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Pooled Variance Calculator

Enter the values of the standard deviation and sample size in the input fields, and the tool will calculate the pooled variance of the dataset.

Type:

Rounding:

Summarized data Sample size

Sample standard deviation (S₁):

Sample standard deviation (S₂):

Raw data (n₁):

Raw data (n₂):

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Pooled Variance Calculator

The pooled variance calculator calculates the pooled variance of two independent samples from a population. It can also compute the pooled standard deviation, standard error, and degrees of freedom for any dataset.

What Is Pooled Variance?

The pooled variance is the weighted average of two sample variances. Each sample variance is weighted by its degrees of freedom, and the sum is divided by the total degrees of freedom. It provides an estimate of the population variance when combining two independent samples.

The formula for pooled variance is:

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

Where:

S = Standard deviation of each sample
n = Sample size

The pooled standard deviation is the square root of the pooled variance:

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

Example Calculation

Suppose we have two samples:

Sample 1: n₁ = 10, S₁ = 2
Sample 2: n₂ = 20, S₂ = 3

Step 1: Compute pooled variance:

\(S_p^2 = \dfrac{(10-1)(2)^2 + (20-1)(3)^2}{10 + 20 - 2} = \dfrac{207}{28} = 7.3929\)

Step 2: Compute pooled standard deviation:

\(S_p = \sqrt{7.3929} \approx 2.719\)

Standard Error (SE)

The standard error of the difference between two sample means is:

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

Substitute values:

\(SE = 2.719 \sqrt{\dfrac{1}{10} + \dfrac{1}{20}} = 2.719 \sqrt{0.15} \approx 1.0531\)

Degrees of Freedom (df)

The degrees of freedom for the pooled variance t-test is:

\(df = n_1 + n_2 - 2 = 10 + 20 - 2 = 28\)

How the Pooled Variance Calculator Works

Input:

Select whether the samples have equal or unequal variance.
Choose summarized data (sample size & standard deviation) or raw data.
Enter the sample standard deviations and sample sizes for both samples.
Click the "Calculate" button.

Output:

Pooled variance (S_p²)
Pooled standard deviation (S_p)
Standard error (SE)
Degrees of freedom (df)

This calculator is helpful for performing t-tests or comparing two independent samples when estimating population variance.

References:

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