Examples of pooled variance in the following topics:

 it can be assumed that the two distributions have the same variance.
 Here, ${ S }{ x }_{ 1 }{ x }_{ 2 }$ is the grand standard deviation (or pooled standard deviation), 1 = group one, 2 = group two.
 Note that in this case ${ { s }_{ { \bar { X } }_{ 1 }{ \bar { X } }_{ 2 } } }^{ 2 }$ is not a pooled variance.
 This is the formula for a pooled variance in a twosample ttest with unequal sample size but equal variances.
 This is the formula for a pooled variance in a twosample ttest with unequal or equal sample sizes but unequal variances.

 Variance between samples: An estimate of σ2 that is the variance of the sample means multiplied by n (when there is equal n).
 Variance within samples: An estimate of σ2 that is the average of the sample variances (also known as a pooled variance).
 MS means "mean square. " MSbetween is the variance between groups and MSwithin is the variance within groups.
 As it turns out, MSbetween consists of the population variance plus a variance produced from the differences between the samples.
 MSwithin is an estimate of the population variance.

 To calculate the $F$ratio, two estimates of the variance are made:
 Variance between samples: An estimate of $\sigma^2$ that is the variance of the sample means multiplied by $n$ (when there is equal $n$).
 Variance within samples: An estimate of $\sigma^2$ that is the average of the sample variances (also known as a pooled variance).
 $MS$ means "mean square. " $MS_{\text{between}}$ is the variance between groups and $MS_{\text{within}}$ is the variance within groups.
 As it turns out, $MS_{\text{between}}$ consists of the population variance plus a variance produced from the differences between the samples.

 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.

 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:
 Caution: Pooling standard deviations should be done only after careful research
 A pooled standard deviation is only appropriate when background research indicates the population standard deviations are nearly equal.

 Notice that the sample variances s 1 2 and s 2 2 are not pooled.
 (If the question comes up, do not pool the variances. )
 Do not pool the variances.
 Arrow down to Pooled: and No.

 Other uses for the F distribution include comparing two variances and TwoWay Analysis of Variance.
 Comparing two variances is discussed at the end of the chapter.
 Next, calculate the variance of the three group means (Calculate the variance of 24.2, 25.4, and 24.4).
 Variance of the group means = 0.413 = $s^2_{\bar{x}}$
 Mean of the sample variances = 15.433 = $s^2_{pooled}$

 When comparing two proportions, it is necessary to use a pooled standard deviation for the $z$test.
 A $t$statistic may be used for one sample, two samples (with a pooled or unpooled standard deviation), or for a regression $t$test.
 $F$tests (analysis of variance, also called ANOVA) are used when there are more than two groups.
 If the variance of test scores of the lefthanded in a class is much smaller than the variance of the whole class, then it may be useful to study lefties as a group.
 The null hypothesis is that two variances are the same, so the proposed grouping is not meaningful.

 The intraclass correlation can be regarded within the framework of analysis of variance (ANOVA), and more recently it has been regarded in the framework of a random effect model.
 The variance of $\alpha_j$ is denoted $\sigma_{\alpha}^2$ and the variance of $\epsilon_{ij}$ is denoted $\sigma_{\epsilon}^2$.
 One key difference between the two statistics is that in the ICC, the data are centered and scaled using a pooled mean and standard deviation; whereas in the Pearson correlation, each variable is centered and scaled by its own mean and standard deviation.
 This pooled scaling for the ICC makes sense because all measurements are of the same quantity (albeit on units in different groups).
 where ${ \mu }_{ x }$ and ${ \mu }_{ y }$ are the means for the two variables and ${ { \sigma }^{ 2 } }_{ x }$ and ${ { \sigma }^{ 2 } }_{ y }$ are the corresponding variances.

 Compute the variance of the sum of two variables if the variance of each and their correlation is known
 Compute the variance of the difference between two variables if the variance of each and their correlation is known
 which is read: "The variance of X plus or minus Y is equal to the variance of X plus the variance of Y."
 The variance of the difference is:
 If the variances and the correlation are computed in a sample, then the following notation is used to express the variance sum law: