Examples of pooled variance in the following topics:

 Here, ${ S }{ x }_{ 1 }{ x }_{ 2 }$ is the grand standard deviation (or pooled standard deviation), 1 = group one, 2 = group two.
 This test is used only when it can be assumed that the two distributions have the same variance.
 This is the formula for a pooled variance in a twosample ttest with unequal or equal sample sizes but unequal variances.
 This is the formula for a pooled variance in a twosample ttest with unequal sample size but equal variances.
 Calculate the t value for different types of sample sizes and variances in an independent twosample ttest



 In such cases, we can make our t distribution approach slightly more precise by using a pooled standard deviation.
 The pooled standard deviation of two groups is a way to use data from both samples to better estimate the standard deviation and standard error.
 If s1 and s2 are the standard deviations of groups 1 and 2 and there are good reasons to believe that the population standard deviations are equal, then we can obtain an improved estimate of the group variances by pooling their data:
 To use this new statistic, we substitute $s^2_{pooled}$ in place of $s^2_1$ and $s^2_2$ in the standard error formula, and we use an updated formula for the degrees of freedom:
 The beneﬁts of pooling the standard deviation are realized through obtaining a better estimate of the standard deviation for each group and using a larger degrees of freedom parameter for the t distribution.

 The method of analysis of variance in this context focuses on answering one question: is the variability in the sample means so large that it seems unlikely to be from chance alone?
 The MSG can be thought of as a scaled variance formula for means.
 To this end, we compute a pooled variance estimate, often abbreviated as the mean square error (MSE), which has an associated degrees of freedom value dfE = n−k.
 The F statistic and the F test Analysis of variance (ANOVA) is used to test whether the mean outcome diﬀers across 2 or more groups.



 Recall that when the variables X and Y are independent, the variance of the sum or difference between X and Y can be written as follows:
 which is read: "The variance of X plus or minus Y is equal to the variance of X plus the variance of Y."
 For example, if the variance of verbal SAT were 10,000, the variance of quantitative SAT were 11,000 and the correlation between these two tests were 0.50, then the variance of total SAT (verbal + quantitative) would be:
 If the variances and the correlation are computed in a sample, then the following notation is used to express the variance sum law:

