6.6: The Starting Point
Now that you have a basic understanding of the writing process, what are the first steps when starting to write a paper? First step: read the assignment guidelines carefully . Second step: think about YOU in relation to the topic .
Step 1: Review assignment guidelines
- Highlight the key points when you read the guidelines and the marking rubric, if provided.
- Look for the verbs in the assignment and on the rubric. Are you being asked to reflect, analyze, argue, or something else? These verbs will help you determine what is expected and the type of writing that is required from you.
- Schedule the timing of when you will engage in each phase of the writing process. See Figure 6.2 . Many factors will affect how long it will take you to write a paper: the length of the paper itself, the complexity of the ideas in the paper, the amount of research that is required before writing a paper, and all of your other competing obligations. For example, if you are expected to conduct a literature review, that will take you longer than writing a two-page reflection where you are only asked to incorporate one reference.
Figure 6.2: Scheduling your timing
Step 2: Think about YOU in relation to the topic
Before beginning the writing process, think about your positionality in relation to the topic. Positionality means how you position yourself in relation to a topic, informed by your identity. If you are writing about an experience, consider the emotions and feelings that have informed your thoughts about what has transpired. Every decision you make in writing (what you choose to write, how you describe the event, what you compare/contrast) conveys meaning. It is helpful to be aware of the meaning that underpins your writing so that you can be consistent and intentional in how you write. Early in your writing journey, you may be more comfortable with a more neutral or objective tone. As your writing evolves, you will become more skilled at integrating yourself into writing via tone, style, and delivery.
Student Tip
Time Management
Time management is important because you will have many nursing assignments. Start by writing the due date on your calendar, and then work backward from the due date to set aside blocks of time when you will work on the assignment. Always plan multiple writing sessions per assignment. Don’t try to move through all phases of the writing process in one evening: trying to work that fast is stressful and does not yield great results. You will plan better, think better, and write better if you space out the steps.
Activity: Check Your Understanding
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15.1C: Malaria
- Reconstruct the route of transmission and life cycle for Plasmodium species that cause malaria and describe its symptoms
Malaria is a parasitic disease that is caused by the bite of an infected Anopheles mosquito. Malaria can be transmitted from mother to baby and by blood transfusions. The Anopheles mosquito transmits the parasites, called sporozoites, upon biting the hosts, into the bloodstream to the liver, where the parasites continue their life cycle. In the liver, the parasites mature and release another form called merozoites, which enter the bloodstream and infect the red blood cells. In the red blood cells, they develop into ring forms called trophozoites and schizonts that in turn, produce further merozoites. Upon infection of the red blood cells, the parasite is able to multiply within the cell, break open and continue infecting additional red blood cells. The symptoms occur in a cyclical manner every 48-72 hours. Malaria is characterized by the development of symptoms that include high fevers, shaking chills, flu-like symptoms, and anemia. The symptoms that persist due to parasitic infection are a result of the release of merozoites into the bloodstream, destruction of the red blood cells and the free circulation of large amounts of hemoglobin in the red blood cells due to disruption.
The five types of malaria parasites include species of Plasmodium. The fives species include: Plasmodium falciparum, Plasmodium vivax, Plasmodium ovale, Plasmodium malariae, and Plasmodium knowlesi . Plasmodium falciparum is responsible for the majority of deaths caused by infection and Plasmodium vivax, ovale and malariae cause a milder form of malaria. The species, Plasmodium knowlesi , commonly causes malaria in macaques but can also cause severe infections in humans.
Malaria is common in temperate climates and the Centers for Disease Control and Prevention (CDC) estimates 300-500 million cases each year. In addition, it is estimated that 1 million people die from it each year as well. Malaria is typically diagnosed by microscopic examination of blood or with antigen-based rapid diagnostic tests. Disease transmission can be reduced by preventing mosquito bites through the use of mosquito nets and insect repellents. However, the mosquitoes which transmit malaria have begun to develop resistance to insecticides and the parasite itself has developed resistance to commonly used antibiotics. As a result of increased resistance, it is extremely difficult to contain the spread of this disease.
Key Points
- The five common species of Plasmodium that cause malaria include: Plasmodium falciparum, Plasmodium vivax, Plasmodium ovale, Plasmodium malariae and Plasmodium knowlesi.
- Mosquitoes transmit the protists by injecting sporozoites into the bloodstream of humans.
- The sporozoites injected into the bloodstream, travel to the liver where they multiply into merozoites, rupture the liver cells, and then return to the bloodstream.

- The symptoms of malaria can present in a cyclic manner.
Key Terms
- merozoites : the organisms formed by multiple fission of a sporozoite within the body of the host.
- antigen : A substance that induces an immune response, usually foreign.
- sporozoites : Any of the minute active bodies into which a sporozoan divides just before it infects a new host cell.


11.2: Statistical Functions
By the end of this section, you will be able to:
- Use basic statistical functions to describe a dataset
- Describe other useful functions for data analysis in Microsoft Excel
- Use the Data Analysis tool for advanced statistical functions
- Understand the Solver add-in and its use for data analysis
Data surrounds us. In business, there is always a need to better understand the company, the competitors, and the customers. You gain this understanding through data collection and data analysis. The field of statistics is centered on analyzing and making sense out of data. You may have taken a statistics class at some point, but even if you have not taken a statistics class, you have probably used some statistical concepts—maybe more often than you think. Concepts such as averages and probabilities are often applicable to everyday life. For example, you may have been exposed to some statistical concepts when learning about political candidates prior to an election. This could include the percentage of registered voters of a particular demographic or the correlation between geographic location and political party. Statistics can be applied across nearly all fields and situations, and there is almost always a need to know more and dig deeper into what the data are showing.
There are two branches of statistics: descriptive and inferential. Descriptive statistics , as the name implies, describe situations. For example, the average transaction amount for purchases in a store at a specific location is descriptive information about that location. Another example of descriptive information is the turnover rate of employees in a manufacturing facility.
Inferential statistics relate to predicting and forecasting. With inferential statistics , you are using historical data and trends to forecast future performance. The regional sales manager might use the last five years of sales data to make a forecast for this year’s total sales. A nonprofit organization might use historical fundraising efforts to make predictions about fundraising for the next two years. Both branches are useful in understanding data. Information garnered from statistics can be used in a variety of ways in businesses to improve performance and track progress. Microsoft Excel has a variety of tools to make statistical analysis easier.
Basic Statistical Functions
There are several commonly used statistical functions that can give you a basic summary and understanding of the information in your spreadsheet. You can access the functions through the Formulas tab. From this tab, you can either choose Insert Function on the far left of the tab or use the grouped functions in the Function Library (Figure 11.28). The Function Library is helpful if you are not quite sure of the appropriate function for your needs. The functions are organized by application (e.g., financial) and include the ones that are most commonly used.
Using the Insert Function command has some advantages. When you click on Insert Function, a window appears with the functions grouped by category like in the Function Library, and then listed alphabetically within each category. When you click on a function, a brief description of the function appears in the window. There is also an option to search for functions by typing a description or keyword. This is useful if you are unsure of the function’s purpose (Figure 11.29).
To use functions, first, click in the empty cell where you want to put the function. Then, use either the Insert Function option or the Function Library to select the function. For this example, you will choose the AVERAGE function to calculate the average of a list of data. When you select a function from the Insert Function window, an input window will appear where you will select the range of data for analysis (Figure 11.30). After you select the range, click OK. Notice that the result of the function is displayed in the lower left of the input window. Also notice that a dotted line will surround the data identified for analysis. The function will also be visible in the Formula Bar.
When you select a function from one of the groups in the Function Library, the function will appear in the spreadsheet (Figure 11.31). You can select the data for analysis and press Enter. A dotted line will still appear around the selected range of cells, and the function still displays in the Formula Bar.
One of the most commonly used functions is SUM . It even has its own button on the Home tab as the AutoSum feature (Figure 11.32). The function gives the total for a selected range of cells. Other commonly used functions in data analysis are MODE , RANGE , MEDIAN , and AVERAGE , which returns the mean of a selected range of data.
Other Useful Functions
There are several other functions available that are not specifically statistical functions, but they can be quite useful when dealing with large datasets or with text. Some of these were covered in Advanced Excel Formulas, Functions, and Techniques. These functions can help you manage and sort datasets of any size. Table 11.1 summarizes some useful functions for data analysis.
| Function | Definition |
|---|---|
| AVERAGEIF | Returns the average of a range that meets specific criteria |
| COUNT | Counts the number of entries in a range |
| COUNTIF | Returns the count of a range that meets specific criteria |
| CORR | Returns the correlation between two ranges of data |
| IF | Tests a condition and returns a predetermined value |
| MAX | Returns the maximum value in a range |
| MIN | Returns the minimum value in a range |
| SUMIF | Returns the sum of a range that meets a specific criterion |
| SUMIFS | Returns the sum of a range that meets multiple specific criteria |
Let us look closer at the AVERAGEIF function. It returns the average of a selected range according to a set of criteria. As with other functions, it is accessed through the Insert Function button on the Data tab or from the Function Library. When selected from the Insert Function tool, an input window will appear with three arguments to fill in (Figure 11.33). In the first box, input the range to examine against the criteria. In the second box, set the criteria, and in the third box, input the range of numbers to be averaged. Both formulas and text can be entered into the criteria. Be sure to put text elements in quotation marks when establishing criteria.
Let’s revisit the employee vacation data and use it to find the average number of vacation days employees have remaining for the North location. The criteria range is the Location column (D2:D21), the criteria is “North,” and the average range is the Days Left column (C2:C21) (Figure 11.34). The function then returns the average number of vacation days remaining for all the employees in the list at the North location.
