upgrade

📊honors statistics review

Pooled Standard Deviation

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

The pooled standard deviation is a measure of the combined variability of two or more populations when comparing their means. It is calculated as a weighted average of the individual standard deviations of the populations, and is used in statistical tests that involve comparing the means of two or more groups.

5 Must Know Facts For Your Next Test

  1. The pooled standard deviation is used when the standard deviations of the two populations are unknown, but are assumed to be equal.
  2. The pooled standard deviation is calculated as the square root of the weighted average of the individual sample variances.
  3. The pooled standard deviation is used in the two-sample t-test to compare the means of two populations when the standard deviations are unknown.
  4. The degrees of freedom for the pooled standard deviation is the sum of the degrees of freedom for the two samples.
  5. The pooled standard deviation is also used in one-way ANOVA to compare the means of more than two populations when the standard deviations are unknown.

Review Questions

  • Explain the purpose of using the pooled standard deviation in the context of comparing the means of two populations with unknown standard deviations.
    • When comparing the means of two populations, and the standard deviations of the populations are unknown, the pooled standard deviation is used to estimate the common variability of the two groups. This allows for a more accurate and powerful statistical test, such as the two-sample t-test, to be conducted to determine if the means of the two populations are significantly different. The pooled standard deviation takes into account the variability within each group, rather than assuming equal variances, which can lead to more reliable conclusions about the differences between the population means.
  • Describe how the pooled standard deviation is calculated and how the degrees of freedom are determined for the statistical test.
    • The pooled standard deviation is calculated as the square root of the weighted average of the individual sample variances. Specifically, the formula is: $\sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}$, where $n_1$ and $n_2$ are the sample sizes, and $s_1^2$ and $s_2^2$ are the sample variances for the two populations. The degrees of freedom for the pooled standard deviation is the sum of the degrees of freedom for the two samples, which is $n_1 + n_2 - 2$. This degrees of freedom is then used in the statistical test, such as the two-sample t-test, to determine the appropriate critical value and p-value for the hypothesis test.
  • Explain how the pooled standard deviation is used in the context of one-way ANOVA to compare the means of more than two populations with unknown standard deviations.
    • In one-way ANOVA, the pooled standard deviation is used to compare the means of more than two populations when the standard deviations are unknown and assumed to be equal across the groups. The pooled standard deviation is calculated as the square root of the weighted average of the individual sample variances, just as in the two-sample t-test. This pooled standard deviation is then used to compute the F-statistic, which is used to determine if there are any significant differences among the population means. The degrees of freedom for the pooled standard deviation is the sum of the degrees of freedom for all the samples, which is used to determine the appropriate critical value for the F-distribution. By using the pooled standard deviation, the one-way ANOVA can provide a more accurate and powerful test of the equality of the population means, even when the standard deviations are unknown.

"Pooled Standard Deviation" also found in: