{np}{n}, \dfrac{\sqrt{npq}}{n}\right)\]
The confidence interval has the form
\[(p′ – EBP, p′ + EBP).\]
where
- \(EBP\) is error bound for the proportion.
- \(p′ = \dfrac{x}{n}\)
- \(p′ =\) the estimated proportion of successes ( p′ is a point estimate for p , the true proportion.)
- \(x =\) the number of successes
- \(n =\) the size of the sample
The error bound (EBP) for a proportion is
\[EBP = \left(z_{\frac{\alpha}{2}}\right)\left(\sqrt{\dfrac{p'q'}{n}}\right)\]
where \(q\ = 1 - p'\).
This formula is similar to the error bound formula for a mean, except that the "appropriate standard deviation" is different. For a mean, when the population standard deviation is known, the appropriate standard deviation that we use is \(\dfrac{\sigma}{\sqrt{n}}\). For a proportion, the appropriate standard deviation is
\[\sqrt{\dfrac{pq}{n}}.\]
However, in the error bound formula, we use
\[\sqrt{\dfrac{p'q'}{n}}\]
as the standard deviation, instead of
\[\sqrt{\dfrac{pq}{n}}.\]
In the error bound formula, the sample proportions p′ and q′ are estimates of the unknown population proportions p and q . The estimated proportions \(p′\) and \(q′\) are used because \(p\) and \(q\) are not known. The sample proportions \(p′\) and \(q′\) are calculated from the data: \(p′\) is the estimated proportion of successes, and \(q′\) is the estimated proportion of failures.
The confidence interval can be used only if the number of successes \(np′\) and the number of failures \(nq′\) are both greater than five.
Normal Distribution of Proportions
For the normal distribution of proportions, the \(z\)-score formula is as follows.
If
\[P' - N\left(p, \sqrt{\dfrac{pq}{n}}\right)\]
then the \(z\)-score formula is
\[z = \dfrac{p'-p}{\sqrt{\dfrac{pq}{n}}}\]
Example \(\PageIndex{1}\)
Suppose that a market research firm is hired to estimate the percent of adults living in a large city who have cell phones. Five hundred randomly selected adult residents in this city are surveyed to determine whether they have cell phones. Of the 500 people surveyed, 421 responded yes - they own cell phones. Using a 95% confidence level, compute a confidence interval estimate for the true proportion of adult residents of this city who have cell phones.
Solution A
- The first solution is step-by-step (Solution A).
- The second solution uses a function of the TI-83, 83+ or 84 calculators (Solution B).
Let \(X =\) the number of people in the sample who have cell phones. \(X\) is binomial.
\[X \sim B(500,\dfrac{421}{500}).\]
To calculate the confidence interval, you must find \(p′\), \(q′\), and \(EBP\).
- \(n = 500\)
- \(x =\) the number of successes \(= 421\)
\[p′ = \dfrac{x}{n} = \dfrac{421}{500} = 0.842\]
- \(p′ = 0.842\) is the sample proportion; this is the point estimate of the population proportion.
\[q′ = 1 – p′ = 1 – 0.842 = 0.158\]
Since \(CL = 0.95\), then
\[\alpha = 1 – CL = 1 – 0.95 = 0.05\left(\dfrac{\alpha}{2}\right) = 0.025.\]
Then
\[z_{\dfrac{\alpha}{2}} = z_{0.025 = 1.96}\]
Use the TI-83, 83+, or 84+ calculator command invNorm(0.975,0,1) to find \(z_{0.025}\). Remember that the area to the right of \(z_{0.025}\) is \(0.025\) and the area to the left of \(z_{0.025}\) is \(0.975\). This can also be found using appropriate commands on other calculators, using a computer, or using a Standard Normal probability table.
\[EBP = \left(z_{\dfrac{\alpha}{2}}\right)\sqrt{\dfrac{p'q'}{n}} = (1.96)\sqrt{\dfrac{(0.842)(0.158)}{500}} = 0.032\]
\[p' – EBP = 0.842 – 0.032 = 0.81\]
\[p′ + EBP = 0.842 + 0.032 = 0.874\]
The confidence interval for the true binomial population proportion is \((p′ – EBP, p′ +EBP) = (0.810, 0.874)\).
Interpretation
We estimate with 95% confidence that between 81% and 87.4% of all adult residents of this city have cell phones.
Explanation of 95% Confidence Level
Ninety-five percent of the confidence intervals constructed in this way would contain the true value for the population proportion of all adult residents of this city who have cell phones.
Solution B
Press
STAT
and arrow over to
TESTS
.
Arrow down to
A:1-PropZint
. Press
ENTER
.
Arrow down to xx and enter 421.
Arrow down to nn and enter 500.
Arrow down to
C-Level
and enter .95.
Arrow down to
Calculate
and press
ENTER
.
The confidence interval is (0.81003, 0.87397).
Exercise \(\PageIndex{1}\)
Suppose 250 randomly selected people are surveyed to determine if they own a tablet. Of the 250 surveyed, 98 reported owning a tablet. Using a 95% confidence level, compute a confidence interval estimate for the true proportion of people who own tablets.
Answer
(0.3315, 0.4525)
Example \(\PageIndex{2}\)
For a class project, a political science student at a large university wants to estimate the percent of students who are registered voters. He surveys 500 students and finds that 300 are registered voters. Compute a 90% confidence interval for the true percent of students who are registered voters, and interpret the confidence interval.
Answer
- The first solution is step-by-step (Solution A).
- The second solution uses a function of the TI-83, 83+, or 84 calculators (Solution B).
Solution A
- \(x = 300\) and
- \(n = 500\)
\[p' = \dfrac{x}{n} = \dfrac{300}{500} = 0.600\]
\[q′ = 1 − p′ = 1 − 0.600 = 0.400\]
Since \(CL = 0.90\), then
\[\alpha = 1 – CL = 1 – 0.90 = 0.10\left(\dfrac{\alpha}{2}\right) = 0.05\[
\[z_{\dfrac{\alpha}{2}} = z_{0.05} = 1.645\]
Use the TI-83, 83+, or 84+ calculator command invNorm(0.95,0,1) to find \(z_{0.05}\). Remember that the area to the right of \(z_{0.05}\) is 0.05 and the area to the left of \(z_{0.05}\) is 0.95. This can also be found using appropriate commands on other calculators, using a computer, or using a standard normal probability table.
\[EBP = \left(z_{\dfrac{\alpha}{2}}\right)\sqrt{\dfrac{p'q'}{n}} = (1.645)\sqrt{\dfrac{(0.60)(0.40)}{500}} = 0.036\]
\[p′ – EBP = 0.60 − 0.036 = 0.564\]
\[p′ + EBP = 0.60 + 0.036 = 0.636\]
The confidence interval for the true binomial population proportion is \((p′ – EBP, p′ +EBP) = (0.564,0.636)\).
Interpretation
- We estimate with 90% confidence that the true percent of all students that are registered voters is between 56.4% and 63.6%.
- Alternate Wording: We estimate with 90% confidence that between 56.4% and 63.6% of ALL students are registered voters.
Explanation of 90% Confidence Level
Ninety percent of all confidence intervals constructed in this way contain the true value for the population percent of students that are registered voters.
Solution B
Press
STAT
and arrow over to
TESTS
.
Arrow down to
A:1-PropZint
. Press
ENTER
.
Arrow down to xx and enter 300.
Arrow down to nn and enter 500.
Arrow down to
C-Level
and enter 0.90.
Arrow down to
Calculate
and press
ENTER
.
The confidence interval is (0.564, 0.636).
Exercise \(\PageIndex{2}\)
A student polls his school to see if students in the school district are for or against the new legislation regarding school uniforms. She surveys 600 students and finds that 480 are against the new legislation.
a. Compute a 90% confidence interval for the true percent of students who are against the new legislation, and interpret the confidence interval.
Answer
(0.7731, 0.8269); We estimate with 90% confidence that the true percent of all students in the district who are against the new legislation is between 77.31% and 82.69%.
b. In a sample of 300 students, 68% said they own an iPod and a smart phone. Compute a 97% confidence interval for the true percent of students who own an iPod and a smartphone.
Answer
Solution A
Sixty-eight percent (68%) of students own an iPod and a smart phone.
\[p′ = 0.68\]
\[q′ = 1–p′ = 1 – 0.68 = 0.32\]
Since \(CL = 0.97\), we know
\[\alpha = 1 – 0.97 = 0.03\]
and
\[\dfrac{\alpha}{2} = 0.015.\]
The area to the left of \(z_{0.05}\) is 0.015, and the area to the right of \(z_{0.05}\) is 1 – 0.015 = 0.985.
Using the TI 83, 83+, or 84+ calculator function InvNorm(0.985,0,1),
\[z_{0.05} = 2.17\]
\[EPB = \left(z_{\dfrac{\alpha}{2}}\right)\sqrt{\dfrac{p'q'}{n}} = 2.17\sqrt{\dfrac{0.68(0.32)}{300}} \approx 0.0269\]
\[p′ – EPB = 0.68 – 0.0269 = 0.6531\]
\[p′ + EPB = 0.68 + 0.0269 = 0.7069\]
We are 97% confident that the true proportion of all students who own an iPod and a smart phone is between 0.6531 and 0.7069.
Solution B
Press STAT and arrow over to TESTS.
Arrow down to A:1-PropZint. Press ENTER.
Arrow down to x and enter 300*0.68.
Arrow down to n and enter 300.
Arrow down to C-Level and enter 0.97.
Arrow down to Calculate and press ENTER.
The confidence interval is (0.6531, 0.7069).
"Plus Four" Confidence Interval for \(p\)
There is a certain amount of error introduced into the process of calculating a confidence interval for a proportion. Because we do not know the true proportion for the population, we are forced to use point estimates to calculate the appropriate standard deviation of the sampling distribution. Studies have shown that the resulting estimation of the standard deviation can be flawed.
Fortunately, there is a simple adjustment that allows us to produce more accurate confidence intervals. We simply pretend that we have four additional observations. Two of these observations are successes and two are failures. The new sample size, then, is \(n + 4\), and the new count of successes is \(x + 2\). Computer studies have demonstrated the effectiveness of this method. It should be used when the confidence level desired is at least 90% and the sample size is at least ten.
