Pooled Variance Calculator

The Pooled variance calculator calculates the pooled variance of two independent samples of the population. You can also calculate pooled standard deviation, standard error, and degree of freedom of dataset values.

What is a Pooled Variance?

The pooled variance is the weighted average of two sample variances drawn from two populations. The pooled variance is determined by taking the weighted mean of both sample sets. The square of the mean of both sample sets is divided by the degree of freedom of both sample sets. The pooled variance calculator calculates the pooled variance of the dataset, standard deviation, standard error, and degree of freedom.

The Pooled Variance Formula:

The formula for the estimated population variance is given by:

\(S_p^2 = \dfrac{(n -1)S_1^2 + (n_2 - 1)S_2^2}{n_1 + n_2 - 2}\)

Where:

S = Variance of the dataset

n= Number of data points 

The pooled standard deviation calculator finds the pooled variance of the two sample set variance and raw data.

How to Calculate Pooled Variance?

Let two pooled sample standard deviations of two populations be 2 and 3 respectively.  The size of both the sample dataset are 10 and 20, then and standard deviation of the dataset, and the standard deviation of the dataset?

Given:

Sample size (n1) = 10

Sample size (n2) = 20

Sample standard deviation (S1) = 2

Sample standard deviation (S2) = 3

Pooled Variance:

\(S_p^2 = \dfrac{(n -1)S_1^2 + (n_2 - 1)S_2^2}{n_1 + n_2 - 2}\)

\(S_p^2 = \dfrac{(10 -1)(2)^2 + (20 - 1)(3)^2}{10 + 20 - 2}\)

\(S_p^2 = \dfrac{207}{28}\)

\(S_p^2 = 7.3929\)

The estimated population variance of the two samples is  7.3929. You can calculate pooled standard deviation by taking the square root of the pooled variance.

Standard Deviation:

\(S_p^2 = 7.3929\)

Taking under the root of both sides 

\(sqrt{(S_p)^2} = sqrt{7.3929}\)

S_p = 2.719

The pool estimate calculator is used to calculate pooled standard deviation after getting the values of variables. The pooled sample standard deviation represents the variation in the dataset values.

Standard Error:

\(SE = S_{{\bar x_1 - \bar x_2}} = S_p \sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}\)

\(SE = 2.719 \sqrt{\dfrac{1}{10} + \dfrac{1}{20}}\)

\(SE = 2.719 \sqrt{0.15}\)

\(SE = 1.0531\)  

The standard error is calculated by the Pooled Variance Calculator to find the estimated error in the dataset values.

Degree of Freedom:

\(df = n_1 + n_2 - 2\)

\(df = 10 + 20 - 2\)

\(df = 28\)

The degree of freedom of the dataset is the variable that can affect dataset values. The pooled variance t test calculator identifies the degree of freedom of dataset values with respect to the sample values.

Working of Pooled Variance Calculator:

Let's look at the working procedure of the pool estimate calculator. A couple of steps are simple to understand for users.

Input:

• Select the equal or unequal variance from the list
• Choose the summarized data or raw data 
• Enter the sample standard deviation and sample size of two samples
• Hit the Calculate Button

Ouput:

• Pooled variance, standard deviation, standard error, and degree of freedom

References:

From the source of blogs.sas.com: Pooled Variance, How to find sample variance in statcrunch?  From the source of wikipedia.org: what is a Pooled Variance, Estimated population variance