In some cases, you might need to evaluate data using more than a single criterion. The AVERAGEIFS, COUNTIFS, and SUMIFS functions give the flexibility to add additional criteria. Just remember that each criterion must be matched with a range where the function will search (Figure 11.35). To add an additional criterion with its corresponding range to the analysis, simply press the Tab key on your keyboard and additional criteria will be added to the function arguments.
The functions we have discussed so far deal with numbers, but there are some functions that are useful for dealing with text. These can be particularly helpful when you are importing data into a spreadsheet. These functions can separate out parts of text or take the first several characters and put them in a separate column. Table 11.2 summarizes some useful functions for managing text in a spreadsheet.
| Function | Description |
|---|---|
| CONCATENATE | Combines information in multiple cells into one cell |
| LEFT | Will extract characters from the left of text |
| RIGHT | Will extract characters from the right of text |
| TRIM | Removes spaces from cells except the single space between words |
The CONCATENATE function is useful when data comes from a database or other files. It combines information from multiple columns into a single column. For example, data imported into an Excel spreadsheet often contains first and last names in separate columns. With the CONCATENATE function, you can easily combine those two columns into a single column. To use the function, select the text elements that you want to combine into one cell (Figure 11.36). To include a space, in the Text2 argument insert “ ” (quotation marks with a space between). This will then output the first name last name correctly spaced (Figure 11.37). Use AutoFill to copy the CONCATENATE function through the entire list. In newer versions of Excel, the function has been replaced with CONCAT , which includes additional features.
The Text to Columns tool on the Data tab is the opposite of the CONCATENATE function. It takes a single column of text and separates it into two or more columns. This tool is also a helpful tool if you are importing information. Sometimes data imported from another program might not be in the format you desire or need for further analysis. For example, you can use this tool to place First Name and Last Name in a separate column if they are imported into a single column. Text to Columns can also be used to separate out City, State, and Zip into columns.
Data Analysis Tools Add-in
Excel has many advanced data analysis tools available. These include a wide variety of simple and more complex statistical functions that are available as individual functions in Excel, but it is easier to access the tools through the add-in called Data Analysis. This add-in puts all the statistical tools together in one place. As this section discusses these tools, it will keep the focus on using the tools within Excel and not on statistical techniques, their assumptions, and usage for data analysis. (You can learn more about those topics in a statistics course or resource.) Some of the most commonly used built-in tools include Correlation , Descriptive Statistics , Histogram , and Regression . Additional advanced tools include Analysis of variance (ANOVA) , and several tools for testing hypothesis, including the F-Test , t-Test , and z-Test .
Excel also offers a helpful feature to guide you through ideas for analyzing your dataset, called the Analyze Data help feature, accessed on the far-right side of the Home tab (Figure 11.38). This tool allows you to search for helpful functions or Excel tools to analyze your dataset. A pop-out will appear on the right side of the screen, listing a few examples of ways to understand the data better. If you are finding it difficult to get started on analyzing a dataset, this might be an effective way to generate ideas as you work through the information. Suggested items for data analysis might include charts, graphs, or a PivotTable.
Link to Learning
GCFGlobal.org provides 100 percent free online courses in a number of topics relevant to career preparation. This includes a variety of technology and computer skills courses. They currently offer a number of micro-courses in Excel. Go to this page to see a list of these courses, as well as others offered on the site.
Getting the Data Analysis Tool
Because the Data Analysis tools are an add-in, they are included with Excel but generally not installed by default. To obtain the add-in, go to the File tab, select Options, and select Add-Ins. From there, click on Analysis ToolPak and select Go at the bottom left of the window (Figure 11.39a). Then, select the checkbox for Analysis ToolPak, Analysis ToolPak - VBA, and Solver Add-in. Click OK (Figure 11.39b). This will add both the Data Analysis tool and the Solver add-in to the far-right side of the Data tab (Figure 11.40).
Built-in Data Analysis Tools
The Data Analysis tool add-in offers a variety of statistical tools. You likely won’t use many of them unless you are doing in-depth statistical analysis. However, there are several that are quite helpful for evaluating most datasets. These tools are some of the more frequently used statistical analysis tools regardless of the industry or application. The functions and tools are listed in the Data Analysis library in alphabetical order. Detailed explanations of the context, analysis, and use of these statistical analysis tools is outside the scope of this text. Rather than focus on the statistics here, this section will walk through the steps involved in using each tool.
Correlation
Correlation is used to analyze the relationship between two variables. The function in Excel computes the correlation between two or more variables at one time. The output is in the form of a correlation matrix or grid that shows how the variables are connected to each other. The closer the correlation value is to either +1 or −1, the more strongly connected the variables are to each other. A value closer to zero is a sign that the two values are not strongly connected. A positive correlation indicates that the variables will move in the same direction. This means if one variable is higher, the other variable will also be higher, for instance, when a company’s increased sales also increase their overall company profit. A negative correlation value indicates that the data move in opposite directions, for example, if an increase in price decreases the quantity sold. Only numerical data can be analyzed using correlation. If you have categorical data that you are interested in analyzing, you will need to code it using numbers.
To use the tool, go to the Data tab and select the Data Analysis tool and then Correlation. Then, select the data of interest and click OK (Figure 11.42). Be sure to select Labels in First Row if your dataset contains column headings. It is generally good practice to include these in your analysis as it helps to identify the information and has a more professional look. If there are no headings, the output will list the data by column number. You can also select an area on the page to display the output or have the output on a new tab in the current worksheet or a new worksheet entirely. The output will be static. That means if you change any of the data points in the selection, the correlation will not be updated. You will need to run the Correlation tool again to update it based on new or different data.
WorldCorp would like to investigate the relationship between the price of their products and the quantity sold. Management thinks that the items with the lower prices sell more. For this example, the selected range is the Quantity column and the Price column. Be sure to check Labels in First Row to tell Excel that there are column headings in your selection.
Using the dataset from WorldCorp’s product sales, you can create the correlation matrix in Figure 11.43 as the output. The default for Excel is to include several decimal points. Often, this level of detail is not necessary, so you may need to clean the output by reducing the number of decimal points and changing the column widths. You might also consider additional formatting such as bolding certain numbers or filling cells with color. According to the output, the correlation between sales and price is not very strong. The value −0.092 is closer to zero than it is to one, which indicates that management’s suspicion that more products at the lower price range are sold is not correct.
Descriptive Statistics
The Descriptive Statistics tool provides a summary of the dataset in a table format. The summary information includes items such as the mean, variance, standard deviation, and range. You can select a single column of data or multiple columns at once. You can also analyze data that is grouped in rows rather than columns. Be sure to check the appropriate box when prompted if your data is in rows. As with correlation, you can display the information on the current worksheet, on a new worksheet, or in an entirely new workbook. To access Descriptive Statistics, go to the Data tab and then Data Analysis. Choose Descriptive Statistics (Figure 11.44). When the input window appears, select the data you want to analyze. Be sure to check the box to indicate that the data has a header row. Next, determine the desired location for the output result. Finally, select Summary Statistics at the bottom of the input window and click OK.
Notice that you also have an option to select confidence levels and both the K th largest and smallest values in the output. These are more advanced level statistical analyses. If desired, choose those options to be included in the output. The default setting for confidence intervals in Excel is 95 percent, but you can customize that setting. If you use the dataset for WorldCorp, you obtain the following output when you use the Descriptive Statistics tool (Figure 11.45).
Notice with this output, you will again need to adjust the number of displayed decimal points to give it a more professional look. You can also choose to format the numbers with commas or as currency, if applicable.
Histograms
A histogram is a chart that displays the number of occurrences of data within a defined range. This is a convenient way to easily see the ranges of most of the data. It is not an in-depth statistical analysis of the data; instead, it provides a summary of the overall distribution of the data. Histograms can be used with data that is at least at the ordinal level of measurement. The histogram function can be accessed through the Data Analysis library. When selected, as with others in the analysis library, an input window will appear (Figure 11.46). You will need to input some information such as the input range and the bin range .
The input range represents the data being divided into the various “bins,” or category ranges. These bins must be mutually exclusive and collectively exhaustive. In other words, each data point needs to have only one bin into which it fits. This means the bin ranges must be set up to accommodate the maximum and minimum values. Excel will sort the data correctly into the bins. Other options for the Histogram tool are to include the cumulative percentage, output a sorted histogram, or have the output be in the form of a chart in addition to sorting the data into each of the bins.
Let us look at the WorldCorp sales data again. To construct the histogram, establish the number of bins and the size of the bins you will be using. Be sure to include enough bins to show differences that may exist in the dataset but not so many that it is difficult to see trends or that several bins have no data in them. A general rule of thumb is to divide the range of the data (highest data point – lowest data point) by the width of the categories. For example, if the range is 800 and you want each category or bin to increase by 50, you will need approximately 16 bins. You can also determine the range of each bin by first determining how many bins you want. For your range of 800, if you want 20 bins, each bin will have a range of 40.
The column of data you will use for the histogram is column E, Unit Price. When using the Descriptive Statistics tool, you determined that the minimum number is $7.64 and the maximum number is $14,353. The output also gave the range of the dataset, $14,345.36. If you divide the range by the number of desired bins, you get the approximate size of the bins you should use. In this example, let’s use 15 bins. That would be 14,345.36/15 or 95.62. You then use this as a basis to establish the bin range that you will use in the Histogram tool. You can use whole number categories to make the histogram easier to follow and look cleaner. You could set the first bin at $100 since the value calculated was almost $100 ($95.62). This means that all data up to, but not including $100 will be included in that bin. Each subsequent bin will increase by 100 until you reach 16 bins. You record these bins in the spreadsheet manually. Excel does not create the bins for you. These bins can be either on the dataset worksheet or a separate worksheet. You will use these cells as your bin range when creating the histogram.