Example \(\PageIndex{3}\)
A random sample of 25 statistics students was asked: “Have you smoked a cigarette in the past week?” Six students reported smoking within the past week. Use the “plus-four” method to find a 95% confidence interval for the true proportion of statistics students who smoke.
Solution A
Six students out of 25 reported smoking within the past week, so \(x = 6\) and \(n = 25\). Because we are using the “plus-four” method, we will use \(x = 6 + 2 = 8\) and \(n = 25 + 4 = 29\).
\[p' = \dfrac{x}{n} = \dfrac{8}{29} \approx 0.276\]
\[q′ = 1 – p′ = 1 – 0.276 = 0.724\]
Since \(CL = 0.95\), we know \(\alpha = 1 – 0.95 = 0.05\) and \(\dfrac{\alpha}{2} = 0.025\).
\[z_{0.025} = 1.96\]
\(EPB = \left(z_{\dfrac{\alpha}{2}}\right)\sqrt{\dfrac{p'q'}{n}} = (1.96)\sqrt{\dfrac{0.276(0.724)}{29}} \approx 0.163\)
\[p′ – EPB = 0.276 – 0.163 = 0.113\]
\[p′ + EPB = 0.276 + 0.163 = 0.439\]
We are 95% confident that the true proportion of all statistics students who smoke cigarettes is between 0.113 and 0.439.
Solution B
Press STAT and arrow over to TESTS.
Arrow down to A:1-PropZint. Press ENTER.
REMINDER
Remember that the plus-four method assume an additional four trials: two successes and two failures. You do not need to change the process for calculating the confidence interval; simply update the values of x and n to reflect these additional trials.
Arrow down to \(x\) and enter eight.
Arrow down to \(n\) and enter 29.
Arrow down to C-Level and enter 0.95.
Arrow down to Calculate and press ENTER.
The confidence interval is (0.113, 0.439).
Exercise \(\PageIndex{3}\)
Out of a random sample of 65 freshmen at State University, 31 students have declared a major. Use the “plus-four” method to find a 96% confidence interval for the true proportion of freshmen at State University who have declared a major.
Solution A
Using “plus four,” we have \(x = 31 + 2 = 33\) and \(n = 65 + 4 = 69\).
\[p′ = 3369 \approx 0.478\]
\[q′ = 1 – p′ = 1 – 0.478 = 0.522\]
Since \(CL = 0.96\), we know \(\alpha = 1 – 0.96 = 0.04\) and \(\dfrac{\alpha}{2} = 0.02\).
\[z_{0.02} = 2.054\]
\[EPB = \left(z_{\dfrac{\alpha}{2}}\right)\sqrt{\dfrac{p'q'}{n}} = (2.054)\left(\sqrt{\dfrac{(0.478)(0.522)}{69}}\right) - 0.124\]
\[p′ – EPB = 0.478 – 0.124 = 0.354\]
\[p′ + EPB = 0.478 + 0.124 = 0.602\]
We are 96% confident that between 35.4% and 60.2% of all freshmen at State U have declared a major.
Solution B
Press STAT and arrow over to TESTS.
Arrow down to A:1-PropZint. Press ENTER.
Arrow down to \(x\) and enter 33.
Arrow down to \(n\) and enter 69.
Arrow down to C-Level and enter 0.96.
Arrow down to Calculate and press ENTER.
The confidence interval is (0.355, 0.602).
Example \(\PageIndex{4}\)
The Berkman Center for Internet & Society at Harvard recently conducted a study analyzing the privacy management habits of teen internet users. In a group of 50 teens, 13 reported having more than 500 friends on Facebook. Use the “plus four” method to find a 90% confidence interval for the true proportion of teens who would report having more than 500 Facebook friends.
Solution A
Using “plus-four,” we have \(x = 13 + 2 = 15\) and \(n = 50 + 4 = 54\).
\[p′ = 1554 \approx 0.278\]
\[q′ = 1 – p′ = 1 − 0.241 = 0.722\]
Since \(CL = 0.90\), we know \(\alpha = 1 – 0.90 = 0.10\) and \(\dfrac{\alpha}{2} = 0.05\).
\[z_{0.05} = 1.645\]
\[EPB = \left(z_{\dfrac{\alpha}{2}}\right)\left(\sqrt{\dfrac{p'q'}{n}}\right) = (1.645)\left(\sqrt{\dfrac{(0.278)(0.722)}{54}}\right) \approx 0.100\]
\[p′ – EPB = 0.278 – 0.100 = 0.178\]
\[p′ + EPB = 0.278 + 0.100 = 0.378\]
We are 90% confident that between 17.8% and 37.8% of all teens would report having more than 500 friends on Facebook.
Solution B
Press STAT and arrow over to TESTS.
Arrow down to A:1-PropZint. Press ENTER.
Arrow down to \(x\) and enter 15.
Arrow down to \(n\) and enter 54.
Arrow down to C-Level and enter 0.90.
Arrow down to Calculate and press ENTER.
The confidence interval is (0.178, 0.378).
Exercise \(\PageIndex{4}\)
The Berkman Center Study referenced in Example talked to teens in smaller focus groups, but also interviewed additional teens over the phone. When the study was complete, 588 teens had answered the question about their Facebook friends with 159 saying that they have more than 500 friends. Use the “plus-four” method to find a 90% confidence interval for the true proportion of teens that would report having more than 500 Facebook friends based on this larger sample. Compare the results to those in Example.
Answer
Solution A
Using “plus-four,” we have \(x = 159 + 2 = 161\) and \(n = 588 + 4 = 592\).
\[p′ = 161592 \approx 0.272\]
\[q′ = 1 – p′ = 1 – 0.272 = 0.728\]
Since CL = 0.90, we know \(\alpha = 1 – 0.90 = 0.10\) and \(\dfrac{\alpha}{2} = 0.05\)
\[EPB = \left(z_{\dfrac{\alpha}{2}}\right)\left(\sqrt{\dfrac{p'q'}{n}}\right) = (1.645)\left(\sqrt{\dfrac{(0.272)(0.728)}{592}}\right) \approx 0.030\]
\[p′ – EPB = 0.272 – 0.030 = 0.242\]
\[p′ + EPB = 0.272 + 0.030 = 0.302\]
We are 90% confident that between 24.2% and 30.2% of all teens would report having more than 500 friends on Facebook.
Solution B
- Press STAT and arrow over to TESTS.
- Arrow down to A:1-PropZint. Press ENTER.
- Arrow down to \(x\) and enter 161.
- Arrow down to \(n\) and enter 592.
- Arrow down to C-Level and enter 0.90.
- Arrow down to Calculate and press ENTER.
- The confidence interval is (0.242, 0.302).
Conclusion: The confidence interval for the larger sample is narrower than the interval from Example. Larger samples will always yield more precise confidence intervals than smaller samples. The “plus four” method has a greater impact on the smaller sample. It shifts the point estimate from 0.26 (13/50) to 0.278 (15/54). It has a smaller impact on the EPB , changing it from 0.102 to 0.100. In the larger sample, the point estimate undergoes a smaller shift: from 0.270 (159/588) to 0.272 (161/592). It is easy to see that the plus-four method has the greatest impact on smaller samples.
Calculating the Sample Size \(n\)
If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size. The error bound formula for a population proportion is
\[EBP = \left(z_{\frac{\alpha}{2}}\right)\left(\sqrt{\dfrac{p'q'}{n}}\right)\]
Solving for \(n\) gives you an equation for the sample size.
\[n = \dfrac{\left(z_{\frac{\alpha}{2}}\right)^{2}(p'q')}{EBP^{2}}\]
Example \(\PageIndex{5}\)
Suppose a mobile phone company wants to determine the current percentage of customers aged 50+ who use text messaging on their cell phones. How many customers aged 50+ should the company survey in order to be 90% confident that the estimated (sample) proportion is within three percentage points of the true population proportion of customers aged 50+ who use text messaging on their cell phones.
Answer
From the problem, we know that \(\bf{EBP = 0.03}\) (3%=0.03) and \(z_{\dfrac{\alpha}{2}} z_{0.05} = 1.645\) because the confidence level is 90%.
However, in order to find \(n\), we need to know the estimated (sample) proportion \(p′\). Remember that \(q′ = 1 – p′\). But, we do not know \(p′\) yet. Since we multiply \(p′\) and \(q′\) together, we make them both equal to 0.5 because \(p′q′ = (0.5)(0.5) = 0.25\) results in the largest possible product. (Try other products: \((0.6)(0.4) = 0.24\); \((0.3)(0.7) = 0.21\); \((0.2)(0.8) = 0.16\) and so on). The largest possible product gives us the largest \(n\). This gives us a large enough sample so that we can be 90% confident that we are within three percentage points of the true population proportion. To calculate the sample size \(n\), use the formula and make the substitutions.
\[n = \dfrac{z^{2}p'q'}{EBP^{2}}\]
gives
\[n = \dfrac{1.645^{2}(0.5)(0.5)}{0.03^{2}} = 751.7\]
Round the answer to the next higher value. The sample size should be 752 cell phone customers aged 50+ in order to be 90% confident that the estimated (sample) proportion is within three percentage points of the true population proportion of all customers aged 50+ who use text messaging on their cell phones.
Exercise \(\PageIndex{5}\)
Suppose an internet marketing company wants to determine the current percentage of customers who click on ads on their smartphones. How many customers should the company survey in order to be 90% confident that the estimated proportion is within five percentage points of the true population proportion of customers who click on ads on their smartphones?
Answer
271 customers should be surveyed. Check the Real Estate section in your local
Glossary
- Binomial Distribution
- a discrete random variable (RV) which arises from Bernoulli trials; there are a fixed number, \(n\), of independent trials. “Independent” means that the result of any trial (for example, trial 1) does not affect the results of the following trials, and all trials are conducted under the same conditions. Under these circumstances the binomial RV \(X\) is defined as the number of successes in \(n\) trials. The notation is: \(X \sim B(\mathbf{n},\mathbf{p})\). The mean is \(\mu = np\) and the standard deviation is \(\sigma = \sqrt{npq}\). The probability of exactly \(x\) successes in \(n\) trials is \(P(X = x = \left(\binom{n}{x}\right))p^{x}q^{n-x}\).
