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

 To calculate the F ratio, two estimates of the variance are made.
 We used sum of squares to calculate the sample variance and the sample standard deviation in Descriptive Statistics.
 " 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.
 Since variances are always positive, if the null hypothesis is false, MSbetween will generally be larger than MSwithin.

 The latter condition is guaranteed if the data values are independent and normally distributed with a common variance .
 To calculate the Fratio, two estimates of the variance are made:
 " 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.
 Since variances are always positive, if the null hypothesis is false, MSbetween will generally be larger than MSwithin.

 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.

 Notice that the sample variances s 1 2 and s 2 2 are not pooled.

 Notice that each group has the same number of plants so we will use the formula $\displaystyle F = \frac {n \cdot s^2_{\bar{ x}}}{s^2_{pooled}}$ .
 First, calculate the sample mean and sample variance of each group.
 Next, calculate the variance of the three group means (Calculate the variance of 24.2, 25.4, and 24.4).
 Calculate the mean of the three sample variances (Calculate the mean of 11.7, 18.3, and 16.3).

 When comparing two proportions, it is necessary to use a pooled standard deviation for the ztest .
 A tstatistic may be used for one sample, two samples (with a pooled or unpooled standard deviation), or for a regression ttest .
 Ftests (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.

 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:

 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 αj is denoted σα2 and the variance of εij is denoted σε2.The population ICC in this framework is shown in .
 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.