Open the Histogram tool in Data Analysis. Select the input range, the Unit Price column, and then select the bin range. Be sure to select the Labels checkbox if you are including the column header in the input data selection. You must also include a column heading for the bin range if the data range has a column heading (Figure 11.47a).
The output for the histogram Data Table shows the count in each of the established bins for the input range selected (Figure 11.47b). Excel will automatically add a “More” bin at the end to include all data that falls outside of the range that you establish. Even if you have a bin to accommodate the maximum data point, Excel will add the “More” bin at the end of the histogram Data Table.
From this table of data, you can create the histogram using the options on the Insert tab. Alternatively, you can select the chart output and create the chart at the same time you create the data by checking the Chart Output option at the bottom of the window. You can also include cumulative percentages in the output by checking the Cumulative Percentage (Figure 11.48). Creating a histogram chart from a dataset not using the Data Analysis tools will be covered in Histogram.
Regression
Regression is a tool used in inferential statistics. It determines the best fit line or data model for a given dataset. There are many types of regressions, but linear regression is the method most used in business. Linear regression can be used to forecast sales and expenses for the next year, to better understand the drivers of customer purchasing levels, or to investigate how price changes might impact profit. With the regression process, you have one dependent variable you are predicting ( Y ) and one or more independent variables ( X ) you are using to predict Y . With the tool in Excel, you can have as many X variables as you need.
Regression is the primary tool used by businesses to determine if a relationship between variables exists, as well as the magnitude and nature of this relationship should it exist. In business, regression can be helpful in optimization and business forecasting across a variety of industries. For instance, you can investigate the increase in new home sales based on economic growth. You can also make determinations about how much inventory to purchase.
Regression is one of the Data Analysis tools. As with the other tools, when you open it, an input window will appear (Figure 11.49). Select the column to use for the Y variable and then select the column(s) to use for the independent variable(s) X . If using more than one independent variable, be sure that all the X variable columns are next to each other. Also note that you must have equal numbers of rows for all variables you select for analysis.
Be sure to check the Labels box if you included the column headings. It is good practice to include the headings, as the output for regression will be more specific to your dataset. Other checkboxes relate to residuals and line fit plots, so check them if they apply to your situation. These options are more advanced statistical tools that can be used. As with the other tools, you can decide where to place the output of the function. The output will be a detailed regression analysis from which you can extract the relevant information, such as R 2 and the regression equation, using the coefficients column.
Using the same dataset you’ve been using, you can run a single regression using Quantity as the independent variable ( X ) to predict the FOB $ as the dependent variable ( Y ). These values were chosen to investigate the impact that Quantity has on FOB $. Management believes that although Unit Price and Quantity are not strongly connected, the Quantity value is a good predictor of FOB $ (Figure 11.50). To begin, go to the Data tab, and choose Data Analysis. Then, select Regression from the list. Select the data in column F, FOB $, as the Y input range and column D, Quantity, as the X input range. Check the Labels box if you also included the column headings. Then, click OK. Make sure to clean up the data, such as by reducing the number of decimal places, and format it to give it a professional look (Figure 11.51).
If the dataset has more than one independent variable, the information for each variable will be listed in the output as an additional row under the Intercept row.
Advanced Data Analysis Tools
The Data Analysis library of tools has many other statistical analysis tools. These are used in business less frequently than the tools discussed so far, but they do offer some opportunities for more in-depth analysis. You might see these tools used in more scientific disciplines such as chemistry or biology. This section will briefly introduce how to use these functions in Excel.
Analysis of Variance
Analysis of variance (ANOVA) is a statistical technique used to compare more than two groups at the same time. It involves conducting an analysis to determine if differences are present in the data based on their respective variances.
There are three types of ANOVA analyses available in Excel that are easily accessed through the Data Analysis tool. The input windows for the ANOVA functions look very similar, so be sure to check the title at the top of the input window.
ANOVA: Single Factor focuses on analyzing information for a single variable (Figure 11.52). For example, perhaps you want to do some price checking on your favorite brands at three retailers. You could use ANOVA to determine if there are price differences between the retailers.
To use the Single Factor function, simply select the data of interest and choose the appropriate grouping of the data (columns or rows). Be sure to check the Labels box if you included the headings in your selection. You also have the option to change the alpha (significance level) for your analysis. The output will be in the form of a typical ANOVA table, which you may have seen in a statistics course.
ANOVA: Two-Factor With Replication is used to test two variables and to determine if there is any interaction present in the variables (Figure 11.53). The input window looks like that of the Single Factor, except you will need to add the number of rows per sample. This number will need to be the same for all categories. You must also have row and column headings for this option. Excel assumes that the first column and the first row in the selection will include the headings for the data, so there is no Labels checkbox.
The final type of ANOVA in Excel is ANOVA: Two-Factor Without Replication , which allows you to test two variables. Unlike With Replication, here you will not be testing for the interaction between the two variables. Instead, you are testing the two variables independent of each other, but simultaneously. Notice in Figure 11.54 that the window looks the same as the window for Single Factor, except for the title. There is no box to fill in the Rows per Sample, as there is in Two-Factor With Replication.
Link to Learning
Through the use of the advanced statistical tools in Excel, you can better analyze data and discover differences among various groups of consumers in your target market. Watch this video on how ANOVA can be used to understand consumers in the craft beer industry to learn more.
F-Test
The F-Test is used to determine if two datasets are from the same population based on their variances. It is another type of hypothesis test that can be used for in-depth statistical analysis. Select F-Test: Two Sample for Variances from the Data Analysis tool library. As with the other functions, an input window will appear (Figure 11.55). You will need to select the range of each variable separately. The ranges do not need to be the same size. Check the Labels box if you included the column headings. Click OK when you are ready to analyze the data.
T-Tests
T-Tests are also hypothesis tests. T-Tests are used when data does not follow the normal distribution (a concept in statistics), when the sample sizes are small (generally less than 30), or when you do not have a population standard deviation. It is often used in statistics to test for statistical differences between two groups of data. There are three t-Tests available in Excel. The input windows look the same, except for the title of the input window, so make sure you have selected the appropriate test. The first type, t-Test: Paired Two Sample for Means is for dependent datasets, such as if you are determining the success of a coupon campaign. You could track sales for a group of customers before the coupon and then track the sales for the same group of customers after the coupon campaign. You can then use this test to determine if there is a statistical difference between the means of the two groups, or you can input a specific level for the difference in the Hypothesized Mean Difference input argument (Figure 11.56). If there is no set difference level you are investigating, then type “0” in the Hypothesized Mean Difference box.
The two remaining t-Tests in Excel are for analysis with independent datasets. The tests are t-Test: Two-Sample Assuming Equal Variances and t-Test: Two-Sample Assuming Unequal Variances . You could use the two-sample t-Test to determine if there is a difference between purchase amounts for males and females. You could also use the test to determine if sales have changed between last year and this year. The difference between the two tests is that one deals with equal variances, and the other deals with unequal variances. You will need to determine before selecting the test if the variances are nearly equal or not. You can determine if the variances of two datasets are equal by using the Descriptive Statistics tool discussed in Descriptive Statistics. When you choose the appropriate function, an input window will appear (Figure 11.57). As with the Paired test, you will need to select the two ranges of data and a Hypothesized Mean Difference. If there is no difference, then type “0” into the Hypothesized Mean Difference box.
Using the WorldCorp data file with unit price, you can determine if there is a difference in the unit price between Home Stereo Systems and Computer Servers Accessories. The t-Test is appropriate for this analysis because the population standard deviation is not known. In other words, you do not know whether the data follows the normal distribution. Upon verification, you determine that the variances of the two groups are very different, so you should choose the t-Test: Two-Sample Assuming Unequal Variances (Figure 11.58).
The output from the function gives all the information needed to make the decision for the hypothesis test. As with previous functions, clean up the decimal points and reformat. The output obtained from using the other t-Test functions will look like Figure 11.59.
Z-Test
The z-Test: Two Sample for Means (Figure 11.60) tests for differences between two independent datasets that follow the normal distribution. It is like the t-Test except that the data is normal.
You will need to select the two ranges for analysis and include a Hypothesized Mean Difference if desired. In addition, you will need to enter the variance for each of the variable ranges. If you don’t already know the variance, you can use the Descriptive Statistics tool previously discussed or the variance function in the Insert Function library to obtain it. The variance you enter must be a number, not a cell reference. Check the Labels box, if appropriate, and then click OK. The output obtained from the z-Test will be much like the output for the t-Tests.
The Solver Add-in
The Solver add-in offers additional ways to analyze a dataset. This section will just cover the basics, but you can research the tool further if you want to know more about Solver and its uses for data analysis. Solver appears with Data Analysis on the Data tab (Figure 11.61).
The Solver tool allows you to change levels in a dataset according to specific constraints you set up (Figure 11.62). Then, you can see the various solutions under those constraints. This is a helpful tool if you have multiple variables within a dataset. The output provides an optimal solution with the dataset given the constraints you establish.
To use the tool, you need to determine the target cell for optimization and the criteria (maximum, minimum, or a specific value). Then, you add the various constraints to the dataset. You can add up to 100 constraints to the Solver tool. Solver will find a solution to meet the criteria for the target cell assuming there is a solution. You can apply the solution to your dataset by choosing “Keep Solver Solution” with the output. There are many applications in business for Solver. For example, you could use the tool to determine the optimal level needed to minimize costs or maximize revenues in a business. You can use it to find the optimal solution for allocations in a retirement fund to enable you to retire at your desired age. You can also use it to determine the conditions necessary to pay off a mortgage loan in half the time. Take a look at the data in the "solvsamp" tabs of the downloadable Chapter 11 data file ("solvsamp1-7") for some great examples of Solver and applications of the tool to various problems.
