# Population Variance Calculator

Created by Tibor Pál, PhD candidate and Jasmine J Mah
Reviewed by Dominik Czernia, PhD and Jack Bowater
Last updated: Dec 08, 2022

Table of contents:

The population variance calculator is a tool designed to not only estimate the variance of a population, but also to explain step-by-step how to find population variance. Therefore, you can also use this tool to study the computational procedure.

If you are more interested in a sample population, you can check the variance calculator, where you can also learn the difference between these two concepts.

Read on to learn:

• What is population in statistic;
• How to calculate population variance;
• The population variance formula; and
• The population variance symbol.

## What is a population in statistics and population variance?

A population in statistics refers to a set of comparable items or events which is of interest for some investigation or experiment. A statistical population can be a group of existing objects or a hypothetical and potentially infinite group of objects perceived as a generalization from experience.

In the present context, the population is a set of similar data points or values, which is based on variance analysis.

On the other hand, variance is a measure of the variability of the values in a dataset. A high variance implies that a dataset is more spread out. A low variance suggests that the data is more tightly clustered around the mean, or less spread out.

Population variance, therefore (with a population variance symbol, σ2), tells us how these data points are spread out in a specific population. It is the average distance from each data point in the population to the mean squared.

## How to calculate population variance? Population variance formula

We define variance (denoted with the population variance symbol $\sigma^2$) as the average squared difference from the mean for all data points. For the population variance, we write it as:

$\sigma^2 = \frac{\sum(x_i - \mu)^2}{N}$

where,

• $\sigma^2$ is the variance;
• $\mu$ is the mean; and
• $x_i$ represents the $i^{th}$ data point out of $N$ total data points.

And this is how to find population variance. There are three steps you need to follow:

1. Find the difference from the mean for each point. Use the formula: $x_i - \mu$.

2. Square the difference from the mean for each point: $(x_i - \mu)^2$.

3. Find the adjusted average of the squared differences from the mean which you found in step 2:

$\sum(x_i - \mu)^2/N$.

Knowing how to calculate population variance isn't enough if you want to do it quickly, especially with a large data sample. So, if saving time is critical, we recommend that you to use the population variance calculator.

If you are interested in measuring the diversity of a community, you can use our Simpson's diversity index calculator.

## Population vs. sample variance

For practical reasons, most scientific experiments make inferences about the population only from a sample of the population. However, when we use sample data to estimate the variance of a population, the regular population variance formula, $\sum(x_i - \mu)^2/N$, underestimates the variance of the population.

How to find a population variance that is more reliable? To avoid underestimating the variance of a population (and consequently, the standard deviation), we replace $N$ with $N - 1$ in the variance formula when we use sample data. This adjustment is known as Bessels' correction.

Therefore the sample variance formula becomes the following:

$s^2 = \frac{\sum(x_i - \bar x)^2}{N-1}$

where,

• $s^2$ is the estimate of variance;
• $\bar x$ (pronounced as "x-bar") is the sample mean; and
• $x_i$ is the $i^{th}$ data point out of $N$ total data points.
Tibor Pál, PhD candidate and Jasmine J Mah
Data (You may enter up to 30 numbers)
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Steps to show
Summary
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 Observations (N): 0 Mean (μ): 0 Variance (σ²): 0 Standard deviation (σ): 0
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