The pooled variance between two samples is typically denoted as s p 2 and is calculated as:. When the two sample sizes n 1 and n 2 are equal, the formula simplifies to:. When we want to compare two population means, there are two statistical tests we could potentially use:.
Two sample t-test : This test assumes the variances between the two samples are approximately equal. If we use this test, then we calculate the pooled variance. If we use this test, we do not calculate the pooled variance. Instead, we use a different formula. To determine which test to use, we use the following rule of thumb:. Rule of Thumb: If the ratio of the larger variance to the smaller variance is less than 4, then we can assume the variances are approximately equal and use the two sample t-test.
For example, suppose sample 1 has a variance of The following graph shows a box plot of the ozone levels at each site.
The height of each box gives a rough estimate of the variance in each group. The mean and variance for sites A and B appear to be similar to each other. However, the mean at site C appears to be larger and the variance seems smaller, compared to the other sites.
You can use your favorite statistical software to estimate the variance of the data for each site. The table shows an estimate for the variance of the data within each group. Although the smallest sample variance Group C: 1. A parameter value such as 2. If we assume that the variance of the groups are equal, the pooled variance formula provides a way to estimate the common variance.
The graph at the top of this article visualizes the information in the table and uses a reference line to indicate the pooled variance. The blue markers indicate the sample variances of each group. The confidence intervals for the population variances are shown by the vertical lines. The pooled variance is indicated by the horizontal reference line.
It is the weighted average of the sample variances. If you think that all groups have the same variance, the pooled variance estimates that common variance. In two-sample t tests and ANOVA tests, you often assume that the variance is constant across different groups in the data. The pooled variance is an estimate of the common variance.
It is a weighted average of the sample variances for each group, where the larger groups are weighted more heavily than smaller groups. You can download a SAS program that computes the pooled variance and creates the graphs in this article.
Although the pooled variance provides an estimate for a common variance among groups, it is not always clear when the assumption of a common variance is valid.
The visualization in this article can help, or you can perform a formal "homogeneity of variance" test as part of an ANOVA analysis. His areas of expertise include computational statistics, simulation, statistical graphics, and modern methods in statistical data analysis. Hi, I am a graduate student in South Korea. Pooled variance is required to get a better estimate of population variance, from two samples that have been randomly taken from that population and come up with different variance estimates.
Example, you are trying to gauge variance in the smoking habits of males in London. You sample two times, males from London. You end up getting two variances probably a bit different! Now since, you did a fair random sampling best to your capability! But how is that possible? Thus, we go ahead and find a common point estimate which is pooled variance. It is nothing but weighted average of two point estimates, where the weights are the degree of freedom associated with each sample.
As far as I am aware the main practical need for this kind of dispersion measure arises from wanting to compare means of sub- groups: so if I want to compare the average nose length for 1 people who did not undergo a gene therapy, 2 people who underwent gene therapy A and 3 people who underwent gene therapy B.
Depending on the size of the square root of pooled variance pooled standard deviation we can better judge the size of the 2mm difference between those groups e.
And if so, how much? This is the idea of effect size first theoretically introduced by Neyman and Pearson as far as I know, but in one kind or another used well before, see Stigler, , for example.
So what I am doing is comparing the mean difference between groups with the mean differences within those same groups, i. This makes more sense than to compare the mean difference between sub- groups with the mean difference within the "whole" group, because, as you Hanciong have shown, the variance and standard deviation of the whole group contains the difference s of the group means as well.
Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. What does pooled variance "actually" mean? Ask Question. Asked 4 years, 2 months ago. Active 1 year, 2 months ago. Viewed 28k times. My question is the following: What does pooled variance actually mean? Thank you in advance. Improve this question.
Hanciong Hanciong 1 1 gold badge 1 1 silver badge 7 7 bronze badges. However, the second formula is the variance of concatenation of two samples. If we want to distincguish them, no only from literal aspect but also from their functionality. Add a comment. Active Oldest Votes.
Improve this answer. Jake Westfall Jake Westfall Pooled simply means averaged , omnibus see my comment to Tim. If the population variances are not assumed equal, then it's unclear what we could consider the pooled variance to be an estimate of. Of course, we could just think about it as being an amalgamation of the two variances and leave it at that, but that's hardly enlightening in the absence of any motivation for wanting to combine the variances in the first place.
Thank you. Although I am still not clear about one thing. According to Wikipedia, pooled variance is a method for estimating variance of several different populations when the mean of each population may be different , but one may assume that the variance of each population is the same. Because the first variance is measuring the variation with respect to the first mean, and the second variance is with respect to the second mean.
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