pooled variance equation

HaC 2025

3 min read 18-03-2025

The pooled variance is a crucial concept in statistics, particularly when dealing with hypothesis testing involving two or more groups. It's a weighted average of the variances of those groups, providing a single, combined estimate of the population variance. This is especially useful when we assume the populations have equal variances (homoscedasticity). This article will dissect the pooled variance equation, explain its applications, and provide examples to clarify its use.

What is Pooled Variance?

Pooled variance is an estimate of the common variance shared by two or more groups. It's used when we believe the different groups are all samples from populations with the same variance, even if their means might differ. Instead of using the individual variance estimates from each group, we combine them into a single, more precise estimate. This increased precision stems from using more data points in the calculation.

The Pooled Variance Equation

The equation for calculating pooled variance (denoted as s_p²) is:

s_p² = [(n₁ - 1)s₁² + (n₂ - 1)s₂² + ... + (n_k - 1)s_k²] / (n₁ + n₂ + ... + n_k - k)

Where:

s_p²: The pooled variance.
n_i: The sample size of group i.
s_i²: The sample variance of group i.
k: The number of groups.

For the common case of two groups, the equation simplifies to:

s_p² = [ (n₁ - 1)s₁² + (n₂ - 1)s₂² ] / (n₁ + n₂ - 2)

This simplified version is frequently encountered in t-tests comparing the means of two independent groups.

Why Use Pooled Variance?

Using the pooled variance offers several advantages:

Increased Precision: Combining data from multiple groups leads to a more precise estimate of the population variance, especially when individual group sample sizes are small. More data generally means a more reliable estimate.
Improved Statistical Power: In hypothesis testing (like a t-test), using the pooled variance can increase the statistical power of the test. This means you're more likely to detect a true difference between groups if one exists.
Assumption of Homoscedasticity: The pooled variance is only appropriate when the assumption of equal variances across groups (homoscedasticity) holds. Tests like Levene's test can be used to assess this assumption before using the pooled variance.

Example Calculation: Pooled Variance for Two Groups

Let's say we have two groups of students who took the same exam.

Group 1: n₁ = 10, s₁² = 25 Group 2: n₂ = 15, s₂² = 36

Using the simplified equation:

s_p² = [(10 - 1) * 25 + (15 - 1) * 36] / (10 + 15 - 2) = (225 + 504) / 23 ≈ 31.7

Therefore, the pooled variance is approximately 31.7.

When Not to Use Pooled Variance

It's crucial to remember that pooled variance should only be used when the assumption of equal variances across groups is met. If Levene's test or a similar test indicates unequal variances (heteroscedasticity), using the pooled variance is inappropriate. In such cases, alternative methods that don't assume equal variances should be employed.

Conclusion

The pooled variance equation is a valuable tool in statistical analysis, providing a more efficient estimate of the population variance when the assumption of equal variances across groups is valid. Understanding its calculation and appropriate application is vital for accurate hypothesis testing and drawing reliable conclusions from your data. Remember to always check the assumption of homoscedasticity before using the pooled variance.