How to Calculate Pooled Variance

Pooled Variance Formula:

\[ s_p^2 = \frac{\sum_{i=1}^{k} (n_i - 1) s_i^2}{\sum_{i=1}^{k} (n_i - 1)} \]

Pooled Variance (s_p²):

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1. What is Pooled Variance?

Pooled variance is a method for estimating the combined variance of several different groups or samples. It's commonly used in statistical analysis when we assume that different groups have the same variance but potentially different means.

2. How Does the Calculator Work?

The calculator uses the pooled variance formula:

\[ s_p^2 = \frac{\sum_{i=1}^{k} (n_i - 1) s_i^2}{\sum_{i=1}^{k} (n_i - 1)} \]

Where:

\( s_p^2 \) — Pooled variance
\( n_i \) — Sample size of group i
\( s_i^2 \) — Variance of group i
\( k \) — Number of groups

Explanation: The formula calculates a weighted average of the group variances, where the weights are the degrees of freedom (n_i - 1) for each group.

3. Importance of Pooled Variance

Details: Pooled variance is essential in various statistical tests, including the two-sample t-test and ANOVA, where we need to estimate a common variance across multiple groups. It provides a more reliable estimate than using individual group variances separately.

4. Using the Calculator

Tips: Enter sample counts and variances as comma-separated values. Make sure both lists have the same number of elements corresponding to the same groups. All counts should be greater than 1, and variances should be non-negative.

5. Frequently Asked Questions (FAQ)

Q1: When should I use pooled variance?
A: Use pooled variance when you have multiple groups that you believe have the same variance, and you want to combine them into a single estimate.

Q2: What's the difference between pooled variance and regular variance?
A: Regular variance measures spread within a single group, while pooled variance combines information from multiple groups assuming equal variances.

Q3: Can I use pooled variance if group sizes are very different?
A: Yes, the formula appropriately weights each group's variance by its degrees of freedom, so it works with different group sizes.

Q4: What statistical tests use pooled variance?
A: The independent two-sample t-test (assuming equal variances) and ANOVA both use pooled variance in their calculations.

Q5: What if my groups have different variances?
A: If groups have significantly different variances, you should not use pooled variance. Consider using tests that don't assume equal variances, like Welch's t-test.