There are many ways that information and data can be manipulated to tell a different story from what is factual or to influence your interpretation of the information to slant a certain way. Simple changes such as adjusting the scaling on a chart or adding formatting changes to numbers can alter the way people interpret the information. For instance, you may want to show a larger than actual impact. You might decide to use a baseline number other than zero along the x -axis to show a much larger trend in the data than what is real. Or, perhaps you can choose to emphasize information and impacts by using very small units of measure. With today’s endless access to data, the influence of the media, and the ever-present “fake news” online, it is important to be diligent about making sure that data is accurately reported, analyzed, and presented to the intended audience.


11.6.3: What Should the Nurse Do?
What Should the Nurse Do?
2 .
What considerations should the nurse give when a medication is ordered on a scheduled and PRN basis?
3 .
Which medication should be administered if the patient complains of pain?
4 .
How might the patient’s medical history potentially impact the administration of medications?
6 .
How should the nurse respond to the provider?
7 .
What additional steps might the nurse take to ensure patient safety?
9 .
What is the Q8 hour therapeutic range (in milligrams) based on the patient’s weight?
10 .
What is the Q8 hour therapeutic range (in grams) based on the patient’s weight?
11 .
Is the prescribed dose within the therapeutic range?
12 .
How many milliliters should the patient receive per dose?
13 .
How many milliliters should the patient receive per day?
15 .
What evaluations are needed after administering these medications?
16 .
How long after administration should the nurse perform these evaluations?
17 .
How should the nurse document these evaluations?


6.2: Types of Memory
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In the following section, we will discuss the three different types of memory and their respective characteristics: Sensory Memory , Short Term (STM) or Working Memory (WM) and Long Term Memory (LTM) .
Sensory Memory
This type of memory has the shortest retention time, only milliseconds to five seconds. Roughly, Sensory Memory can be subdivided into two main kinds:
Sensory Memory
- Iconic Memory (visual input)
- Echoic Memory (auditory input)
While Iconic and Echoic Memory have been well researched, there are other types of Sensory Memory, like
haptic
,
olfactory
, etc., for which no sophisticated theories exist so far.
It should be noted, though, that according to the Atkinson and Shiffrin (1968)
[2]
Sensory Memory was considered to be the same thing as Iconic Memory. Echoic Memory was added to the concept of Sensory Memory due to research done by Darwin and others (1972).
[3]
Let us consider the following intuitive example for Iconic Memory: Probably we all know the phenomenon that it seems possible to draw lines, figures or names with lighted sparklers by moving the sparkler fast enough in a dark environment. Physically, however, there are no such things as lines of light. So why can we nevertheless see such figures? This is due to Iconic Memory. Roughly speaking, we can think of this subtype of memory as a kind of photographic memory, but one which only lasts for a very short time (milliseconds, up to a second). The image of the light of a sparkler remains in our memory (persistence of vision) and thus makes it seem to us like the light leaves lines in the dark. The term "Echoic Memory", as the name already suggests, refers to auditory input. Here the persistence time is a little longer than with Iconic Memory (up to five seconds).
At the level of Sensory Memory no manipulation of the incoming information occurs, it is transferred to the
Working Memory
. By ‘transfer’ it is meant that the amount of information is reduced because the capacity of the working memory is not large enough to cope with all the input coming from our sense organs. The next paragraph will deal with the different theories of selection when transferring information from Sensory Memory to Working Memory.
One of the first experiments researching the phenomenon of
Attention
was the
Shadowing Task
(Cherry et al., 1953).
[4]
This experiment deals with the filtering of auditory information. The subject is wearing earphones, getting presented a different story on each ear. He or she has to listen to and repeat out loud the message on one ear (shadowing). When asked for the content of the stories of both ears only the story of the shadowed side can be repeated; participants do not know about the content of the other ear’s story. From these results Broadbent concluded the
Filter Theory
(1958).
[5]
This theory proposes that the filtering of information is based on specific physical properties of stimuli. For every frequency there exists a distinct nerve pathway. The attention control selects which pathway is active and can thereby control which information is passed to the Working Memory. This way it is possible to follow the utterance of one person with a certain voice frequency even though there are many other sounds in the surrounding. But imagine a situation in which the so called cocktail party effect applies: having a conversation in a loud crowd at a party and listening to your interlocutor you will immediately switch to listening to another conversation if the content of it is semantically relevant to you, e.g. if your name is mentioned.
So it is found that filtering also happens semantically. The above mentioned Shadowing Task was changed so that the semantic content of a sentence was split up between the ears, and the subject, although shadowing, was able to repeat the whole sentence because he or she was following the semantic content unconsciously.
Reacting to the effect of semantic filtering, new theories were developed. Two important theories are the
Attenuation Theory
(Treisman, 1964)
[6]
and the
Late Selection Theory
(Deutsch & Deutsch, 1963).
[7]
The former proposes that we attenuate information which is less relevant, but do not filter it out completely. Thereby also semantic information of ignored frequencies can be analyzed but not as efficiently as those of the relevant frequencies. The Late Selection Theory presumes that all information is analyzed first and afterwards the decision of the importance of information is made. Treisman and Geffen did an experiment to find out which one of the theories holds. The experiment was a revision of the Shadowing Task. Again the subjects have to shadow one ear but in contrast they also have to pay attention to a certain sound which could appear on either ear. If the sound occurs the subject has to react in a certain way (for example knock on the table). The result is that the subject identifies the sound on the shadowed ear in 87% of all cases and can only do this in 8% of the cases on the ignored side. This shows that the information on the ignored side must be attenuated since the rate of identification is lower. If the Late Selection Theory were to hold then the subject would have to analyze all information and would have to be able to identify the same amount on the ignored side as on the shadowed side. Since this is not the case the Attenuation Theory by Treisman explains the empirical results more accurately.
Illustration of the Attention Control Model by a) Treisman - Attenuation Theory and b) Deutsch & Deutsch – Late Selection Theory.
Short Term Memory
The Short Term Memory (STM) was initially discussed by Attkinson and Shiffrin (1968). [8] The Short Term Memory is the link between Sensory Memory and Long Term Memory (LTM). Later Baddeley proposed a more sophisticated approach and called the interface Working Memory (WM). We will first look at the classical Short Term Memory Model and then go on to the concept of Working Memory.
As the name suggests, information is retained in the Short Term Memory for a rather short period of time (15–30 seconds).
Short Term Memory
If we look up a phone number in the phone book and hold it in mind long enough for dialling the number, it is stored in the Short Term Memory. This is an example of a piece of information which can be remembered for a short period of time. According to George Miller (1956)
[9]
the capacity of the Short Term Memory is five to nine pieces of information (The
magical number seven, plus or minus two
). The term "pieces of information” or, as it is also called,
chunk
might strike one as a little vague. All of the following are considered as chunks: single digits or letters, whole words or even sentences and the like. It has been shown by experiments also done by Miller that
chunking
(the process of bundeling information) is a useful method to memorize more than just single items in the common sense. Gobet et al. defined a chunk as "a collection of elements that are strongly associated with one another but are weakly associated with other chunks" (Goldstein, 2005).
[10]
A very intuitive example of chunking information is the following:
Try to remember the following digits:
- 0 3 1 2 1 9 8 2
But you could also try another strategy to remember these digits:
- 03. 12. 1982.
With this strategy you bundeled eight pieces of information (eight digits) to three pieces with help to remember them as a date schema.
A famous experiment concerned with chunking was conducted by Chase and Simon (1973)
[11]
with novices and experts in chess playing. When asked to remember certain arrangements of chess pieces on the board, the experts performed significantly better that the novices. However, if the pieces were arranged arbitrarily, i.e. not corresponding to possible game situations, both the experts and the novices performed equally poorly. The experienced chess players do not try to remember single positions of the figures in the correct game situation, but whole bundles of figures as already seen before in a game. In incorrect game situations this strategy cannot work which shows that chunking (as done by experienced chess players) enhances the performance only in specific memory tasks.
- From Short Term Memory to Baddeley’s Working Memory Model
Baddeley and Hitch (1974) [12] drew attention to a problem with the Short Term Memory Model. Under certain conditions it seems to be possible to do two different tasks simultaneously, even though the STM, as suggested by Atkinson and Shiffrin, should be regarded as a single, undivided unit. An example for the performance of two tasks simultaneously would be the following: a person is asked to memorize four numbers and then read a text (unrelated to the first task). Most people are able to recall the four numbers correctly after the reading task, so apparently both memorizing numbers and reading a text carefully can be done at the same time. According to Baddeley and Hitch the result of this experiment indicates that the number-task and the reading-task are handled by two different components of Short Term Memory. So they coined the term "Working Memory" instead of "Short Term Memory" to indicate that this kind of Memory enables us to perform several cognitive operations at a time with different parts of the Working Memory.
