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# Population Variance Calculator

By Tibor Pal, PhD candidate and Jasmine J Mah
Last updated: Oct 22, 2020

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.

• 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 σ2) as the average squared difference from the mean for all data points. For the population variance, we write it as:

`σ2 = ∑(xi - μ)2 / N`

where,

• σ2 is the variance;
• μ is the mean; and
• xᵢ represents the ith 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: `xi - μ`.

2. Square the difference from the mean for each point: `(xi - μ)2`.

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

`∑(xi - μ)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.

## 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, `σ2 = ∑(xi - μ)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:

`s2 = ∑(xi - x̄)2 / (N - 1)`

where,

• s2 is the estimate of variance;
• (pronounced as "x-bar") is the sample mean; and
• xi is the ith data point out of N total data points.
Tibor Pal, PhD candidate and Jasmine J Mah
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