numbers from the earlier example (where you had DBP data on seven participants, with the values 84,

84, 89, 91, 110, 114, and 116 mmHg), the equation looks like this:

Even with technology, this formula is computationally challenging. By using logarithms (which turn

multiplications into additions and roots into divisions), you can develop a numerically stable

alternative formula, which is:

This formula may look complicated, but it really just says, “The geometric mean is the antilog of the

mean of the logs of the values in the sample.” In other words, to calculate the GM using this formula,

you take the log of each value in your sample, then average all those logs together, and then take the

antilog of that average. You can choose to use either natural or common logarithms, but make sure that

whatever you choose, you use same type of antilog. (Flip to Chapter 2 for the basics of logarithms.)

Describing the spread of your data

After central tendency (described earlier in “Locating the center of your data”), the second most

important set of summary statistics for numerical values refers to how tightly or loosely they tend to

cluster around a central value, meaning how they are dispersed. There are several common measures

of dispersion, as you find out in the following sections.

Standard deviation, variance, and coefficient of variation

The standard deviation (usually abbreviated SD, sd, or just s) of a set of numerical values tells you

how much the individual values tend to differ from the mean in either direction (see “Locating the

center of your data” for a discussion of the mean). The SD is calculated as follows:

This formula is saying that you calculate the SD of a set of N numbers by first subtracting the

mean from each value (

) to get the deviation (

) of each value from the mean. Then, you take

the square each of these deviations and add up the

terms. After that, you divide that number by

N – 1, and finally, you take the square root of that number to get your answer, which is the SD.

For the sample of diastolic blood pressure (DBP) measurements for seven study participants in the

example used earlier in this chapter, where the values are 84, 84, 89, 91, 110, 114, and 116 mmHg and

the mean is 98.3 mmHg, you calculate the SD as follows:

Several other useful measures of dispersion are related to the SD:

Variance: The variance is just the square of the SD. For the DBP example, the variance

.