Working Memory
According to Baddeley, Working Memory is limited in its capacity (the same limitations hold as for Short Term Memory) and the Working Memory is not only capable of storage, but also of the manipulation of incoming information. Working Memory consists of three parts:
- Phonological Loop
- Visuospatial Sketch Pad
- Central Executive
We will consider each module in turn:
The
Phonological Loop
is responsible for auditory and verbal information, such as phone numbers, people’s names or general understanding of what other people are talking about. We could roughly say that it is a system specialized for language. This system can again be subdivided into an active and a passive part. The storage of information belongs to the passive part and fades after two seconds if the information is not rehearsed explicitly. Rehearsal, on the other hand, is regarded as the active part of the Phonological Loop. The repetition of information deepens the memory. There are three well-known phenomena that support the idea that the Phonological Loop is specialized for language: The
phonological similarity effect
, the
word-length effect
and
articulatory suppression
. When words that sound similar are confused, we speak of the phonological similarity effect. The word-length effect refers to the fact that it is more difficult to memorize a list of long words and better results can be achieved if a list of short words is memorized. Let us look at the phenomenon of articulatory suppression in a little more detail. Consider the following experiment:
Participants are asked to memorize a list of words while saying "the, the, the ...“ out loud. What we find is that, with respect to the word-length effect, the difference in performance between lists of long and short words is levelled out. Both lists are memorized equally poorly. The explanation given by Baddeley et al. (1986),
[13]
who conducted this experiment, is that the constant repetition of the word "the" prevents the rehearsal of the words in the lists, independent of whether the list contains long or short words. The findings become even more drastic if we compare the memory-performance in the following experiment (also conducted by Baddeley and his co-workers in 1986):
Participants were again asked to say out loud "the, the, the ..." But instead of memorizing words from a list of short or long words, their task was to remember words that were either spoken to them or shown to them written on paper. The results indicated that the participants’ performances were significantly better if the words were presented to them and not read out aloud. Baddeley concluded from this fact that the performance in a memory task is improved if the two stimuli can be dealt with in distinct components of the Working Memory. In other words, since the reading of words is handled in the Visuospatial Sketch Pad, whereas the saying of "the" belongs to the Phonological Loop, the two tasks do not "block" each other. The rather bad performance of hearing words while speaking could be explained by the fact that both hearing and speaking are dealt with in the Phonological Loop and thus the two tasks conflict with each other, decreasing the performance of memorization.
In the
Visuospatial Sketch Pad
, visual and spatial information is handled. This means that information about the position and properties of objects can be stored. As we have seen above, performance decreases if two tasks that are dealt with in the same component are to be done simultaneously. Let us consider a further example that illustrates this effect. Brandimonte and co-workers (1992)
[14]
conducted an experiment where participants were asked to say out loud "la, la, la...“ At the same time they were given the task of subtracting a partial image from a given whole image. The subtraction had to be done mentally because the two images were presented only for a short time. The interesting result was that the performance not only didn't decrease while saying "la, la, la ..." when compared to doing the subtraction-task alone, but the performance even increased. According to Brandimonte this was due to the fact that the subtraction task was easier if handled in the Visuospatial Sketch Pad as opposed to the Phonological Loop (both the given and the resulting pictures were such that they could also be named, i.e. verbalized, a task that belongs to the Phonological Loop). As mentioned above, because of the fact that the subtraction of a partial image from a whole given image is easier if done visually, the performance increased if participants were forced to visually perform that task, i.e. if they were forced to use the component that is suited best for the given task. We have seen that the Phonological Loop and the Visuospatial Sketch Pad deal with rather different kinds of information which nonetheless have to somehow interact in order to do certain tasks. The component that connects those two systems is the Central Executive. The
Central Executive
co-ordinates the activity of both the Phonological Loop and the Visuospatial Sketch Pad. Imagine the following situation: You are driving a car and your friend in the passenger seat has the map and gives you directions. The directions are given verbally, i.e. they are handled by the Phonological Loop, while the perception of the traffic, street lights, etc. is obviously visual, i.e. dealt with in the Visuospatial Sketch Pad. If you now try to follow the directions given to you by your friend it is necessary to somehow combine both kinds of information, the verbal and the visual information. This important connection of the two components is done by the Central Executive. It also links the Working Memory to Long Term Memory, controls the storage in Long Term Memory and the retrieval from it. The process of storage is influenced by the duration of holding information in Working Memory and the amount of manipulation of the information. The latter is stored for a longer time if it is semantically interpreted and viewed with relation to other information already stored in Long Term Memory. This is called Deep Processing. Pure syntactical processing (reading a text for typos) is called Shallow Processing. Baddeley proposes also further capabilities for the Central Executive:
- Initiating movement
- Control of conscious attention
- Problems which arise with the Working Memory approach
In theory, all information has to pass the Working Memory in order to be stored in the Long Term Memory. However, cases have been reported where patients could form Long Term Memories even though their STM-abilities were severely reduced. This clearly poses a problem to the modal model approach. It was suggested by Shallice and Warrington (1970) [15] that there must be another possible way for information to enter Long Term Memory than via Working Memory.
Long Term Memory
As the name already suggest, Long Term Memory is the system where memories are stored for a long time. "Long" in this sense means something between a few minutes and several years or even decades to lifelong.
Long Term Memory
Similar to Working Memory, Long Term Memory can again be subdivided into different types. Two major distinctions are made between Declarative (conscious) and Implicit (unconscious) Memory . Those two subtypes are again split into two components each: Episodic and Semantic Memory with respect to Declarative Memory and Priming Effects , and Procedural Memory with respect to Implicit Memory. In contrast to Short Term or Working Memory, the capacity of Long Term Memory is theoretically infinite. The opinions as to whether information remains in the Long Term Memory forever or whether information can get deleted differ. The main argument for the latter opinion is that apparently not all information that was ever stored in LTM can be recalled. However, theories that regard Long Term Memories as not being subject to deletion emphasize that there might be a useful distinction between the existence of information and the ability to retrieve or recall that information at a given moment. There are several theories about the “forgetting” of information. These will be covered in the section “Forgetting and False Memory” .
Declarative Memory
Let us now consider the two types of Declarative Memory . As noted above, those two types are Episodic and Semantic Memory . Episodic Memory refers to memories for particular events that have been experienced by somebody (autobiographical information). Typically, those memories are connected to specific times and places. Semantic Memory, on the other hand, refers to knowledge about the world that is not connected to personal events. Vocabularies, concepts, numbers or facts would be stored in the Semantic Memory. Another subtype of memories stored in Semantic Memory is that of the so called Scripts . Scripts are something like blueprints of what happens in a certain situation.  Semantic and Episodic Memory are usually closely related to one another, i.e. memory of facts might be enhanced by interaction with memory about personal events and vice versa. For example, the answer to the factual question of whether people put vinegar on their chips might be answered positively by remembering the last time you saw someone eating fish and chips. The other way around, good Semantic Memory about certain things, such as football, can contribute to more detailed Episodic Memory of a particular personal event, like watching a football match. A person that barely knows the rules of that game will most probably have a less specific memory for the personal event of watching the game than a football-expert will.
Implicit Memory
We now turn to the two different types of
Implicit Memory
. As the name suggests, both types are usually active when unconscious memories are concerned. This becomes most evident for
Procedural Memory
, though it must be said that the distinction between both types is not as clearly cut as in the case of Declarative Memory and that often both categories are collapsed into the single category of Procedural Memory. But if we want to draw the distinction between
Priming Effects
and
Procedural Memory
, the latter category is responsible for highly skilled activities that can be performed without much conscious effort. Examples would be the tying of shoelaces or the driving of a car, if those activities have been practiced sufficiently. It is some kind of movement plan. As regards the
Priming Effect
, consider the following experiment conducted by Perfect and Askew (1994):
[16]
Participants were asked to read a magazine without paying attention to the advertisements. After that, different advertisements were presented to them; some had occurred in the magazine, others had not. The participants were told to rate the presented advertisement with respect to different criteria such as how appealing, how memorable or eye-catching they were. The result was that in general those advertisements that had been in the magazine received higher rankings than those that had not been in the magazine. Additionally, when asked which advertisements the participants had actually seen in the magazine, the recognition was very poor (only 2.8 of the 25 advertisements were recognized). This experiment shows that the participants performed
implicit learning
(as can be seen from the high rankings of advertisements they had seen before) without being conscious of it (as can be seen from the poor recognition rate). This is an example of the Priming Effect.
Final overview of all different types of memory and their interaction


5.3: Present Value of Annuities and Installment Payment
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Learning Objectives
In this section, you will learn to:
- Find the present value of an annuity.
- Find the amount of installment payment on a loan.
- Find the outstanding balance on a loan.
Prerequisite Skills
Before you get started, take this prerequisite quiz.
1. What will the final amount be in 4 years if $8,000 is invested at 9.2% compounded monthly?
- Click here to check your answer
-
\($11,542.53\)
If you missed this problem, review Section 6.2 . (Note that this will open in a new window.)
2. How much should be invested at 10.3% compounded quarterly for it to amount to $10,000 in 6 years?
- Click here to check your answer
-
\($5,432.55\)
If you missed this problem, review Section 6.2 . (Note that this will open in a new window.)
3. Find the future value of an annuity of $200 per month for 5 years at 6% compounded monthly.
- Click here to check your answer
-
\($13,954.01\)
If you missed this problem, review Section 6.3 . (Note that this will open in a new window.)
4. How much money should be deposited at the end of each month in an account paying 7.5% for it to amount to $10,000 in 5 years?
- Click here to check your answer
-
\($137.88\)
If you missed this problem, review Section 6.3 . (Note that this will open in a new window.)
PRESENT VALUE OF AN ANNUITY
In the previous two sections, we learned to find the future value of a lump sum and the future value of an annuity. With these two concepts in hand, we will now learn to amortize a loan, and to find the present value of an annuity.
The present value of an annuity is the amount of money we would need now in order to be able to make the payments in the annuity in the future. In other words, the present value is the value now of a future stream of payments.
We start by breaking this down step by step to understand the concept of the present value of an annuity. After that, the examples provide a more efficient way to do the calculations by working with concepts and calculations we have already explored in the last two sections.