- Error Bound for a Population Proportion (\(EBP\))
- the margin of error; depends on the confidence level, the sample size, and the estimated (from the sample) proportion of successes.


4.7: Apportionment of Legislative Districts
In most states, there are a fixed number of representatives to the state legislature. Rather than apportioning each county a number of representatives, legislative districts are drawn so that each legislator represents a district. The apportionment process, then, comes in the drawing of the legislative districts, with the goal of having each district include approximately the same number of constituents. Because of this goal, a geographically small city may have several representatives, while a large rural region may be represented by one legislator.
When populations change, it becomes necessary to redistrict the regions each legislator represents (Incidentally, this also occurs for the regions that federal legislators represent). The process of redistricting is typically done by the legislature itself, so not surprisingly it is common to see gerrymandering .
Gerrymandering is when districts are drawn based on the political affiliation of the constituents to the advantage of those drawing the boundary.
Consider three districts, simplified to the three boxes below. On the left there is a college area that typically votes Democratic. On the right is a rural area that typically votes Republican. The rest of the people are more evenly split. The middle district has been voting 50% Democratic and 50% Republican.
Solution
As part of a redistricting, a Democratic led committee could redraw the boundaries so that the middle district includes less of the typically Republican voters, thereby making it more likely that their party will win in that district, while increasing the Republican majority in the third district.
The map to the right shows the 38 th congressional district in California in 2004[1]. This district was created through a bi-partisan committee of incumbent legislators. This gerrymandering leads to districts that are not competitive; the prevailing party almost always wins with a large margin.
Solution
The map to the right shows the 4 th congressional district in Illinois in 2004.[2] This district was drawn to contain the two predominantly Hispanic areas of Chicago. The largely Puerto Rican area to the north and the southern Mexican areas are only connected in this districting by a piece of the highway to the west.
[1] http://en.Wikipedia.org/wiki/File:Ca...ct_38_2004.png
[2] en.Wikipedia.org/wiki/File:Il...ict_4_2004.png


8.5: Individual and Group Incentive Plans
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5. How do managers and organizations use incentives and rewards effectively to secure the best possible performance from employees?
We now turn to an examination of various employee incentive programs used by organizations. First, we consider the relative merits of individuals versus group incentive programs. Next, we focus on several relatively new approaches to motivation and compensation. Finally, we suggest several guidelines for effective incentive systems.
Individual versus Group Incentives
Companies usually have choices among various compensation plans and must make decisions about which is most effective for its situation. Incentive systems in organizations are usually divided into two categories on the basis of whether the unit of analysis—and the recipient of the reward—is the individual or a group. Among individual incentive plans, several approaches can be identified, including merit-based compensation (commonly known as merit compensation), piece-rate incentive programs (where people are paid according to the quantity of output), bonus systems of various sorts, and commissions. In each case, rewards are tied fairly directly to the performance level of the individual.
Although individual incentive systems often lead to improved performance, some reservations have been noted. In particular, these programs may at times lead to employees competing with one another, with undesirable results. For instance, department store salespeople on commission may fight over customers, thereby chasing the customers away. After all, customers don’t care who they deal with, only that the service is good. Second, these plans typically are resisted by unions, which prefer compensation to be based on seniority or job classification. Third, where quality control systems are lax, individual incentives such as piece rates may lead employees to maximize units of output while sacrificing quality. And, finally, in order for these programs to be successful, an atmosphere of trust and cooperation is necessary.
In order to overcome some of these shortcomings, many companies have turned to group or organizational incentive plans. Group incentive programs base at least some of an employee’s rewards on group or organization performance. Hence, employees are encouraged to cooperate with one another and with the corporation so that all employees can benefit. Programs such as profit-sharing or gain-sharing plans (discussed below) are designed to tie the employees’ future rewards and prosperity to that of the company and reduce the age-old antagonism between the two. The results are often dramatic.
Creative Pay Practices
Recently, we have seen several innovations in the way corporations approach reward systems. These efforts are designed to facilitate the integration of employee and company interests in a way that maximizes both productivity and quality of working life. Five such creative pay practices should be noted: (1) gain-sharing plans, (2) skills-based incentives, (3) lump-sum pay increases, (4) participative pay decisions, and (5) flexible benefits programs. These approaches, along with their major advantages and drawbacks, are summarized in Table 8.7 .
Gain-Sharing Plans. Giving executives and senior managers bonuses to reflect their contributions to organizational effectiveness is commonplace. In fact, in some companies executive bonuses are often larger than salaries. Recently, companies have increasingly applied this same principle to all employees in the form of gain-sharing (profit-sharing) plans. Here, employees are given a chance to share in corporate productivity gains through increased earnings. The greater the productivity gains, the greater the earnings. Several variations on this theme can be found, including the Scanlon Plan, IMPROSHARE, the Ruker Plan, and the Lincoln Electric Plan. Regardless of the title, the basic plan is similar.
For example, under the Scanlon Plan (probably the oldest such program), three operating guidelines are used: (1) each department or division is treated as a business unit for purposes of performance measurement, (2) specific cost measures associated with the production process are identified and agreed to by all parties, and (3) bonuses are paid to all employees according to a predetermined formula tying the amount of the bonus to the actual cost savings realized during the time period. Under such a plan, it is clearly in the employees’ best interest to contribute to cost savings, thereby increasing their own incomes.
expanding around the GLobe
Providing Feedback in Different Countries
Global workplaces are increasing within the world businesses, and it has become a trend to have managers from one country, most likely the country in which the headquarters arise, manage employees abroad. An important consideration when managing globally is how cultural differences can have a profound effect on performance evaluations, negotiations, and criticisms.
For example, oftentimes in the United States, a method of critical feedback in the “hamburger method” (Step 1: Identify tasks. As a group, identify technical steps that would be involved in implementing. Step 2: Identify options for tasks. Split the team into several small groups. Step 3: Combine results.) is acceptable, while other countries give their feedback with just the meal alone. This strategy in the Netherlands and Germany can be off-putting to other cultures, and when you read into another culture’s technique with your own lens of reference, it can feel wrong.
Managing globally means that you need to do your research on which approach for feedback is best received for the employee’s cultural differences. For example, being direct is key when communicating with a Dutch person. In contrast, in England or the United States, criticism is not delivered directly, but with positive pieces wrapped around the negative. In Asian countries, feedback is often avoided or the message is blurred in order to “save face.” With all of these complications and considerations, it is ever more important to acutely understand the culture, the cultural understandings of employees who are direct reports, and also the lens through which feedback is being viewed as well.
Questions:
- How can a new manager that is working with international employees ensure she is providing reviews in an appropriate manner?
- What methods can a manager employ in her preparation for the review to be successful when providing feedback to employees of different cultures?
Sources: C. Solbach. “Feedback through cultural looking glass.” Krauthammer , September 16, 2015, www.krauthammer.com/en/publi...2/07/feedback- through-cultural-looking-glass; M. Abadi. “The exact same sentence from your boss can mean 'yes,' 'no,' or 'maybe' depending on the country where you work.” Business Insider , December 7, 2017, https://www.businessinsider.com/dire...ulture-2017-12 ; J. Windust. “An International Approach to 360-Degree Feedback.” Cognology , July 26, 2016, https://www.cognology.com.au/interna...gree-feedback/ ; “Giving Employee Feedback To A Culturally Diverse Workforce.” Impraise Blog , accessed January 26, 2019, blog.impraise.com/360-feedba...ences-between- branchescountries-in-feedback-behavior-performance-review.
Skills-Based Incentives. Typical compensation programs are tied to job evaluations. In these, jobs are analyzed to assess their characteristics, and then salary levels are assigned to each job on the basis of factors such as job difficulty and labor market scarcity. In other words, pay levels are set on the basis of the job, not the individual. This approach fails to encourage employees to improve their skills on the job, because there is no reward for the improvement. This thinking also keeps all employees in their places and minimizes the possibility of inter-job transfers.
Under the skills-based incentive program , employees are paid according to their skills level (that is, the number of jobs they can perform), regardless of the actual tasks they are allowed to perform. This approach has proved successful in organizations such as Procter & Gamble and General Foods. Employees are encouraged to learn additional skills and are appropriately rewarded. The organization is provided with a more highly trained and more flexible workforce. However, training and compensation costs are necessarily increased, so the program is appropriate only in some situations. The technique is most often seen as part of a larger quality-of-working-life program, where it is associated with job redesign efforts.
Lump-Sum Pay Increases. Another technique that has received some attention is to allow employees to decide how (that is, in what amounts) they wish to receive their pay raises for the coming year. Under the traditional program, pay raises are paid in equal amounts in each paycheck over the year. Under the alternate plan, employees can elect to receive equal amounts during the year, or they can choose to take the entire raise in one lump-sum pay increase . This plan allows employees greater discretion over their own financial matters. If an employee wants to use the entire pay raise for a vacation, it can be paid in a lump sum in June. Then, if the employee quits before the end of the year, the unearned part of the pay raise is subtracted from the final paycheck. This plan increases the visibility of the reward to the employee. The employee receives, for example, a $600 pay raise (a rather sizable amount) instead of twelve $50 monthly pay raises. As with the flexible rewards system discussed below, however, the administration costs of the lump-sum plan are greater than those of the traditional method.
Participative Pay Decisions. In addition, of concern to many managers is the extent to which employees should be involved in decisions over pay raises. This is the issue of participative pay decisions . Recently, several organizations have been experimenting with involving employees in pay raise decisions, and the results seem to be quite positive. By allowing employees to participate either in the design of the reward system or in actual pay raise decisions (perhaps through a committee), it is argued that decisions of higher quality are made on the basis of greater information. Also, employees then have greater reason to place confidence in the fairness of the decisions. On the negative side, this approach requires considerably more time for both the manager and the participating subordinates. Costs must be weighed against benefits to determine which approach is most suitable for the particular organization and its goals.
