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56 UNIT 1 Exploring One-Variable Data

Measuring Variability: The Standard Deviation

To avoid the problem illustrated by the film strip example, we prefer a measure of vari-
ability that uses more than just the minimum and maximum values of a quantitative
data set. If we summarize the center of a distribution with the mean, then we should
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use the standard deviation to describe the variation of data values around the mean.

DEFINITION Standard deviation
The standard deviation measures the typical distance of the values in a distribution
from the mean. It is calculated by finding an “average” of the squared deviations of
AP® EXAM TIP the individual data values from the mean, and then taking the square root.
Formula sheet If the values in a quantitative data set are xx, , … x , the sample standard devia-

,
The formula for the sample tion s is given by the formula 1 2 n
x

standard deviation s is
x
(
included on the formula sheet x ( 1 − x ) 2 + x( 2 − x ) 2 + …+ x n − x ) 2 Σ x ( i − x ) 2
provided on both sections of s x = n −1 = n −1
the AP® Statistics exam.
The value obtained before taking the square root in the standard deviation calcu-
2
lation is called the sample variance, denoted by s . Because variance is measured in

x
squared units, it is not a very helpful way to describe the variability of a distribution.

HOW TO CALCULATE THE SAMPLE STANDARD DEVIATION s x
1. Find the mean of the distribution.

2. Calculate the deviation of each value from the mean:
deviation = value − mean

3. Square each of the deviations.

4. Add all of the squared deviations, then divide by –1n to get the sample
2
variance s .

x
5. Take the square root to return to the original units.

EXAMPLE How many friends? Skill 2.C
Measuring variability: The standard deviation

© 2024 1 2 2 2 3 3 3 3 4 4 6
PROBLEM: A random sample of 11 high school students are asked how many

“close” friends they have. Here are their responses, along with a dotplot:

1 2 3 4 5 6
Number of close friends
Ian Allenden/Alamy Stock Photo
Calculate the sample standard deviation. Interpret this value.

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