Suppose Carlos owns a small business and employs an assistant manager to help him run the business. Assume it is January 1 now. Carlos plans to pay his assistant manager a $1000 bonus at the end of this year and another $1000 bonus at the end of the following year. Carlos’ business had good profits this year so he wants to put the money for his assistant’s future bonuses into a savings account now. The money he puts in now will earn interest at the rate of 4% per year compounded annually while in the savings account.
How much money should Carlos put into the savings account now so that he will be able to withdraw $1000 one year from now and another $1000 two years from now?
At first, this sounds like a sinking fund. But it is different. In a sinking fund, we put money into the fund with periodic payments to save to accumulate to a specified lump sum that is the future value at the end of a specified time period.
In this case we want to put a lump sum into the savings account now, so that lump sum is our principal, \(\mathrm{P}\). Then we want to withdraw that amount as a series of period payments; in this case the withdrawals are an annuity with $1000 payments at the end of each of two years.
We need to determine the amount we need in the account now, the present value, to be able to make withdraw the periodic payments later.
We use the compound interest formula from Section 6.2 with \(r\) = 0.04 and \(n\) = 1 for annual compounding to determine the present value of each payment of $1000.
Consider the first payment of $1000 at the end of year 1. Let P 1 be its present value
\[\$ 1000=P_{1}(1.04)^{1} \text { so } P_{1}=\$ 961.54 \nonumber\]
Now consider the second payment of $1000 at the end of year 2. Let P 2 is its present value
\[\$ 1000=P_{2}(1.04)^{2} \text { so } P_{2}=\$ 924.56 \nonumber\]
To make the $1000 payments at the specified times in the future, the amount that Carlos needs to deposit now is the present value \(P=P_{1}+P_{2}=\$ 961.54+\$ 924.56=\$ 1886.10\)
The calculation above was useful to illustrate the meaning of the present value of an annuity.
But it is not an efficient way to calculate the present value. If we were to have a large number of annuity payments, the step by step calculation would be long and tedious.
Example \(\PageIndex{1}\) investigates and develops an efficient way to calculate the present value of an annuity, by relating the future (accumulated) value of an annuity and its present value.
Example \(\PageIndex{1}\)
Suppose you have won a lottery that pays $1,000 per month for the next 20 years. But, you prefer to have the entire amount now. If the interest rate is 8%, how much will you accept?
Solution
This classic present value problem needs our complete attention because the rationalization we use to solve this problem will be used again in the problems to follow.
Consider, for argument purposes, that two people Mr. Cash, and Mr. Credit have won the same lottery of $1,000 per month for the next 20 years. Mr. Credit is happy with his $1,000 monthly payment, but Mr. Cash wants to have the entire amount now.
Our job is to determine how much Mr. Cash should get. We reason as follows:
If Mr. Cash accepts P dollars, then the P dollars deposited at 8% for 20 years should yield the same amount as the $1,000 monthly payments for 20 years. In other words, we are comparing the future values for both Mr. Cash and Mr. Credit, and we would like the future values to equal.
Since Mr. Cash is receiving a lump sum of \(x\) dollars, its future value is given by the lump sum formula, and it is
\[\mathrm{A}=\mathrm{P}(1+.08 / 12)^{240} \nonumber\]
Since Mr. Credit is receiving a sequence of payments, or an annuity, of $1,000 per month, its future value is given by the annuity formula This value is
\[\mathrm{A}=\frac{\$ 1000\left[(1+.08 / 12)^{240}-1\right]}{.08 / 12} \nonumber\]
The only way Mr. Cash will agree to the amount he receives is if these two future values are equal. So we set them equal and solve for the unknown.
\[\begin{array}{l}
\mathrm{P}(1+.08 / 12)^{240}=\frac{\$ 1000\left[(1+.08 / 12)^{240}-1\right]}{.08 / 12} \\
\mathrm{P}=\frac{\frac{\$ 1000\left[(1+.08 / 12)^{240}-1\right]}{.08 / 12}}{(1+.08 / 12)^{240}} \\
\mathrm{P}=\$ 119,554.36
\end{array} \nonumber\]
The present value of an ordinary annuity of $1,000 each month for 20 years at 8% is $119,554.36
The reader should also note that if Mr. Cash takes his lump sum of \(\mathrm{P}\) = $119,554.36 and invests it at 8% compounded monthly, he will have an accumulated value of \(\mathrm{A}\)=$589,020.41 in 20 years.
If a person or business needs to buy or pay for something now (a car, a home, college tuition, equipment for a business) but does not have the money, they can borrow the money as a loan.
They receive the loan amount called the principal (or present value) now and are obligated to pay back the principal in the future over a stated amount of time (term of the loan), as regular periodic payments with interest.
We can see the formula begin to develop in the example above.
Definition: Present Value of an Annuity
If a payment of \(m\) dollars is made in an account \(n\) times a year at an interest \(r\), then the present value \(\mathrm{P}\) of the annuity after \(t\) years is
\[\mathbf{P}(\mathbf{1}+\mathbf{r} / \mathbf{n})^{\mathbf{n} \mathbf{t}}=\frac{\mathbf{m}\left[(\mathbf{1}+\mathbf{r} / \mathbf{n})^{\mathbf{n} \mathbf{t}}-\mathbf{1}\right]}{\mathbf{r} / \mathbf{n}}\]
When used for a loan, the amount \(\mathrm{P}\) is the loan amount , and \(m\) is the periodic payment needed to repay the loan over a term of \(t\) years with \(n\) payments per year.
Note that the formula assumes that the payment period is the same as the compounding period. If these are not the same, then this formula does not apply.
Example \(\PageIndex{2}\)
You determine that you can afford $550 per month for a car. At the dealership, the finance department offers you a 3-year loan with a 7% interest rate, compounded monthly.
- What is the highest priced car you can afford in this scenario?
- How much will you pay over the 3 years?
- What is the difference between the answers for part a and part b, and what does it represent?
Solution
a. Using the formula above, we are looking for the total amount of the loan, or \(P\). We know that \(m=550\), \(r=0.07\), \(n=12\), and \(t=3\).
\(\mathbf{P}(\mathbf{1}+\mathbf{0.07} / \mathbf{12})^{\mathbf{(12 \cdot 3)}}=\frac{\mathbf{550}\left[(\mathbf{1}+\mathbf{0.07} / \mathbf{12})^{\mathbf{(12 \cdot 3)}}-\mathbf{1}\right]}{\mathbf{0.07} / \mathbf{12}}\)
\(\mathbf{P}=\frac{\frac{\mathbf{550}\left[(\mathbf{1}+\mathbf{0.07} / \mathbf{12})^{\mathbf{(12 \cdot 3)}}-\mathbf{1}\right]}{\mathbf{0.07} / \mathbf{12}}}{(\mathbf{1}+\mathbf{0.07} / \mathbf{12})^{\mathbf{(12 \cdot 3)}}}\)
\(P=17812.55545\)
You can take a loan of $17,812.55 to pay off within 3 years.
b. Over the course of 3 years, you would pay $550 every month, 12 months each year, for 3 years.
Total paid \(=$550 \cdot 12 \cdot 3=$19,800\).
c. The difference between $17,812.55 as the purchase price and $19,800 as the repayment price is \($19800-$17812.55=$1987.45\). This means that you'll pay $1,987.45 in interest over the 3 years.
Alternate Method to find Present Value of an Annuity
We should note that many finite mathematics and finance books develop the formula for the present value of an annuity differently.
Instead of using the formula:
\[\mathrm{P}(1+\mathrm{r} / \mathrm{n})^{\mathrm{nt}}=\frac{\mathrm{m}\left[(1+\mathrm{r} / \mathrm{n})^{\mathrm{nt}}-1\right]}{\mathrm{r} / \mathrm{n}} \label{8.4.1}\]
and solving for the present value \(\mathrm{P}\) after substituting the numerical values for the other items in the formula, many textbooks first solve the formula for \(\mathrm{P}\) in order to develop a new formula for the present value. Then the numerical information can be substituted into the present value formula and evaluated, without needing to solve algebraically for \(\mathrm{P}\).
Starting with formula \ref{8.4.1}: \(\mathrm{P}(1+\mathrm{r} / \mathrm{n})^{\mathrm{nt}}=\frac{\mathrm{m}\left[(1+\mathrm{r} / \mathrm{n})^{\mathrm{nt}}-1\right]}{\mathrm{r} / \mathrm{n}}\)
Divide both sides by \((1+r / n)^{n t}\) to isolate \(\mathrm{P}\), and simplify
\[P=\frac{m\left[(1+r / n)^{n t}-1\right]}{r / n} \cdot \frac{1}{(1+r / n)^{n t}} \nonumber\]
\[P=\frac{m\left[1-(1+r / n)^{-n t}\right]}{r / n} \label{8.4.2} \]
The authors of this book believe that it is easier to use formula \ref{8.4.1} at the top of this page and solve for \(\mathrm{P}\) or \(m\) as needed. In this approach there are fewer formulas to understand, and many students find it easier to learn. In the problems the rest of this chapter, when a problem requires the calculation of the present value of an annuity, formula \ref{8.4.1} will be used.
However, formula \ref{8.4.2} is ideal when used to solve for \(t\), and some people prefer to use this formula to find present value. It is a mathematically correct option to do so. Note that if you choose to use formula \ref{8.4.2}, you need to be careful with the negative exponents in the formula.
INSTALLMENT PAYMENTS ON A LOAN
Example \(\PageIndex{3}\) examines how to calculate the loan payment, using reasoning similar to Example \(\PageIndex{1}\).
Example \(\PageIndex{3}\)
Daria purchased a truck costing $45,000 and accepted a 60-month loan at an interest rate of 9%, compounded monthly.