Flexible Benefits Systems . A typical fringe benefit package provides the same benefits—and the same number of benefits—to all employees. As a result, individual differences or preferences are largely ignored. Studies by Lawler indicate variations in benefit preferences. 18 For instance, young unmarried men prefer more vacation time, whereas young married men prefer to give up vacation time for higher pay. Older employees want more retirement benefits, whereas younger employees prefer greater income. Through a flexible benefits program (also called a “cafeteria benefits program”), employees are allowed some discretion in the determination of their own packages and can make trade-offs, within certain limits. Organizations such as PepsiCo, TRW, and the Educational Testing Service already use such programs. Although certain problems of administration exist with the programs, efforts in this direction can lead to increased need satisfaction among employees.
We have seen a number of different creative solutions to the compensation dilemma. Which approaches are most effective in motivating employees? This is obviously a difficult question to answer. However, one way to get relevant information on this question is to see what corporations actually use. One such study asked major employers which of a variety of approaches had been used with a high success level. The results are shown in Table 8.8 . As can be seen, skills-based compensation, earned time off, and gain sharing all received high marks from personnel executives, although other programs are also widely supported. It would appear from these results that many approaches can be useful; the choice of which one to use would depend upon the circumstances and goals of a particular organization.
Guidelines for Effective Incentive Programs
Whatever incentive plan is selected, care must be taken to ensure that the plan is appropriate for the particular organization and workforce. In fact, a simple test of the effectiveness of an incentive plan would be as follows: 19
- Does the plan capture attention? Do employees discuss the plan and take pride in their early successes?
- Do employees understand the plan? Can employees explain how the plan works, and do they understand what they must do to earn the incentive?
- Does the plan improve communication? As a result of the plan, do employees understand more about corporate mission, goals, and objectives?
- Does the plan pay out when it should? Are incentives being paid for desired results, and are they withheld for undesirable results?
- Is the company performing better as a result of the plan? Are profits or market share up or down? Have any gains resulted in part from the incentive plan?
If a new (or existing) pay plan can meet these tests, it is probably fairly effective in motivating employee performance and should be retained by the organization. If not, perhaps some other approach should be tried. On the basis of such a test, several specific guidelines can be identified to increase the effectiveness of the programs. These include the following: 20
- Any reward system or incentive plan should be as closely tied to actual job performance as possible. This point was discussed earlier in this chapter.
- If possible, incentive programs should allow for individual differences . They should recognize that different people want different outcomes from a job. Flexible benefits programs such as the ones discussed here make an effort to accomplish this.
- Incentive programs should reflect the type of work that is done and the structure of the organization. This simply means that the program should be tailored to the particular needs, goals, and structures of a given organization. Individual incentive programs, for example, would probably be less successful among unionized personnel than would group programs such as the Scanlon plan. This point has been clearly demonstrated in research by Lawler, which points out that organizations with traditional management and those with more participative management might approach reward systems quite differently in order to be effective.21 As shown in Table 8.9 , both types of company can be effective as long as their reward systems are congruent with their overall approach to management.
- The incentive program should be consistent with the culture and constraints of the organization. Where trust levels are low, for example, it may take considerable effort to get any program to work. In an industry already characterized by high levels of efficiency, basing an incentive system on increasing efficiency even further may have little effect, because employees may see the task as nearly impossible.
- Finally, incentive programs should be carefully monitored over time to ensure that they are being fairly administered and that they accurately reflect current technological and organizational conditions. For instance, it may be appropriate to offer sales clerks in a department store an incentive to sell outdated merchandise because current fashion items sell themselves. Responsibility falls on managers not to select the incentive program that is in vogue or used “next door,” but rather to consider the unique situations and needs of their own organizations. Then, with this understanding, a program can be developed and implemented that will facilitate goal-oriented performance.
concept check
- What are the differences between individual and group incentives?
- What is the variety of reward incentives available to organizations?


7.6: Volume, Orientation, and the Levi-Civita Tensor
Learning Objectives
- Introduction of some geometrical machinery that is used in both special and general relativity
This optional section introduces some geometrical machinery that is used in both special and general relativity.
Volume
Desirable properties
In \(3 + 1\) dimensions, we have a natural way of defining four dimensional volume, which is to pick a frame of reference and let the element of volume be \(dt dx dy dz\) in the Minkowski coordinates of that frame. Although this definition of \(4\)-volume is stated in terms of certain coordinates, it turns out to be Lorentz-invariant (section 2.5). It also has the following desirable properties, which we state for an arbitrary value of \(m\) from \(1\) to \(4\):
- V1. Any two \(m\)-volumes can be compared in terms of their ratio.
- V2. For any \(m\) nonzero vectors, the \(m\)-volume of the parallelepiped they span is nonzero if and only if the vectors are linearly independent (that is, if none of them can be expressed in terms of the others using scalar multiplication and vector addition).
We would also like to have convenient methods for working with three-volume, two-volume (area), and one-volume (length). But the \(m\)-volumes for \(m < 4\) give us headaches if we try to define them so that they obey both V1 and V2 . For example, the obvious way to define length (\(m = 1\)) is to use the metric, but then lightlike vectors would violate V2 .
Affine measure
If we’re willing to abandon V1 , then the following approach succeeds. Consider the \(m = 1\) case. We ignore the metric completely and exploit the fact that in special relativity,
spacetime is ﬂat (postulate P2), so that parallelism works the same way as in Euclidean geometry. Let \(l\) be a line, and suppose we want to define a number system on this line that measures how far apart events are. Depending on the type of line, this could be a measurement of time, of spatial distance, or a mixture of the two.First we arbitrarily single out two distinct points on \(l\) and label them \(0\) and \(1\), as in figure \(\PageIndex{1}\). Next, pick some auxiliary point \(q_0\) not lying on \(l\). Construct \(q_0 q_1\) and parallel to \(01\) and \(1q_1\) parallel to \(0q_0\), forming the parallelogram shown in the figure. Continuing in this way, we have a scaffolding of parallelograms adjacent to the line, determining an infinite lattice of points \(1, 2, 3, ...\) on the line, which represent the positive integers. Fractions can be defined in a similar way. For example, \(\tfrac{1}{2}\) is defined as the point such that when the initial lattice segment \(0\tfrac{1}{2}\) is extended by the same construction, the next point on the lattice is \(1\). The continuously varying variable constructed in this way is called an affine parameter . The time measured by a freefalling clock is an example of an affine parameter, as is the distance measured by the tick marks on a free-falling ruler. An affine parameter can only be defined along a straight world-line, not an arbitrary curve. The affine measurement of \(1\)-volume violates V1 , because it only allows us to compare distances that lie on \(l\) or parallel to it. On the other hand, it has the advantage over metric measurement that it allows us to measure lengths along lightlike lines.
Figure \(\PageIndex{2}\) shows how to define an affine measure of \(2\)-volume, and a similar method works for \(3\)-volume.
Linearity
Suppose that a parallelogram is formed with vectors a and b as two of its sides. It we double a, then the area doubles as well,
\[area(2a,b) = 2area(a,b)\]
In general, if we scale either of the vectors by a factor \(c\), the area scales by the same factor, provided that we set some rule for handling signs — an issue that we’ll postpone until the Orientation section below. Something similar happens when we add two vectors, e.g.,
\[area(a,b + c) = area(a,b) + area(a,c)\]
again postponing issues with signs. We refer to these properties as linearity of the affine \(2\)-volume. Any sensible measure of m-volume should have similar linearity properties.
Change of basis
Because we have not made use of the metric so far, all of our measures of area have been relative rather than absolute. As shown in figure \(\PageIndex{4}\), they depend on what parallelogram we choose as our unit of area. The unit cell in figure \(\PageIndex{4}\) (2) is smaller than the one in figure \(\PageIndex{1}\) (1), for two reasons: the vectors defining the edges are shorter, and the angle between them is smaller. Words like “shorter” and “angle” show us resorting to metric measurement, but we could also perform the comparison without using the metric, simply by using parallelogram \(1\) to measure parallelogram \(2\), or \(2\) to measure \(1\). If we think of such a pair of vectors as basis vectors for the plane, then switching our choice of unit parallelogram is equivalent to a change of basis. Areas change in proportion to the determinant of the matrix specifying the change of basis.
Example \(\PageIndex{1}\): A halFLing basis
Suppose that \(a' = a/2\), and \(b' = b/2\). The change of basis from the unprimed pair to the primed pair is given by the matrix
\[\begin{pmatrix} 2 & 0\\ 0 & 2 \end{pmatrix}\]
which has determinant \(4\). Scaling down both basis vectors by a factor of \(2\) has caused a reduction by a factor of \(4\) in the area of the unit parallelogram. If we use the primed parallelogram to measure other areas, then all the areas will come out bigger by a factor of \(4\).
Rotations and Lorentz boosts are changes of basis. They have determinants equal to \(1\), i.e., they preserve spacetime volume.
Orientation
As shown in figure \(\PageIndex{5}\), linearity of area requires that some areas be assigned negative values. If we compare the areas \(+1\) and \(-1\), we see that the only difference is one of orientation, or handedness. In the case to which we have arbitrarily assigned area \(+1\), vector b lies counterclockwise from vector a, but when a is ﬂipped, the relative orientation becomes clockwise.
If you’ve had the usual freshman physics background, then you’ve seen this issue dealt with in a particular way, which is that we assume a third dimension to exist, and define the area to be the vector cross product \(a×b\), which is perpendicular to the plane inhabited by \(a\) and \(b\). The trouble with this approach is that it only works in three dimensions. In four dimensions, suppose that a lies along the \(x\)-axis, and \(b\) along the \(t\)-axis. Then if we were to define \(a×b\), it should be in a direction perpendicular to both of these, but we have more than one such direction. We could pick anything in the \(y-z\) plane.
To get started on this issue in m dimensions, where \(m\) does not necessarily equal \(3\), we can consider the \(m\)-volume of the \(m\)-dimensional parallelepiped spanned by \(m\) vectors. For example, suppose that in \(4\)-dimensional spacetime we pick our \(m\) vectors to be the unit vectors lying along the four axes of the Minkowski coordinates, \(\hat{t},\hat{x},\hat{y}\; \text{and}\; \hat{z}\). From experience with the vector cross product, which is anticommutative, we expect that the sign of the result will depend on the order of the vectors, so let’s take them in that order. Clearly there are only two reasonable values we could imagine for this volume: \(+1\) or \(-1\). The choice is arbitrary, so we make an arbitrary choice. Let’s say that it’s \(+1\) for this order. This amounts to choosing an orientation for spacetime.
