Examples of pooled variance in the following topics:

 A twosample ttest for unequal sample sizes and equal variances is used only when 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.
 The tstatistic to test whether the means are different can be calculated as follows:
$t=\frac { { \bar { X } }_{ 1 }{ \bar { X } }_{ 2 } }{ { S }{ x }_{ 1 }{ x }_{ 2 }\cdot \sqrt { \frac { 1 }{ { n }_{ 1 } } +\frac { 1 }{ { n }_{ 2 } } } }$,
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.
 Note that in this case ${ { s }_{ { \bar { X } }_{ 1 }{ \bar { X } }_{ 2 } } }^{ 2 }$ is not a pooled variance.
 pooled variance (noun) A method for estimating variance given several different samples taken in different circumstances where the mean may vary between samples but the true variance is assumed to remain the same.

 To calculate the F ratio, two estimates of the variance are made.
 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).
 As it turns out, MS_{between} consists of the population variance plus a variance produced from the differences between the samples.
 If the groups are the same size, the calculations simplify somewhat and the F ratio can be written as: FRatio Formula when the groups are the same size
$\displaystyle F = \frac{n \cdot s^2_ x}{ s^2_{pooled}}$
where ...
n =the sample size
df_{numerator }= k−1
df_{denominator} = n−k
$s^2_{pooled}$ = the mean of the sample variances (pooled variance)
$s^2_{\bar{x}}$ = the variance of the sample means
The data is typically put into a table for easy viewing.

 To calculate the Fratio, two estimates of the variance are made: variance between samples and variance within samples.
 To calculate the Fratio, two estimates of the variance are made:
Variance between samples: An estimate of σ^{2} that is the variance of the sample means multiplied by n (when there is equal n).
 Variance within samples: An estimate of σ^{2} that is the average of the sample variances (also known as a pooled variance).
 As it turns out, MS_{between} consists of the population variance plus a variance produced from the differences between the samples.
 MS_{within} is an estimate of the population variance.
 pooled variance (noun) A method for estimating variance given several different samples taken in different circumstances where the mean may vary between samples but the true variance is assumed to remain the same.

 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 s_{1} and s_{2} 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:
$s^2_{pooled}=\frac{s^2_1 \times (n_11)+s^2_2 \times (n_21)}{n_1+n_22}$
where n1 and n2 are the sample sizes, as before.
 A pooled standard deviation is only appropriate when background research indicates the population standard deviations are nearly equal.
 When the sample size is large and the condition may be adequately checked with data, the beneﬁts of pooling the standard deviations greatly diminishes.

 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.
)
NOTE : It is not necessary to compute this by hand.
 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.
 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).
 Mean of the sample variances = 15.433 = $s^2_{pooled}$
Then MS_{within} = $s^2_{pooled}$ = 15.433.

 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.

 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 α_{j} is denoted σ_{α}^{2} and the variance of ε_{ij} is denoted σ_{ε}^{2}.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).
 Whereas Pearson's correlation coefficient is immune to whether the biased or unbiased version for estimation of the variance is used, the concordance correlation coefficient is not.

 Learning Objectives
State the variance sum law when X and Y are not assumed to be independent
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
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:
$\sigma{_{X \pm Y}^2}=\sigma{_X^2}+\sigma{_Y^2}$
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:
$\sigma{_{\text{verbal} + \text{quant}}^2}=10,000+11,000+(2)(0.5)\sqrt{10,000} \sqrt{11,000}$
which is equal to 31,488.
 The variance of the difference is:
$\sigma{_{\text{verbal}  \text{quant}}^2}=10,000+11,000(2)(0.5)\sqrt{10,000} \sqrt{11,000}$
which is equal to 10,512.
 If the variances and the correlation are computed in a sample, then the following notation is used to express the variance sum law:
$s{_{X \pm Y}^2}=s{_X^2}+s{_Y^2} \pm 2 rs_X s_Y$