In statistics, pooled variance simply refers to the average of two or more group variances.
We use the word “pooled” to indicate that we’re “pooling” two or more group variances to come up with a single number for the common variance between the groups.
In practice, pooled variance is used most often in a two sample t-test, which is used to determine whether or not two population means are equal.
The pooled variance between two samples is typically denoted as sp2 and is calculated as:
sp2 = ( (n1-1)s12 + (n2-1)s22 ) / (n1+n2-2)
When the two sample sizes (n1 and n2) are equal, the formula simplifies to:
sp2 = (s12 + s22 ) / 2
When to Calculate the Pooled Variance
When we want to compare two population means, there are two statistical tests we could potentially use:
1. Two sample t-test: This test assumes the variances between the two samples are approximately equal. If we use this test, then we calculate the pooled variance.
2. Welch’s t-test: This test does not assume the variances between the two samples are approximately equal. If we use this test, we do not calculate the pooled variance. Instead, we use a different formula.
To determine which test to use, we use the following rule of thumb:
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 and use the two sample t-test.
For example, suppose sample 1 has a variance of 24.5 and sample 2 has a variance of 15.2. The ratio of the larger sample variance to the smaller sample variance would be calculated as:
Ratio: 24.5 / 15.2 = 1.61
Since this ratio is less than 4, we could assume that the variances between the two groups are approximately equal. Thus, we would use the two sample t-test which means we would calculate the pooled variance.
Example of Calculating the Pooled Variance
Suppose we want to know whether or not the mean weight between two different species of turtles is equal. To test this, we collect a random sample of turtles from each population with the following information:
- Sample size n1 = 40
- Sample variance s12 = 18.5
- Sample size n2 = 38
- Sample variance s22 = 6.7
Here is how to calculate the pooled variance between the two samples:
- sp2 = ( (n1-1)s12 + (n2-1)s22 ) / (n1+n2-2)
- sp2 = ( (40-1)*18.5 + (38-1)*6.7 ) / (40+38-2)
- sp2 = (39*18.5 + 37*6.7 ) / (76) = 12.755
The pooled variance is 12.755.
Notice that the value for the pooled variance is located between the two original variances of 18.5 and 6.7. This makes sense considering the pooled variance is just a weighted average of the two sample variances.