A hidden and nontrivial assumption was that once we made this choice at one point in spacetime, it could be carried over to other regions of spacetime in a consistent way. This need not be the case, as suggested in figure \(\PageIndex{6}\).
However, our topic at the moment is special relativity, and as discussed brieﬂy in
section 2.4, it is usually assumed in special relativity that spacetime is topologically trivial, so that this issue arises only in general relativity, and only in spacetimes that probably are not realistic models of our universe.Since \(4\)-volume is invariant under rotations and Lorentz transformations, our choice of an orientation suffices to fix a definition of \(4\)-volume that is a Lorentz invariant. If vectors \(a\), \(b\), \(c\), and \(d\) span a \(4\)-parallelepiped, then the linearity of volume is expressed by saying that there is a set of coefficients \(\epsilon _{ijkl}\) such that
\[V = \epsilon _{ijkl} a^i b^j c^k d^l\]
Notating it this way suggests that we interpret it as abstract index notation, in which case the lack of any indices on \(V\) means that it is not just a Lorentz invariant but also a scalar.
Example \(\PageIndex{2}\): HaLFLing coordinates
Let \((t,x,y,z)\) be Minkowski coordinates, and let \((t',x',y',z') = (2t,2x,2y,2z)\). Let’s consider how each of the factors in our volume equation is affected as we do this change of coordinates.
\[\overset{\underbrace{V}}{\text{no change}}\; \; = \; \; \overset{\underbrace{\epsilon _{\kappa \lambda \mu \nu }}}{\times 1/16}\; \overset{\underbrace{a^{\kappa }}}{\times 2}\; \overset{\underbrace{b^{\lambda }}}{\times 2}\; \overset{\underbrace{c^{\mu }}}{\times 2}\; \overset{\underbrace{d^\nu }}{\times 2}\]
Since our convention is that \(V\) is a scalar, it doesn’t change under a change of coordinates. This forces us to say that the components of change by a factor of \(1/16\) in this example.
The result of Example \(\PageIndex{2}\) tells us that under our convention that volume is a scalar, the components of must change when we change coordinates. One could argue that it would be more logical to think of the transformation in this example as a change of units, in which case the value of \(V\) would be different in the new units; this is a possible alternative convention, but it would have the disadvantage of making it impossible to read off the transformation properties of an object from the number and position of its indices. Under our convention, we can read off the transformation properties in this way. Although
section 7.4 only presented these properties in the case of tensors of rank \(0\) and \(1\), deferring the general description of higher rank tensors to section 9.2, \(\epsilon\)’s transformation properties are, as implied by its four subscripts, those of a tensor of rank \(4\). Different authors use different conventions regarding the definition of \(\epsilon\), which was originally described by the mathematician Levi-Civita.Since by our convention \(\epsilon\) is a tensor, we refer to it as the Levi-Civita tensor. In other conventions, where \(\epsilon\) is not a tensor, it may be referred to as the Levi-Civita symbol. Since the notation is not standardized, I will occasionally put a reminder next to important equations containing \(\epsilon\) stating that this is the tensorial \(\epsilon\).
The Levi-Civita tensor has lots and lots of indices. Scary! Imagine the complexity of this beast. (Sob.) We have four choices for the first index, four for the second, and so on, so that the total number of components is \(256\). Wait, don’t reach for the kleenex. The following example shows that this complexity is illusory.
Example \(\PageIndex{3}\): Volume in Minkowski coordinates
We’ve set up our definitions so that for the parallelepiped \(\hat{t},\hat{x},\hat{y},\hat{z}\), we have \(V = +1\). Therefore
\[\epsilon _{txyz} = +1\]
by definition, and because \(4\)-volume is Lorentz invariant, this holds for any set of Minkowski coordinates.
If we interchange \(x\) and \(y\) to make the list \(\hat{t},\hat{y},\hat{x},\hat{z}\), then as in figure \(\PageIndex{5}\), the volume becomes \(-1\), so
\[\epsilon _{tyxz} = -1\]
Suppose we take the edges of our parallelepiped to be \(\hat{t},\hat{x},\hat{x},\hat{z}\), with \(y\) omitted and \(x\) duplicated. These four vectors are not linearly independent, so our parallelepiped is degenerate and has zero volume.
\[\epsilon _{txxz} = 0\]
From these examples, we see that once any element of has been fixed, all of the others can be determined as well. The rule is that interchanging any two indices ﬂips the sign, and any repeated index makes the result zero.
Example \(\PageIndex{3}\) shows that the the fancy symbol \(\epsilon _{ijkl}\), which looks like a secret Mayan hieroglyph invoking \(256\) different numbers, actually encodes only one number’s worth of information; every component of the tensor either equals this number, or minus this number, or zero. Suppose we’re working in some set of coordinates, which may not be Minkowski, and we want to find this number. A complicated way to find it would be to use the tensor transformation law for a rank-\(4\) tensor (section 9.2). A much simpler way is to make use of the determinant of the metric, discussed in Example 6.2.1. For a list of coordinates ijkl that are sorted out in the order that we define to be a positive orientation, the result is simply \(\epsilon _{ijkl} = \sqrt{\left | det\; g \right |}\). The absolute value sign is needed because a relativistic metric has a negative determinant.
Example \(\PageIndex{4}\): Cartesian coordinates and their halFLIng versions
Consider Euclidean coordinates in the plane, so that the metric is a \(2×2\) matrix, and \(\epsilon _{ij}\) has only two indices. In standard Cartesian coordinates, the metric is \(g = diag(1,1)\), which has \(det\; g = 1\). The Levi-Civita tensor therefore has \(\epsilon _{xy} = +1\]), and its other three components are uniquely determined from this one by the rules discussed in Example \(\PageIndex{3}\). (We could have ﬂipped all the signs if we had wanted to choose the opposite orientation for the plane.) In matrix form, these rules result in
\[\epsilon = \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix}\]
Now transform to coordinates \((x',y') = (2x,2y)\). In these coordinates, the metric is \(g' = diag(1/4,1/4)\), with \(det\; g = 1/16\), so that \(\epsilon _{x'y'} = 1/4\), or in matrix form,
\[\epsilon '= \begin{pmatrix} 0 & 1/4\\ -1/4 & 0 \end{pmatrix}\]
Example \(\PageIndex{5}\): Polar coordinates
In polar coordinates \((r,θ)\), the metric is \(g = diag(1,r^2)\), which has determinant \(r^2\). The Levi-Civita tensor is
\[\epsilon = \begin{pmatrix} 0 & r\\ -r & 0 \end{pmatrix}\]
(taking the same orientation as in Example \(\PageIndex{4}\)).
Example \(\PageIndex{6}\): Area of a circle
Let’s find the area of the unit circle. Its (signed) area is
\[A = \int \text{2-volume}(dr, d\theta )\]
where the order of \(dr\) and \(dθ\) is chosen so that, with the orientation we’ve been using for the plane, the result will come out positive. Using the definition of the Levi-Civita tensor, we have
\[\begin{align*} A &= \int \epsilon _{r\theta } dx^r dx^\theta \\ &= \int_{r = 0}^{1}\int_{\theta =0}^{2\pi }rdrd\theta \\ &= \pi \end{align*}\]
The 3-volume covector
Consider the volume of a three-dimensional subspace of four-dimensional spacetime. Linearity leads to an especially simple characterization of the \(3\)-volume. Let a \(3\)-volume be defined by the parallelepiped spanned by vectors \(a\), \(b\), and \(c\). If we threw in a fourth vector \(d\), we would have a \(4\)-volume, and \(4\)-volume is a scalar. This \(4\)-volume would depend in a linear way on all four vectors, and in particular it would depend linearly on \(d\). But this means we have a scalar that is a linear function of a vector, and such a function is exactly what we mean by a covector. We can therefore define a volume covector \(S\) according to
\[S_l d^l = \text{4-volume(a,b,c,d)}\]
or
\[S_l = \epsilon _{ijkl} a^i b^j c^k \; \; \; \; [\text{tensorial }\epsilon ]\]
The volume covector collects the information about the volume of the \(3\)-parallelepiped, encapsulating it in a convenient form with known transformation properties. In particular, the statement and proof of Gauss’s theorem in \(3 + 1\) dimensions are greatly simplified by the use of this tool. The \(3\)-volume covector, unlike the affine \(3\)-volume, is
defined in an absolute sense rather than in relation to some parallelepiped arbitrarily chosen as a standard. Both the covector and the affine volume fail to satisfy the ratio comparison property V1, since we can’t compare volumes unless they lie in parallel \(3\)-planes.We’ve been visualizing covectors in \(n\) dimensions as stacks of \((n-1)\)-dimensional planes (figure 6.3.1; figure 6.6.1). The volume three-vector should therefore be visualized as a stack of \(3\)-planes in a four-dimensional space. Since most of us can’t visualize things very well in four dimensions, figure \(\PageIndex{8}\) omits one of the dimensions, so that the \(3\)-surfaces appear as two-dimensional planes. The small hand figure \(\PageIndex{1}\) (1) has a certain \(3\)-volume, and the covector that measures it is represented by the stack of \(3\)-planes parallel to it, figure \(\PageIndex{1}\) (2) . The bigger hand figure \(\PageIndex{1}\) (3) has twice the \(3\)-volume, and its covector is represented by a stack of planes with half the spacing.
If we step down from four dimensions to three, then the volume covector formed by vectors \(u\) and \(v\) becomes the vector cross product \(S = u×v\), i.e., \(S_k = \epsilon _{ijk} u^i v^j\).
