*90*

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 s_{p}^{2} and is calculated as:

s_{p}^{2} = ( (n_{1}-1)s_{1}^{2} + (n_{2}-1)s_{2}^{2} ) / (n_{1}+n_{2}-2)

When the two sample sizes (n_{1} and n_{2}) are equal, the formula simplifies to:

s_{p}^{2} = (s_{1}^{2} + s_{2}^{2} ) / 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 1:**

- Sample size n
_{1}= 40 - Sample variance s
_{1}^{2}= 18.5

**Sample 2:**

- Sample size n
_{2}= 38 - Sample variance s
_{2}^{2}= 6.7

Here is how to calculate the pooled variance between the two samples:

- s
_{p}^{2}= ( (n_{1}-1)s_{1}^{2}+ (n_{2}-1)s_{2}^{2}) / (n_{1}+n_{2}-2) - s
_{p}^{2}= ( (40-1)*18.5 + (38-1)*6.7 ) / (40+38-2) - s
_{p}^{2}= (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.

**Bonus Resource:** Use this Pooled Variance Calculator to automatically calculate the pooled variance between two samples.