- Find the monthly installment payments.
- How much will she pay over the course of the loan?
- How much interest will she pay over the course of the loan?
Solution
a. The monthly installment payments can be found in a similar process to finding the principal in Example \(\PageIndex{2}\). Note that 60 months is the same as 5 years. We can use \(t=5\) or we can use \(nt=60\).
\[\begin{array}{l}
\$ 45,000(1+.09 / 12)^{60}=\frac{m\left[(1+.09 / 12)^{60}-1\right]}{.09 / 12} \\
\end{array} \nonumber\]
We then multiply both sides by the denominator:
\[\begin{array}{l}
(\$ 45,000(1+.09 / 12)^{60})(\frac{.09}{12})=m\left[(1+.09 / 12)^{60}-1\right] \\
\end{array} \nonumber\]
Finally dividing by the entire quantity that is multiplied to the m leaves:
\[\begin{array}{l}
\frac{(\$ 45,000(1+.09 / 12)^{60})(\frac{.09}{12})}{\left[(1+.09 / 12)^{60}-1\right] }=m\\
\$ 934.13=m
\end{array} \nonumber\]
Therefore, the monthly payment needed to repay the loan is $934.13 for five years.
b. Over the course of the loan, she would pay $934.13 every month for 60 months.
Total paid \(=$934.13 \cdot 60=$56047.80\).
c. To find interest, we find the difference between purchase price and repayment price. \($56047.80-$45000=$16047.80\), so she will pay \($16047.80\) in interest over the course of the loan.
OUTSTANDING BALANCE ON A LOAN
One of the most common problems deals with finding the balance owed at a given time during the life of a loan. Suppose a person buys a house and amortizes the loan over 30 years, but decides to sell the house a few years later. At the time of the sale, he is obligated to pay off his lender, therefore, he needs to know the balance he owes. Since most long term loans are paid off prematurely, we are often confronted with this problem.
To find the outstanding balance of a loan at a specified time, we need to find the present value \(\mathrm{P}\) of all future payments that have not yet been paid. In this case t does not represent the entire term of the loan. Instead:
- \(t\) represents the time that still remains on the loan
- \(nt\) represents the total number of future payments.
If the problem does not directly state the amount of time still remaining in the term of the loan, then it must be calculated FIRST using \(t\) = [original term of loan] - [time already passed since the start date of the loan].
Example \(\PageIndex{4}\)
Mr. Jackson bought his house in 1995, and financed the loan for 30 years at an interest rate of 7.8%. His monthly payment was $1260. In 2015, Mr. Jackson decides to pay off the loan. Find the balance of the loan he still owes.
Solution
The reader should note that the original amount of the loan is not mentioned in the problem. That is because we don't need to know that to find the balance.
The original loan was for 30 years. 20 years have passed so there are 10 years still remaining. 12(10) = 120 payments still remain to be paid on this loan.
As for the bank or lender is concerned, Mr. Jackson is obligated to pay $1260 each month for 10 more years; he still owes a total of 120 payments. But since Mr. Jackson wants to pay it all off now, we need to find the present value \(\mathrm{P}\) at the time of repayment of the remaining 10 years of payments of $1260 each month. Using the formula we get for the present value of an annuity, we get
\[\begin{aligned}
\mathrm{P}(1+.078 / 12)^{120} &=\frac{\left.\$ 1260\left[(1+.078 / 12)^{120}-1\right)\right]}{(.078 / 12)} \\
\mathrm{P}(2.17597) &=\$ 227957.85 \\
\mathrm{P} &=\$ 104761.48
\end{aligned} \nonumber\]
To find the outstanding balance of a loan
If a loan has a payment of \(m\) dollars made \(n\) times a year at an interest \(r\), then the outstanding value of the loan when there are \(t\) years still remaining on the loan is given by \(\mathrm{P}\):
\[\mathbf{P}(\mathbf{1}+\mathbf{r} / \mathbf{n})^{\mathbf{n} \mathbf{t}}=\frac{\mathbf{m}\left[(\mathbf{1}+\mathbf{r} / \mathbf{n})^{\mathbf{n} \mathbf{t}}-\mathbf{1} |\right.}{\mathbf{r} / \mathbf{n}}\]
IMPORTANT: Note that \(t\) is not the original term of the loan but instead \(t\) is the amount of time still remaining in the future; \(nt\) is the number of payments still remaining in the future
If the problem does not directly state the amount of time still remaining in the term of the loan, then it must be calculated BEFORE using the above formula as \(t\) = [original term of loan] - [time already passed since the start date of the loan].
Note that there are other methods to find the outstanding balance on a loan, but the method illustrated above is generally considered the easiest.


13.2.4: Organizing
What are the Primary Functions of Managers in Organizing Activities?
A second key function of managers is organizing , which is the process of coordinating and allocating a firm’s resources in order to carry out its plans. Organizing includes developing a structure for the people, positions, departments, and activities within the firm. Managers can arrange the structural elements of the firm to maximize the flow of information and the efficiency of work processes. They accomplish this by doing the following:
- Dividing up tasks (division of labor)
- Grouping jobs and employees (departmentalization)
- Assigning authority and responsibilities (delegation)
These and other elements of organizational structure are discussed in detail elsewhere. In this chapter, however, you should understand the three levels of a managerial hierarchy. This hierarchy is often depicted as a pyramid, as in Figure <IP_ADDRESS>. The fewest managers are found at the highest level of the pyramid. Called top management , they are the small group of people at the head of the organization (such as the CEO, president, and vice president). Top-level managers develop strategic plans and address long-range issues such as which industries to compete in, how to capture market share, and what to do with profits. These managers design and approve the firm’s basic policies and represent the firm to other organizations. They also define the company’s values and ethics and thus set the tone for employee standards of behavior. For example, Jack Welch, the former CEO of General Electric, was a role model for his managers and executives. Admirers say that he had an extraordinary capacity to inspire hundreds of thousands of people in many countries and he could change the direction of a huge organization like General Electric as if it were a small firm. Following his leadership, General Electric’s executives turned in impressive results. During his tenure, General Electric’s average annual shareholder return was 25 percent.
The second and third tiers of the hierarchy are called middle management and supervisory (first-line) management , respectively. Middle managers (such as division heads, departmental managers, and regional sales managers) are responsible for beginning the implementation of strategic plans. They design and carry out tactical plans in specific areas of the company. They begin the process of allocating resources to meet organizational goals, and they oversee supervisory managers throughout the firm. Supervisors, the most numerous of the managers, are at the bottom of the managerial pyramid. These managers design and carry out operational plans for the ongoing daily activities of the firm. They spend a great deal of their time guiding and motivating the employees who actually produce the goods and services.
CONCEPT CHECK
- Explain the managerial function of organizing.
- What is the managerial pyramid?


3.3: Levels of Analysis and the Changing Ambitions of a Discipline
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Apart from making us more critical and discerning readers, being aware of the issue of different levels of analysis can also help us understand the way in which the academic discipline of IR has developed over time. To begin with, in the early days of IR – say, from 1919 until the after the Second World War – a lot of what could be called traditional or conventional IR was not concerned with any potential distinctions between different levels of analysis or theoretical perspectives. J. David Singer (1961, 78) lamented that scholars would simply roam up and down the ladder of organizational complexity with remarkable abandon, focusing upon the total system, international organizations, regions, coalitions, extra-national associations, nations, domestic pressure groups, social classes, elites, and individuals as the needs of the moment required.
Singer’s criticism of this ‘general sluggishness’ (Singer 1961, 78) highlights another value in thinking of IR as something that can be studied from different and distinctive perspectives. Being clear about our level of analysis can prevent us from indulging in analytical ‘cherry-picking’, that is to say, from randomly gathering evidence across different levels in pursuit of an answer to our research questions. This ‘vertical drift’, as Singer calls it, can compromise the accuracy of our observations and undermine the validity of our findings. That in turn can obscure some of the detail that might have otherwise turned out to be the key to a conclusive explanation. This does not mean that any one piece of scholarly work must not consider aspects from different levels of analysis. However, when moving between different levels of analysis, we need to do so openly and explicitly. We also need to acknowledge the analytical consequences of drifting between levels: that our search for evidence will need to be comprehensive and that we might have to look at a different set of data or material for each additional aspect. For example, if you were to explain Germany’s decision to open its borders to hundreds of thousands of refugees in 2015 you might want to look at the external pressures as much as the personal motivations of German chancellor Angela Merkel. You would investigate factors at the system level (such as economic indicators, refugee flows, the attitude of key partners) and at the individual level (such as Merkel’s ideological background, her interests and perceptions of the problem as it emerges from statements and key decisions throughout her career). Each would contribute to an overall explanation, but you would need to be prepared to look at different sets of information.
From the 1950s onwards, more and more IR scholars endeavoured to specify the focus of their analysis more clearly. The most prominent example was Kenneth Waltz’s Man, the State and War: A Theoretical Analysis (1959) which introduced an analytical framework for the study of IR that distinguished between what he referred to as different ‘images’ of an issue: the individual, the state and the international system. Waltz’s contributions to the discipline generated interest in analysing the international system as a place of interactions between states. From this perspective, the global system is conceived of as the structure or context within which states cooperate, compete and confront each other over issues of national interest. You might visualise it as a level above the state. Particularly important in that context is the distribution of power amongst states, meaning, whether there is one main concentration of power (‘unipolarity’), two (‘bipolarity’) or several (‘multipolarity’). Global circumstances are seen to condition the ability and opportunity of individual states and groups of states to pursue their interests in cooperative or competitive ways. The view of states being embedded in a global context traditionally comes with the assumption that our international system is ‘anarchic’. An anarchic system is one that lacks a central government (or international sovereign) that regulates and controls what happens to states in their dealings with each other.
