The meaning of measure

Error and uncertainty in measurements


1  Uncertainty is certain, error is everywhere!





Error in reading scales


digital meter



Parallax error

One form of scale-reading error that often afflicts beginners in the science laboratory is failure to properly align the eye with the part of the scale you are reading. This gives rise to parallax error. Parallax refers to the change in the apparent position of an object when viewed from different points.

meniscus error
The most notorious example encountered in the introductory chemisty laboratory is failure to read the volume of a liquid properly in a graduated cylinder or burette. Getting all of their students trained to make sure their eye is level with the bottom of the meniscus is the lab instructors' hope and dispair.







Random (indeterminate) error

random error

Systematic error


systemtic error

2  The difference between accuracy and precision







3  Dealing with replicate measurements and scatter

When there is no clearly defined "true" value




How many measurements do I need?

One measurement may be enough



... but more are needed when there is no clearly-defined "true" value





Mean, median, and range of a series of observations



The mean and its meaning






The range of a data set is the difference between its smallest and largest values. As such, its value reflects the precision of the result. For example, the following data sets have the same average, but the one having the smaller range is clearly more precise.

If you arrange the list of measured values in order of their magnitude, the median is the one that has as many values above it as below it.

Examples: for the data set [22 23 23 24 26 28] the mode would be 23.
For an odd number of values n, the median is the [(n+1)/2]th member of the set. Thus for [22 23 23 24 24 27], (n+1)/2 =3, so 23 is the median value.


This refers to the value that is observed most frequently in a series of measurements. If two or more values tie for the highest frequency, then there can be multiple modes.  Mode is most useful in describing larger data sets.

Example: for the data set [22 23 23 24 26 26] the modes are 23 and 24.

The more observations, the more reliable the mean value.

If this is not immediately obvious, think about it this way. You would not want to predict the outcome of the next election on the basis of interviews with only two or three voters; you would want a sample of ten to twenty at a minimum, and if the election is an important national one, a fair sample would require hundreds to thousands of people distributed over the entire geographic area and representing a variety of socioeconomic groups.

Similarly, if you were testing the lifetimes of light bulbs, you would want to test a large number of them in order to estimate their mean lifetimes.

Statistical theory tells us that the more samples we have, the greater will be the chance that the mean of the results will correspond to the “true” value, which in this case would be the mean obtained if samples could be taken from the entire population (of people or of light bulbs.)

This point can be better appreciated by examining the two sets of data shown here. Each diagram summarizes the observations made The set on the right was made from of only three points (shown in orange), and gives a mean that is quite far removed from the "true" value, which is arbitrarily chosen for this example.

mean values
Deviation of the mean from the "true value" becomes smaller when more measurements are made.

Plots and points

A similar problem arises when you try to fit a curve to a series of plotted points. Suppose, for example, that curve 1 (red) represents the true relationship between the quantities indicated on the y-axis (dependent variable) and those on the x-axis (independent variable). This curve is derived from the seven points indicated on the plot.

Contrast this curve with the false straight-line relationships 2 or 3 that might be obtained if only four or three points had been recorded.


4  Absolute and relative uncertainty, error in calculations


When expressing the uncertainty of a value given in scientific notation, the exponential part should include both the value itself and the uncertainty. An example of the proper form would be (3.19 ± 0.02) × 104 m.

Although the absolute uncertainties in these two examples are identical, we would probably consider the second measurement to be more precise because the uncertainty is a smaller fraction of the measured value. A quantity calculated in this way is known as the relative uncertainty.



Propagation of Error

We are often called upon to find the value of some quantity whose determination depends on several other measured values, each of which is subject to its own sources of error.

Consider a common laboratory experiment in which you must determine the percentage of acid in a sample of vinegar by observing the volume of sodium hydroxide solution required to neutralize a given volume of the vinegar. You carry out the experiment and obtain a value. Just to be on the safe side, you repeat the procedure on another identical sample from the same bottle of vinegar. If you have actually done this in the laboratory, you will know it is highly unlikely that the second trial will yield the same result as the first. In fact, if you run a number of replicate (that is, identical in every way) determinations, you will probably obtain a scatter of results.

To understand why, consider all the individual measurements that go into each determination; the volume of the vinegar sample, your judgement of the point at which the vinegar is neutralized, and the volume of solution used to reach this point. And how accurately do you know the concentration of the sodium hydroxide solution, which was made up by dissolving a measured weight of the solid in water and then adding more water until the solution reaches some measured volume. Each of these many observations is subject to random error; because such errors are random, they can occasionally cancel out, but for most trials we will not be so lucky — hence the scatter in the results.

More on the many sources of error in titrations.

Rules for estimating errors in calculated results

Suppose you measure the mass and volume of a sample, and are required to calculate its density by dividing one quantity by the other: d = m / V.  Both components of this quotient have uncertainties associated with them, and  you wish to attach an uncertainty to the calculated density. 



1. Addition and subtraction, both numbers have uncertainties
The simplest method is to just add the absolute uncertainties.





2. Multiplication or division, both numbers have uncertainties.
3. Multiplication or division by a pure number

Concept Map