Example \(\PageIndex{7}\): A vector cross product
Consider Euclidean 3-space in Cartesian coordinates. We know from freshman physics that
\[\hat{z} = \hat{x}\times \hat{y}\]
Reexpressing this in the notation above, we have \(u^x = 1\), \(v^y = 1\), and zero for all the other components of \(u\) and \(v\). Since the Levi-Civita tensor vanishes if we have any duplicated indices, its only nonvanishing component that can be relevant here is \(\epsilon _{xyz}= 1\). (Here we assume the standard right-handed orientation for Cartesian coordinates, and we make use of the fact that \(g = diag(1,1,1)\), so that \(detg = 1\).) The result is
\[S_z = \epsilon _{xyz}u^x v^x = 1\]
as expected. (It doesn’t matter here whether we talk about \(S_z\) or \(S^z\), because with this metric, raising and lowering indices doesn’t change the components of a vector.)
Classification of 3-surfaces
A useful application of the \(3\)-volume covector is in classifying \(3\)-surfaces by how they relate to the light cone. If I nail together three sticks, all at right angles to one another, then I can consider them as a set of basis vectors spanning a three-dimensional space of events. This three-space is ﬂat, so we can call it a hyperplane — or just a plane if, as throughout this section, there is no danger of forgetting that it has three dimensions rather than two. All of the events in this plane are simultaneous in my frame of reference. None of these facts depends on the use of right angles; we just need to make sure that the sticks don’t all lie in the same plane.
The business of a physicist is ultimately to make predictions. That is, if given a set of initial conditions, we can say how our system will evolve through time. These initial conditions are in principle measured throughout all of space, and a plane of simultaneity would be a natural choice for the set of points at which to take the measurements. A surface used for this purpose is called a Cauchy surface.
If a plane is a surface of simultaneity according to some observer, then we call it spacelike. Any particle’s world-line must intersect such a plane exactly once, and this is why it works as a Cauchy surface: we are guaranteed to detect the particle, so that we can account for its effect on the evolution of the cosmos. We could take a spacelike plane and reorient it. For a small enough change in the orientation (that is, a change that could be described by small enough changes in the basis vectors), it will remain spacelike.
When a plane is not spacelike, and remains so under any sufficiently small change in orientation, we call it timelike. In Minkowski coordinates, an example would be the \(t-x-y\) plane. A given particle’s world-line might never cross such a surface, and therefore a timelike plane cannot be used as a Cauchy surface.
A plane that is neither spacelike nor timelike is called lightlike. An example is the surface defined by the equation \(x = t\) in Minkowski coordinates.
The above classification can be stated very succinctly by using the \(3\)-volume covector defined in above. A plane is spacelike, lightlike, or timelike, respectively, if the regions it contains are described by \(3\)-volume covectors that are timelike, lightlike, or spacelike. A surface that is smooth but not necessarily ﬂat can be be described locally according to these categories by considering its tangent plane. For example, a light cone is lightlike at each of its points, and since it is lightlike everywhere, we call it a lightlike surface.  Any spacelike surface, whether curved or ﬂat, can be used as a Cauchy surface.
Lightlike surfaces have some funny properties. Using birdtracks notation, suppose that we form such a surface as the space spanned by the three basis vectors \(\to a\), \(\to b\), and \(\to c\), and let \(S \to \) be the corresponding \(3\)-volume covector. The surface is lightlike, so
\[S \to S = 0\]
Because \(S \to \) is defined as the function giving the \(4\)-volume of a parallelepiped spanned by the bases with a fourth vector \(\to d\), and because this volume vanishes when \(\to d\) is tangent to the surface (
property V2), we have,\[S \to a = S \to b = S \to c = 0\]
So in this sense \(S \to \) is perpendicular to the surface. In Euclidean space we are used to describing the orientation of a surface in terms of the unit normal vector, and this is very nearly what \(S \to \) is, except that it’s a covector rather than a vector, and it also can’t be made to have unit length, since its magnitude is zero. We could fix the first of these two problems by constructing the vector \(\to S\) that is dual to \(S\to \), but this has a disconcerting effect. Combining \(\PageIndex{17}\) with the definition of \(S\to \), we find that \(\to S\) spans a vanishing \(4\)-volume with the basis vectors, and therefore by V2 we find that \(\to S\) is tangent to the surface. Thus in some sense we have a vector that is both parallel to and tangent to a surface — which avoids being absurd because we are really referring to two different objects, the covector \(S\to \) and the vector \(\to S\).


3.3: Presenting Remotely
Learning Objectives
- Identify the occasions and circumstances which require remote presentations.
- Establish the additional skills needed for remote presentations.
- Establish the difference between live and recorded presentations.
- Identify potential solutions for various remote circumstances.
It was a Wednesday afternoon in March. I had just arrived to my afternoon Speech 100 class and was preparing to take roll. As most often is the case, students were chatting and scrolling through their phones. Suddenly, a student made the following unsolicited announcement:
“Hey! CUNY just tweeted that all classes will be remote starting Monday, March 23rd.”
Silence fell over the usually chatty crowd. As students continued to file in, they could sense the uncertainty in the air.
“Okay, it’s going to be fine…we can do this…” was all I could seem to say.
We all knew it was coming, to some degree. Although this was early in the COVID-19 epidemic, we were all familiar with the news. None of us could have predicted that we’d soon be wearing masks as a regular part of life and would continue to do so for more than a year; we couldn’t have known that we wouldn’t see each other in a classroom for the next few semesters—even as I write this, it’s difficult to comprehend.
I meant what I said, I did believe it would be fine. Although this particular class was an in-person version of a course in public speaking, I had been teaching an online version of Speech 100 at Borough of Manhattan Community College for several semesters. My students may not have believed it, but I knew together we could figure it out. And we did.
The value of being able to effectively deliver presentations remotely is not specific to the events surrounding the COVID-19 pandemic, but it was certainly reinforced by it. As technology has advanced and tiny cameras have become customary in many of the devices we deal with on a daily basis, presenting remotely has become more and more common. This chapter is not a set of rules or definitive right and wrongs, but a discussion of general approaches that work for remote presentations.
Why present remotely?
Whether it is the choice of the speaker or the audience, a remote presentation may be the best option. While we are all certainly aware of this being a solution to safety concerns, there are other more positive advantages to presenting remotely. There are also some obvious challenges.
| Advantages to Rem ote Presentations | Challenges to remote presentation |
|---|---|
| • Reach a larger, more distant audience. | • Audience disengagement. |
| • Safer in times of epidemic or social distancing. | • Technological limitation for speaker and audience. |
| • Environmentally friendly. | • Visual aids solutions. |
| • Economical in terms of time and travel. | |
| • More streamlined visual aid options. |
Clearly the author of this chapter finds the advantages of remote presentations outweigh the challenges. This, however, does not suggest that remote presentation is always a preferable choice. Nothing truly takes the place of live, in-person presentations. But different occasions call for different solutions; when there is an option, it’s up to the speaker and/or the audience to decide the best solution.
Let’s discuss some of these advantages and challenges a bit more thoroughly.
Advantages to Remote Presentations
Reach a larger, more distant audience. Without having to have a group of people gather in one place such as a lecture hall, theatre, classroom, etc., clearly the reach of a speaker is not held to such strict geographical boundaries. This opens up a world of possibilities: audiences can enjoy a presentation from someone in an entirely different country; potential employees can be interviewed or submit recordings of themselves to potential employers in other states; students can study abroad without having to leave their homes.
Safer in times of epidemic or social distancing. Though this chapter is being written at a time when the global COVID-19 pandemic is increasingly under control and social distancing guidelines are being lifted every day, what we have learned during the past year of continual quarantine and uncertainty will certainly be a part of our lives moving forward. During cold and flu season, some speakers or audience may choose to deliver or receive a message remotely to protect themselves and their loved ones. Those who have not or cannot receive vaccinations may find it in their best interest to seek a remote option rather than risk infection.
Travel accounts for a good deal of the world’s environmental pollution. The environment benefits from reducing the need for speakers and audiences to travel to be physically together.
Environmentally friendly. According to the National Center for Biotechnology Information:
“As the transmission of novel corona virus (COVID-19) increases rapidly, the whole world adopted the curfew/lockdown activity with restriction of human mobility. The imposition of quarantine stopped all the commercial activity that greatly affects the various important environmental parameters which directly connected to human health. As all the types of social, economic, industrial and urbanization activity suddenly shut off, nature takes the advantages and showed improvement in the quality of air, cleaner rivers, less noise pollution, undisturbed and calm wildlife.” – Arora, Bhaukhandi, & Mishra, Science Direct, November 10, 2020
Economical in terms of time and travel. Travel expenses can be overwhelming and can actually limit our ability to experience more of the world outside of our specific geographic location. With the advent of various technologies, a remote presentation in India can now be viewed by someone in the Bronx who can’t afford to take the time off to fly abroad; a person in Beijing can apply for a job in London without having to pay for a plane ticket; a student who wants to study abroad but also needs to work in their home country can consider distance learning in any country that offers it.
More streamlined visual aid options. Most video conferencing applications also have a “screen share” or “file sharing” option. If used correctly, this function can streamline the process of sharing presentation aids. Anything a speaker has on their desktop, hard drive, or personal device can usually be shared easily with their audience. Notice, however, that visual aids are mentioned as a possible challenge as well, which will be discussed further.
Challenges to Remote Presentation
Audience disengagement. Those same devices with which we can see each other from miles and miles away also do their best to distract us. It can be safe to assume that most of us don’t turn off all of our notifications or close all of our open applications before engaging in video conferencing, though this would be a very good practice to adopt. When part of a remote audience, often participants may opt to listen more passively while engaging in other activities.
Technological limitations for speaker and audience. We must keep in mind that we do not all have access to the same technology. As a speaker, when given the opportunity to choose between live and remote presentation, one might consider the demographic of their potential audience and the likelihood that they will have the necessary technology to be reached. Will they have the device necessary to receive a remote presentation? Will they have access to the bandwidth? As someone who determines the circumstances of a presentation (such as a teacher, supervisor, or potential employer), the equitable approach is to consider the technology to which the speaker has access and with which the speaker is familiar—or at least reasonable expectations to acquire them.
Visual Aid Solutions. If a speaker is familiar with video conferencing platforms which allow them to share their screen seamlessly and has access to the technology with which to do so, they are fortunate and should take advantage of them. Without these skills and capabilities, sharing a visual aid with an audience and being certain the audience can easily see and process it can be challenging. The same applies to audio aids. Later in this chapter, we will discuss best approaches when dealing with limited technology.
