Like the range and the interquartile range, standard deviation is a way to measure the dispersion in a given data set—that is, how spread out the values are. Standard deviation is found by calculating the mean of a group of values, then using a type of averaging to determine how far away the other values in the group are from that mean. If the values in the group are all relatively close to the mean, then the standard deviation is low. Conversely, the more spread out the values are relative to the mean, the higher the standard deviation. For example, consider these two groups of values:
Set A: {0, 25, 50, 75, 100}
Set B: {48, 49, 50, 51, 52}
Both sets have five distinct values, and both have a mean of 50. But because the
values in Set A are more widely dispersed (as much as 50 units away from the
mean) than the values in Set B (all of which are within 2 units of the mean),
Set A has a greater standard deviation than does Set B.
The GRE tests this basic
understanding of standard deviation in a variety of ways. For example, they might ask you to compare the standard deviation of data sets that include
variables. While such a question might seem daunting, solve it by focusing on the basic concept of standard deviation—data
sets with values that are more spread out have a higher standard deviation.
Example:
If x is a positive integer, which group of values has the greater standard deviation?
Set A:
{x, 2x, 3x, 4x}
Set B: {x, 3x, 5x, 7x}
Both Set A and Set B
contain the positive integer x. In Set A, the second value is twice as large as x, the third value is three times as
large as x, and the fourth value is
four times as large as x. In Set B,
however, the second value is three times as large as x, the third value is five times as large as x, and the fourth value is seven times as large as x. Without even knowing what x is, we know that the values in Set A
are more closely grouped together than the values in Set B. Therefore, since
the values in Set B are more widely dispersed
than those in Set A, Set B has a greater standard
deviation.
Another way the GRE could test your understanding of standard deviation is to provide the mean and standard deviation of a group of values, then ask you to determine specific values in that group. For example, imagine a data set that has a mean of 100 and a standard deviation of 20. Even without knowing specific values in the data set, you know that one standard deviation above the mean is 120, and one standard deviation below the mean is 80. Two standard deviations above the mean is 140, and two standard deviations below the mean is 60.
Example:
The mean of Data Set A is 40 and the standard deviation is 10. What value is 1.5 standard deviations below the mean?
The
distance from the mean is the number of standard deviations times the value of
one standard deviation:
. And since the question asks for the value that is
standard
deviations below the mean, that is
.
Another variant on the same theme would be to provide the standard deviation of a group of values and one value in that group. If you know how many standard deviations away from the mean that value is, you could calculate the mean itself.
Example:
If 25 is 1.5 standard deviations below the mean of Data Set A and the standard deviation of the data set is 10, what is the mean of Data Set A?
The distance from the mean is the number of standard deviations times the value of one standard deviation: 1.5 × 10 = 15.
Since
25 is below the mean, the value of
the mean is
.
On Test Day, you probably won’t have to calculate the precise standard deviation of a group of values. The types of examples you worked through above were solved with just a basic understanding of standard deviation. That said, knowing how standard deviation is calculated can be useful. For any group of values, follow these steps to calculate standard deviation:
Example:
Calculate the standard
deviation of 1, 3, 8, 11, and 12.
First, find the mean:
Next, determine the difference between each term and 7:
(1 − 7) = −6, (3 − 7) = −4, (8 − 7) = 1, (11 − 7) = 4, and (12 − 7) = 5.
Square each difference, and find the mean of the squared differences:
Finally, find the
non-negative square root of that average to determine the standard deviation:
.
When you see a question on the GRE that involves standard deviation, there’s no need to panic. Standard deviation is merely a measure of dispersion that essentially represents each value’s average deviation from the mean. If you keep this in mind, you can approach the occasional standard deviation question with confidence.