# Pooled Standard Deviation Calculator

Created by Wei Bin Loo
Reviewed by Anna Szczepanek, PhD and Steven Wooding
Last updated: Jan 18, 2024

We have prepared this pooled standard deviation calculator to help you measure the variability or spread of data when combining (i.e., pooling) multiple datasets. This tool can also help you understand the overall level of dispersion or inconsistency in the combined data set, providing insights into the collective variation present in the pooled data. You can check out our standard deviation calculator to understand more about this topic.

After reading this article, you will understand what pooled standard deviation is, what the pooled standard deviation formula is, and how to calculate the pooled standard deviation. We will also demonstrate some examples to help you understand the concept.

## What is pooled standard deviation?

Pooled standard deviation is a statistical measure used to determine the overall level of dispersion or inconsistency in a combined dataset. It provides valuable insights into the collective variation present in the pooled data, offering a single value that represents the overall variability of the combined information. By calculating the pooled standard deviation, we can effectively assess the spread of data when datasets are merged.

## How to find pooled standard deviation?

Now, let's talk about how to find the pooled standard deviation. To understand the pooled standard deviation calculation, let's look at the following example:

• Number of datasets: 2
• Dataset A: [5, 7, 9, 11, 13]
• Dataset B: [4, 6, 8, 10, 12]

You can calculate the pooled standard deviation in 3 steps:

1. Determine the sample size (n₁ and n₂) of the datasets

The first step is to determine the sample size of the datasets. For our example, n₁ = n₂ = 5.

2. Calculate the sample variances (sa² and sb²) for each dataset

Next, you need to calculate the sample variance of each dataset using the formula below.

$\footnotesize \qquad s_{\textrm{sample}}^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$

The mean of dataset A is 9, while the mean of dataset B is 8. You can use our mean calculator and average calculator to speed up this calculation.

Hence,

sa² = (16 + 4 + 0 + 4 + 16) / 5 = 8

and

sb² = (16 + 4 + 0 + 4 + 16) / 5 = 8

1. Calculate the pooled standard deviation

The last step is to calculate the pooled standard deviation using the formula below.

$\footnotesize \ \ s_{\textrm{pooled}}^2 = \sqrt{\frac{(n_1 - 1) s_1^2 + (n_2 - 1) s_2^2}{n_1 + n_2 - 2}}$

Thus, the pooled standard deviation for our example is computed as follows:

√[(4 × 8 + 4 × 8) / 8] = √[64 / 8] = √8 = 3.1623

That's it! Hopefully, never again will you have to wonder how to find the pooled standard deviation of some datasets!

## FAQ

### What is the pooled standard deviation for datasets with the same standard deviation of 2?

The pooled standard deviation will be 2. This occurs because the pooled standard deviation arises from the weighted average of variances of the datasets. If these variances are all equal, then taking the average returns again the same result.

### How do I calculate the pooled standard deviation?

You can calculate the pooled standard deviation in steps:

1. Calculate the sample variances (s₁² and s₂²) for each dataset.

2. Determine the sample sizes (n₁ and n₂) for each dataset.

3. Compute (n₁ − 1) × s₁² + (n₂ − 1) × s₂².

4. Compute n₁ + n₂ − 2.

5. Divide the result from Step 3 by that from Step 4.

6. That's it! You've applied the pooled standard deviation formula:

pooled sd = √[((n₁ − 1) × s₁² + (n₂ − 1) × s₂²) / (n₁ + n₂ − 2)]

### Can the pooled standard deviation be used with more than two datasets?

Yes, the pooled standard deviation formula can be extended to accommodate more than two datasets by incorporating the sample sizes and variances of each dataset in the calculation.

### How does the pooled standard deviation differ from the standard deviation?

The standard deviation measures the variability within a single dataset, whereas the pooled standard deviation quantifies the variability when combining multiple datasets.

Wei Bin Loo
Data set A
a₁
a₂
Data set B
b₁
b₂
Pooled standard deviation
Enter some data.
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