Same Idea, Different Approach
A previous chapter in this book discusses the elements of the communication process. This is a standard discussion in any public speaking text, and usually involves a similar description. These are generally considered the elements of the communication process:
- Speaker
- Message
- Channel
- Listener
- Feedback
- Interference
- Situation
The similarities between in-person presentations and remote presentations are most evident in the elements of speaker , message , and listener . The most basic communication model is still in play: a speaker is sending a message to a listener in the hopes that message will be received. The other elements of our communication process — feedback , interference , situation , and channel —are most affected by a remote presentation format.
| Similarities | Differ ences |
|---|---|
| Basic communication model | Conditional or inconsistent feedback |
| Speaker/audience relationship | More easily distracted audience |
| Importance of presentation aids | Speaker and listeners in different contexts |
| Reliance on strong communication skills | Channel for communication depends on technology |
| Audience-centered delivery |
The elements of public speaking which are shared between live and remote presentation have been and will be covered in this book. The differences are more essential to a discussion on remote presentation.
Conditional or inconsistent feedback/More easily distracted audience. A realistic observation of the “work-from-home” habits many of us have developed over the last few years would likely show that we are less focused in our meetings and appointments than before. As our devices are designed to be exciting and interactive, there are perpetual distractions all around us when we are required to surround ourselves with technology. We are not lowering the lights and focussing on one projected slide presentation in the same meeting room, board room, or classroom. We lack the physical conventions which bring a live audience together. Therefore, audience attention is likely to be less focussed. This is simply another type of interference a speaker must work through. The message must be clear and concise, and the speaker should do all they can to grab and hold their audience’s attention. The same care in which a speaker selects their initial attention grabber at the onset of a presentation can be taken with each moment in the presentation thereafter. Furthermore, messages should be made as concise and clear as possible to make it as easy for an audience to commit their focus.
Speaker and Listener in Different Contexts. As mentioned above, our remote audience is not all in the same room. Sometimes, they may not even be in the same moment in time.
If a presentation is recorded, a speaker not only receives no live, non-verbal feedback, but the speaker may never receive any feedback at all from certain audiences. It is still the responsibility of the speaker to try and maintain their audience’s attention, though they may not be present when the audience receives their message. This must be an underlying motivation while recording the presentation.
During a live remote presentation, the audience is composed of independent receivers in various locations with their own set of distractions. What can we do as speakers to bring a sense of unity to our remote audiences? The idea of establishing common ground discussed earlier in this textbook becomes an important tool not only to bond the speaker and the audience, but to help bond the audience members to each other.
Channel for Communication Depends on Technology. Regardless of the technical resources of the speaker or the audience, some level of technology is required to establish a remote connection. While a live video feed is probably the closest a speaker can come to simulating a live presentation, less degrees of technological support can still be successful means of communication. As long as the speaker or presenter can get a message to the audience, the basic communication model can function.
Technology and Presentation Aids
Work with what you’ve got; work with what you know.
We are all presented with different challenges and have access to different resources. While you undeniably must have some sort of device (computer, electronic tablet, smartphone, etc.) and a reliable internet connection, a remote presentation can be successfully achieved with various levels of technological support.
Some speakers will have a good grasp of how to use PowerPoint and share their screen with their audience; some speakers will provide physical presentations aids and show them over video. Some speakers may record their presentation and then spend time in an editing program adding text and images to support their presentation; some speakers may simply record themselves against a nice, neutral background.
While it is always a good idea to improve our technological skills, learning a new software or platform right before a presentation may not be the best choice. Chances are the speaker is nervous enough without adding the element of relying on something unfamiliar.
Using tangible visual aids such as models, printed images, charts and graphs, or other types of artifacts can be just as effective as a more technologically supported solution.
In fact, because so many presentation aids are presented digitally, an argument might be made that a more “old-fashioned” solution could increase audience engagement and help focus attention.
Audio aids and video work a bit differently and depend on the technological support to function properly during presentations. As a teacher, I must be aware that while I stream a video on my desktop and share it to my virtual classroom, my students are often seeing a grainer, choppier version of the video. Playing a video or audio recording on another device that can be picked up by your camera and/or microphone may be an option, i.e. playing a video on an electronic tablet held by the speaker for the virtual audience to see or playing a recording over a speaker that is not attached to the computer and can be picked up by the computers microphone. Just remember, the important part of using a presentation aid is in the title: presentation aid; it must help the speaker get their point across and not provide additional distractions to the audience or obstacles to the presentation. Whenever possible, put yourself in the audience’s position and see the presentation from their viewpoint. This is part of an audience-centered process.
Virtual Eye Contact
The importance of eye contact has been discussed elsewhere in this textbook. While some presentations may be given orally with the on-screen visual aids being the primary optical focus for the audience (such as turning off the speaker’s video and focussing on a shared slide presentation), this section applies to presentations that are attempting to mimic a live presentation model.
No actual eye contact is possible in a remote presentation. When we look at our screen, we are seeing millions of moving pixels making up very accurate visual representations of our audience members. If our audience is gracious enough (or required) to have their video on during a remote meeting, we can honor them by engaging in virtual eye contact.
Personally speaking, when I was new to a remote classroom setting, I discovered not only the opportunity to explore the skills necessary for successful remote presentation, but also to see myself teaching on camera! Not that I’m particular fascinated with my own looks, but I found it very challenging not to get caught up in watching myself speak rather than connecting with my audience. Suddenly, instead of wondering what I must look like while I was presenting information to my students, I could see in real time right on my screen. I learned to set my “grid” up so that I couldn’t see myself as easily. The students may have not known the difference, but I have no doubt that my connection to them improved once I engaged in more regular virtual eye contact.
Whenever possible, a speaker may consider avoiding reading their speaking notes from the same screen on which they are connecting with their audience. In live, in-person presentations, speakers practice taking information from their notes while returning meaningful and regular eye contact with their audience as often as possible. In a remote presentation, an audience may feel more engaged with a speaker if they feel the speaker is splitting their attention between their notes and the listeners rather than speaking to—or through—their notes. If one has a large enough screen on their desktop, this may not be the case. But many of us are dealing with laptops and handheld devices which would not accommodate a split focus as easily.
Standing vs. Sitting
Traditionally, a speaker would stand for a presentation if they have the ability to do so. Gary Genard, founder of the Genard Method of performance-based public speaking training, states that “if you have the choice to sit or stand when presenting, standing is better.” He goes on to say that “speaking virtually benefits from full-body involvement as well.” (Genard).
In a remote setting, many of us work from a desktop or laptop computer which provides the camera and microphone, part of our channel for communicating remotely. These are generally more easily accessed when seated. A speaker who chooses to invest some time and effort into setting up their equipment to accommodate a standing presentation will certainly improve their audience interest and perhaps even enhance their credibility.
Technical Considerations for Remote Presentations
Background
When choosing a location from which to deliver your remote presentation, the background must be taken into consideration. The background should not provide any interference for the speaker. Choose a neutral background or, at the very least, a background that is not distracting. Remember, we are always trying to hold our audience’s attention against their natural tendency to let it wander; we have to help them focus on us. Not everyone has access to a completely blank wall, but we can all take some time to find a background on which to shoot which will give our audience as little to pay attention to (besides us, the speaker) as possible.
Lighting
Believe it or not, lighting has very transformative powers. Many of us have memories of sitting around a dark room or a campfire, holding a flashlight below our chin to shine straight up our faces while we told a scary story. If you’ve never seen this effect, take some time to find a mirror, a flashlight, and a dark room and prepare to see a very different version of yourself.
Auditoriums and classrooms are generally well-lit spaces. This is intentional. Lighting is part of the context in which a presentation is given, and should be taken into careful consideration. Lighting also has the capability of being an obstacle for the speaker under certain circumstances. If a speaker is lit from behind, their audience will see them in shadow; if lit from only one side, half of their face will be in shadow. We have discussed above what happens when lit from below; being lit from directly above casts different kinds of shadows down our faces, though not nearly as scary.
Even if we are working with basic lighting, we can certainly make sure the light source in the room is supporting our presentation as much as possible. We can make sure that whatever device through which we are communicating is between us and the light source. If the light is overhead, we can adjust to make sure it isn’t casting harsh downward shadows. We should always try and take a look at what our audiences are seeing before we present. Most conferencing software allows us to conduct a “meeting” and record it without having other people present. As with most things, a little preparation goes a long way.
Splicing or Cutting Video
Some of us may be called upon to give job interviews by way of recorded presentations. I have done so myself. While all presentations are important on some level, the stakes for a job interview are fairly high. Aside from the obvious care we would want to take with the technical elements above, we want those viewing our video to perceive us as competent communicators. A video that is recorded in a single take, without splicing different clips together, gives the viewer a sense that the speaker is prepared and confident. It also may allow the viewer to forget for a moment they are watching a video and imagine what it may be like to work in person with this potential employee. Splicing videos together may give the viewer a sense that something immediately before or after each clip has gone awry and needs to be fixed. If the video is on the short side and spliced together, it may give the viewer the impression that the potential employee did not care to take the time to record the video over. Often a video job interview is essentially a short informative speech about ourselves; delivering this presentation in a single take can show potential employers that we are prepared and can communicate a message fluently and clearly from beginning to end.
For other types of recorded presentations, the speaker must decide for themselves whether cutting and splicing feels appropriate to the occasion. A cooking demonstration may require a time jump while that turkey roasts in the oven for five hours. A video presentation about a speaker’s home town may provide an opportunity to cover more geography, and splice clips together in which the speaker delivers their message from different locations.
Practicing the Presentation
The old adage states that “If something can go wrong, it will.” While this is not always the truth, presenting remotely adds an entirely new set of potential mishaps. We must be prepared and anticipate what could go wrong and have a plan to prevent it if possible. And just as with a live presentation, we must be prepared to carry on if and when something does go wrong.
