How to Compare Standard Deviations (With Example)


Often you may want to compare standard deviations between two datasets to determine if they are equal.

The most common way to compare standard deviations is to actually compare the variances (the standard deviation squared) between two datasets using one of the following methods:

Method 1: Use Rule of Thumb

One crude way to determine if two variances are equal is to use the variance rule of thumb.

As a rule of thumb, if the ratio of the larger variance to the smaller variance is less than 4 then we can assume the variances are approximately equal.

Otherwise, if the ratio is equal to or greater than 4, we assume that the variances are not equal.

Method 2: Use Formal Statistical Test

A more formal way to test if two variances are equal is to use an F-test.

This test uses the following hypotheses:

  • H0: σ12 = σ22 (the population variances are equal)
  • H1: σ12 ≠ σ22 (the population variances are not equal)

To perform an F-test, we typically use statistical software such as R, Python, Excel, SPSS, etc.

The following examples show how to use each method in practice to compare the standard deviations between the following two datasets that show the exam scores received by students who used two different study methods to prepare for the exam:

Method 1: Use Rule of Thumb to Compare Standard Deviations

One way to compare the standard deviations between the two datasets is to first calculate the variance of each dataset:

Next, we can calculate the ratio of the larger variance to the smaller variance:

Ratio of Variances: 103.41 / 24.21 = 4.27

Since this ratio is greater than 4, we would assume that the variances are not equal.

Thus, we would assume that the standard deviations between the two datasets are not equal.

Method 2: Use F-Test to Compare Standard Deviations

Another way to compare the standard deviations between the two datasets is to perform an F-test.

Most statistical software is able to perform an F-test, but we will use the following code in R to do so:

#enter exam scores for both groups of students
method1 <- c(68, 70, 71, 72, 74, 74, 78, 82, 83, 88, 90, 92, 93, 96, 97)
method2 <- c(77, 80, 81, 81, 82, 83, 83, 84, 84, 85, 88, 89, 90, 92, 95)

#perform an F-test to determine if the variances are equal
var.test(method1, method2)

	F test to compare two variances

data:  method1 and method2
F = 4.2714, num df = 14, denom df = 14, p-value = 0.01031
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
  1.434049 12.722857
sample estimates:
ratio of variances 
           4.27144 

The F-test returns the following results:

  • F-test statistic: 4.2714
  • p-value: .01031

Recall that the F-test uses the following hypotheses:

  • H0: σ12 = σ22 (the population variances are equal)
  • H1: σ12 ≠ σ22 (the population variances are not equal)

Since the p-value of our test (.01031) is less than .05, we have sufficient evidence to reject the null hypothesis.

We would conclude that the variances are not equal.

Thus, we would conclude that the standard deviations between the two datasets are not equal.

Bonus: You can also use the Statology F-Test for Equal Variances Calculator to perform this F-test.

Additional Resources

The following tutorials provide additional information about using standard deviations in statistics:

Standard Deviation vs. Standard Error: The Difference
Population vs. Sample Standard Deviation: When to Use Each
Coefficient of Variation vs. Standard Deviation: The Difference

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