Although this idea of the global or system level as a context of anarchy features in many contributions to the IR literature, the main focus remains on the state as the dominant unit of analysis. This enduring focus on the state, and therefore, on the state level of analysis, is referred to as the relative ‘state-centrism’ of the discipline. This means that IR scholars would generally not only regard states as the central unit of analysis as such, they also conceive of the state as a point of reference for other types of actors. From this perspective, the state acts as the arena in which state officials, politicians and decision-makers operate. The state is seen as the framework that encapsulates society and as the main point of reference for the individual. This predominant focus on the state is strongly related to an assumption IR scholars have made about the state also being the main location of power within the international sphere. This idea that the state is where power is primarily concentrated and located has to be seen against the historical context within which some of the most prominent IR scholars operated – the Cold War. It was an era in which much of international affairs appeared to be run via state channels and in line with particular state interests. Other actors that we would consider important from today’s perspective, such as those explored in later chapters in this book, seem to have had little leverage during the Cold War. This was because the period was dominated by great power confrontation and overwhelming military might on each side of the systemic conflict.
Although the Cold War has long since passed, a lot of today’s political life remains managed in the state framework, based on issues like national security, domestic cohesion or internal stability. States form the primary kind of actor in major international organisations such as the United Nations, they feature prominently in the global discourse on most of the major challenges of our time, and states still hold what famous German sociologist Max Weber called the monopoly on violence – the exclusive right to the legitimate use of physical force. States continue to matter and thus have to be part of our considerations about what happens in the world and why. The state as a unit of analysis and frame of reference will certainly not go away any time soon, nor will the interactions of states as a key level of analysis in IR.


8.7: Reactivity of Enolate Ions
After completing this section, you should be able to
- identify the two possible ways in which a given enolate anion could conceivably react with an electrophile.
- write an equation to illustrate the haloform reaction.
- identify the products formed from the reaction of a given methyl ketone with a halogen and excess base.
- identify the methyl ketone, the reagents, or both, needed to obtain a specified carboxylic acid through a haloform reaction.
Make certain that you can define, and use in context, the key terms below.
- haloform
- haloform reaction
Because the negative charge on an enolate ion is delocalized, there are two reactive sites and therefore two potential products. The α ‑substituted product is much more common.
A “haloform” is any compound of the type CHX 3 , where X = Cl, Br or I. Of these three compounds, chloroform is the most common.
The haloform reaction described in the reading is usually carried out with iodine. This reaction is called the “iodoform test,” and is one of the reactions carried out in the laboratory as a simple qualitative test for methyl ketones.
How Enolates React
Due to their negative charges, enolates are better and more versatile nucleophiles than enols. The increased reactivity of enolates makes them capable of a wider range of reactions than enols. Also, α-hydrogen containing compounds can be completely converted to an enolate by reaction with a strong base. Enols can only be created in small amounts by manipulating their equilibrium.
Since the negative charge of an enolate anion is delocalized between the α-carbon and an oxygen, electrophiles may bond to either atom. Reactants having two or more reactive sites are called ambident, so this term applies to enolate anions. Either the C of the O reactive site in an enolate may act as a nucleophile depending on the reaction conditions. Reactions with the oxygen would create a new O-E bond and produce an enol derivative. Reactions with the α-carbon creates a new C-E bond and creates an α-substituted carbonyl compound. Although reactions with the nucleophilic oxygen are possible, reactions involving the nucleophilic α-carbon are much more common, partially due to the thermodynamic stability of the C=O bonds in the final products. Also, the enolate counter ion, such as Li + or Na + , is more tightly associated with the negatively charged enolate oxygen which can then block incoming electrophiles, reducing their chance of reaction at the oxygen.
Stereochemical Implication of Enolate Formation
During enolate formation, an α-hydrogen is removed to form a sp 2 -hybridized, trigonal planar C=C bond which removes any chiral information from the original α -carbon. Because the enol alkene is planar, the incoming electrophile can attack from either the top or the bottom. If the α-carbon of the starting material has a defined stereochemistry or if a new stereocenter is formed during the reaction, the product will be a racemic mixture of enantiomers.
Base Promoted α-Halogenation
An enolate reacts rapidly with a halogen to produce α-halogenated carbonyl products. This reaction has the tendency to overreact and create polyhalogenated products. If a monohalogenated product is sought, the acid catalyzed halogenation reaction discussed in section 8.4 is preferred. Because complete formation to the enolate is not necessary, weak bases, such as the hydroxide anion, are sufficient to produce this reaction. Once a small amount of enolate is formed, it quickly reacts with the halogen. This removes the enolate and shifts the equilibrium toward forming more enolate by Le Chatelier's principle.
Overreaction During Base Promoted α-Halogenation
The α-hydrogens of halogenated carbonyl products are usually more acidic than the corresponding non-halogenated compounds. The inductive electron withdrawing effect of the electronegative halogen stabilizes the negative charge of the enolate ion. This promotes further enolate formation and also further halogenation of the α-carbon. Monohalogenated carbonyls form an enolate over 100 times faster than their non-halogenated counterparts, making multiple halogenations of the α-carbon frequent. This effect is exploited to cause the haloform reaction.
The Haloform Reaction
Overall, the haloform reaction represents a method for the conversion of methyl ketones to carboxylic acids. Due to the increased reactivity of α-halogenated products, methyl ketones typically undergo base promoted halogenation three times to give a trihalo-ketone. A halomethyl ion leaving group is then substituted with a hydroxide ion during nucleophilic acyl substitution. The resulting carboxylate can then be protonated to form a carboxylic acid.
Example
Mechanism
Note! This reaction is considered to be base promoted and not base catalyzed because an entire equivalent of base is required for each α-halogenation. Deprotonation of an α-hydrogen with hydroxide produces the nucleophilic enolate ion which subsequently reacts with the halogen. The increasing acidity of α-halogenated ketone causes this reaction to occur two more times. Once formed, the -CX 3 group attached to the carbonyl can act as a leaving group. Nucleophilic acyl substitution with a hydroxide anion causes C-C bond cleavage and eventually produces a haloform (CHCl 3 , CHBr 3 , CHI 3 ) and a carboxylate anion. The carboxylate ion is easily protonated with acid to form a carboxylic acid functional group. Often, this reaction is performed using iodine (I 2 ) because the subsequent iodoform (CHI 3 ) side-product is a bright yellow solid which is easily filtered off.
This reaction represents one of the few examples of a carbanion leaving group. The trihalomethyl ion ( - :CX 3 ) is particularly stabilized due to the inductive electron-withdrawing effects of the three halogens. The stability of the carbanion can be seen when considering the pK a corresponding conjugate acid. In particular, bromoform (CHBr 3 ) has a pK a of 13.7 which is more than 10 20 times more acidic than a typical alkane C-H bond.
| 1) Enolate formation | |
| 2) Nucleophilic attack on the halogen | |
| 3) Repeat the halogenation two more times | |
| 4) Nucleophilic attack on the electrophilic carbonyl carbon | |
| 5) Nucleophilic acyl substitution | |
| 6) Deprotonation | |
| 7) Protonation of the carboxylate |
Biological Haloform Reaction
Interest in the haloform reaction has increased since the discovery that certain plants and marine algae can biosynthesize chloroform, bromoform, and other small halocarbons through an analogous process. Previously it was assumed that these toxic compounds were present in the environment as man-made pollutants.
The likely starting material of the biosynthesis are biogenic methyl ketones, a halogen anion, and oxygen. The enzyme, chloroperoxidase, catalyzes the polyhalogenation of the methyl groups. As in the haloform reaction, the final biosynthesis step involves a nucleophilic acyl substitution with a hydroxide anion to create a haloform and a carboxylate anion.
Please predict the expected products of the following reactions:
a) b)
- Answer
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a) b)


1.88: Priority rule
When more than one kind of symmetry element occurs for a given symmetry direction, the choice for defining the appropriate Hermann–Mauguin symbol for the space group is made in order of descending priority:
- m , e , a , b , c , n , d ;
- rotation axes before screw axes.
There are a few exceptions to this choice:
- For glide planes in centered monoclinic space groups, the priority rule is purposely not followed in order to bring out the relations between the three ‘cell choices’ given for each setting.
- For orthorhombic space groups, the priority rule is applied only to the ‘standard symbol’. The symbols for the other five settings are obtained from the standard symbol by the appropriate transformations, without invoking the priority rule again.
- Space-group types I 222 and I 2 1 2 1 2 1 are two distinct types. Both contain parallel twofold rotation and screw axes and thus would receive the same symbol according to the priority rule. In I 222, the three rotation axes and the three screw axes intersect, whereas in I 2 1 2 1 2 1 neither the three rotation axes nor the three screw axes intersect.
- For space groups of type No. 73, the standard symbol Ibca was adopted, instead of Ibaa according to the rule, because Ibca displays the equivalence of the three symmetry directions clearly.
- In tetragonal space groups with both a and b glide planes containing the [001] direction, the preference was given to b , as in P4bm .
- In cubic space groups where tertiary symmetry planes with glide components 1/2, 0, 0; 0, 1/2, 0; 0, 0, 1/2 and 1/2, 1/2 , 1/2 coexist, the tertiary symmetry element was called n in P groups (instead of a , b or c ) but c in F groups, because these symmetry elements intersect the origin.
- Space groups of type I 23 and I 2 1 3 (199) are two distinct types of space groups. For this pair, the same arguments apply as given above for I 222 and I 2 1 2 1 2 1 .