Practice both with and without your video support. Do the presentation a few times as if your audience is live and in the room with you. Then, record yourself doing the presentation the way your audience will see you. After taking some time away so the presentation isn’t as fresh in your mind as the speaker, watch the video from the audience’s point of view. Do some self-analysis to see how you will be perceived remotely by your audience. Speakers who are allowing their lighting, microphone, or camera to become an obstacle or an element of interference for them will come off as unprepared and unprofessional. Speakers who are seen and heard easily over video automatically gain a few credibility points with their audience.
Presenting Our Best Selves
Regardless of the method or occasion, whenever we rise to the occasion of speaking publicly, we hope to present our best self and should do everything within our power to do so. The time and effort we put into preparing to present remotely directly contributes to our success. Whether you plan to give a remote commencement speech at a graduation, bring an important issue forward at a virtual town hall in your community, or submit a recorded video job interview for your dream job, your best self deserves the opportunity to shine through regardless of whatever remote channel you use.
Review Questions
- What are some of the advantages and disadvantages of delivering remote presentations?
- What are some of the similarities and differences between remote presentations and live, in-person delivery?
- What are some important technical elements a speaker should take into consideration when preparing a remote presentation?
Class Activities
- Deliver and record a short remote presentation without regards to any lighting, sound, background, or other technical considerations. In other words, make no effort to take those things into consideration. Don’t watch it yet! Deliver and record the presentation again taking all of the same elements into great consideration. Now watch both videos back to back. Notice the differences and how they affect you and the audience.
- Set up a video conference with a few classmates. Before the conference, arrange your surroundings in a way in which you think you are able to deliver a message the best as a speaker; set up your lighting, your recording device, your background, etc. Each student can provide some type of visual or audio aid to share with the group. Take turns analyzing each others’ setups from an audience perspective.
Works Cited
Arora, Shefali, et al. “Coronavirus Lockdown Helped the Environment to Bounce Back.” The Science of the Total Environment , Elsevier B.V., 10 Nov. 2020, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7323667/ .
Genard, Gary. “The Genard Method: Public Speaking Coaching and Training.” Genard Method Public Speaking Training , https://www.genardmethod.com/ .


20.8: ADHD Treatment
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- Last updated
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Learning Objectives
- Treatment Effects of a Drug on Cognitive Functioning in Children with Mental Retardation and ADHD
Research conducted by
Pearson et al. (2003, see reference below)
Case study prepared by
David Lane and Emily Zitek
Overview
This study investigated the cognitive effects of stimulant medication in children with mental retardation and Attention-Deficit/Hyperactivity Disorder. This case study shows the data for the Delay of Gratification (DOG) task. Children were given various dosages of a drug, methylphenidate (MPH) and then completed this task as part of a larger battery of tests. The order of doses was counterbalanced so that each dose appeared equally often in each position. For example, six children received the lowest dose first, six received it second, etc. The children were on each dose one week before testing.
This task, adapted from the preschool delay task of the Gordon Diagnostic System (Gordon, 1983), measures the ability to suppress or delay impulsive behavioral responses. Children were told that a star would appear on the computer screen if they waited “long enough” to press a response key. If a child responded sooner in less than four seconds after their previous response, they did not earn a star, and the 4-second counter restarted. The DOG differentiates children with and without ADHD of normal intelligence (e.g., Mayes et al., 2001), and is sensitive to MPH treatment in these children (Hall & Kataria, 1992).
Questions to Answer
Does higher dosage lead to higher cognitive performance (measured by the number of correct responses to the DOG task)?
Design Issues
This is a repeated-measures design because each participant performed the task after each dosage.
Descriptions of Variables
| Variable | Description |
| d0 | Number of correct responses after taking a placebo |
| d15 | Number of correct responses after taking .15 mg/kg of the drug |
| d30 | Number of correct responses after taking .30 mg/kg of the drug |
| d60 | Number of correct responses after taking .60 mg/kg of the drug |
Data Files
ADHD.xls
Links
Methylphenidate
References
- Gordon M (1983), The Gordon Diagnostic System. DeWitt, NY: Gordon Systems
- Hall CW, Kataria S (1992), Effects of two treatment techniques on delay and vigilance tasks with attention deficit hyperactive disorder (ADHD) children. J Psychol 126:17-25
- Mayes SD, Calhoun SL, Crowell, EW (2002), The Gordon Diagnostic System and WISC-III Freedom from Distractibility index: Validity in identifying clinic-referred children with and without ADHD. Psychol Rep ,91, 575-587.
- Pearson DA, Santos CW, Roache JD, Casat CD, Loveland KA, Lachar D, Lane DM, Faria, LF, Cleveland LA (2003), Treatment effects of methylphenidate on behavioral adjustment in children with mental retardation and ADHD. J Am Acad Child Adolesc Psychiatry, 42, 209-216.
- Pearson, D.A., Santos, C.W., Jerger, S.W., Casat, C.D., Roache, J., Loveland, K.A., Lane, D.M., Lachar, D., Faria, L.P., & Getchell, C. (2003) Treatment effects of methylphenidate on cognitive functioning in children with mental retardation and ADHD. Journal of the American Academy of Child and Adolescent Psychiatry, 43, 677-685.


8.15B: Extremely Halophilic Archaea
- Describe the methods employed by halophilic Archaea to prevent water loss
Halophiles are extremophiles that thrive in environments with very high concentrations of salt. In fact, the very name “halophile” comes from the Greek word for “salt-loving. ” Although some halophilic bacteria and eukaryotes exist, the largest classification of halophiles is in the Archaea domain.
Halophiles can be found anywhere with a salt concentration at least five times greater than that of the ocean. They are categorized as slight, moderate, or extreme halophiles based on the extent of their halotolerance. Halophiles thrive in places such as the Great Salt Lake, Owens Lake in California, evaporation ponds, and the Dead Sea – places that provide an inhospitable environment to most lifeforms.
High salinity represents an extreme environment that relatively few organisms have been able to adapt to and occupy. Most halophilic organisms cope with the high concentrations of salt by expending energy to exclude salt from their cytoplasm to avoid protein aggregation, or “salting out. ” “Normal” organisms would desiccate in these conditions, losing water via osmosis out of the cytoplasm. Halophiles prevent this loss of water by increasing the internal osmolarity of the cell. One way halophilic archaea can increase their internal osmolarity is by accumulating organic compounds – called osmoprotectants – in their cytoplasm. These compatible solutes can be accumulated from the environment or synthesized. The most common compatible solutes are neutral or zwitterionic, and include amino acids, sugars, polyols, betaines and ectoines, as well as derivatives of some of these compounds.
A more radical adaptation to preventing water loss employs the selective influx of potassium (K + ) ions into the cytoplasm. In archaea, this adaptation is restricted to the the extremely halophilic family Haloarchaea (often known as Halobacteriaceae). To use this method, the entire intracellular machinery – including enzymes, structural proteins, and charged amino acids that allow the retention of water molecules on their surfaces – must be adapted to high salt levels. In the compatible solute adaptation, little or no adjustment is required of intracellular macromolecules – in fact, the compatible solutes often act as general stress protectants as well as osmoprotectants.
The extremely halophilic Haloarchaea require at least a 2 M salt concentration and are usually found in saturated solutions (about 36% w/v salts). These are the primary inhabitants of salt lakes, inland seas, and evaporating ponds of seawater. The red color of deep salterns is due to the carotenoids (organic pigment) in these archaea. These archaea require salt for growth and they will lyse if they are exposed to less salty environment.
The high concentration of NaCl in halophilic environment limits the availability of oxygen for respiration. Halophiles are chemoheterotrophs, using light for energy and methane as a carbon source under aerobic or anaerobic conditions.
Key Points
- Halophiles can be found anywhere with a salt concentration at least five times greater than that of the ocean.
- Most halophilic organisms cope with the high concentrations of salt by expending energy to exclude salt from their cytoplasm.
- Halophiles prevent this loss of water by increasing the internal osmolarity of the cell by accumulating osmoprotectants or by the selective uptake of potassium ions.
Key Terms
- halotolerance : The adaptation of a living organism to conditions of high salinity (dissolved salt).
- zwitterionic : Pertaining to a neutral molecule containing both positive and negative charge.
- osmoprotectant : Any osmolyte that helps an organism to survive osmotic stress


2.2.17: Robert Browning (1812-1889)
Selected Poems British Early Realism (Victorianism) Robert Browning was an unknown poet when he fell in love with one of the most famous poets of his time, Elizabeth Barrett. Their relationship was the stuff of love poems; he fell in love with her, although he had never met her, by reading her poetry. At the time, she was an invalid who was kept pretty much a prisoner by her tyrannical father, who had decided never to allow any of his twelve children to marry. After a secret correspondence over twenty months, they eloped to Italy. She recovered enough of her health that she was able to give birth to a son, and they lived in Italy until her death in 1861. It was really only after her death that Robert Browning started to acquire fame as a poet, eventually becoming one of the most respected poets of the Victorian era. Although he wrote poems and plays on a range of topics, some of his most famous poems are his dramatic monologues, often with narrators who have extreme or even psychotic personalities. At first, audiences were startled by the dark humor and occasionally grotesque situations in many of the monologues, unaccustomed to reading about the human psyche in this way. In "My Last Duchess," the Duke addresses a silent representative of a Count whose daughter he wants to marry. The Duke is the epitome of an entitled snob, but his monologue slowly reveals that these qualities led him to become an unrepentant murderer, with a terrifying possessiveness that continues long after his first wife's death. "Porphyria's Lover" contains one of the best twists (the pun will become clear after reading it) in the monologues; it is again a study in psychotic self-absorption, as well as a technical triumph (try reading it without stopping at the end of the lines, but instead by stopping only at punctuation marks). The third monologue presented here is "Childe Roland to the Dark Tower Came," a nightmarish quest with an ambiguous ending. The poem inspired many other writers, including Stephen King, who based his Dark Tower series (1978-2012) on the poem. Consider while reading:
- In "My Last Duchess," the Duke is speaking to a representative of a Count, and he is trying to marry the Count's daughter. Imagine the entire monologue from the perspective of the Count's representative. Why does he want to go downstairs before the Duke does?
- Look up "porphyria" in a dictionary. How does the meaning of the word affect your reading of the poem?
- Where does Childe Roland find the Dark Tower? In what ways can the ending be interpreted?
Written by Laura Getty