Variance Calculator

A variance calculator measures how spread out a set of numbers is around its mean. Variance is commonly used in statistics, data analysis, probability, finance, research, and scientific calculations.

What Is Variance?

Variance describes the average squared difference between each value and the mean of a dataset. A small variance means the values are relatively close to the mean, while a larger variance indicates greater spread.

Variance Formula

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For a population, variance is commonly calculated as:

σ² = Σ(x − μ)² ÷ N

For a sample, the formula is:

s² = Σ(x − x̄)² ÷ (n − 1)

Here, x represents an individual value, μ is the population mean, x̄ is the sample mean, N is the population size, and n is the sample size.

How to Calculate Population Variance

Consider the dataset 2, 4, 6, 8.

First find the mean:

(2 + 4 + 6 + 8) ÷ 4 = 5

Now find each squared deviation from the mean:

(2 − 5)² = 9

(4 − 5)² = 1

(6 − 5)² = 1

(8 − 5)² = 9

Add the squared deviations:

9 + 1 + 1 + 9 = 20

Divide by the number of values:

20 ÷ 4 = 5

Therefore, the population variance is 5.

Sample Variance

When the data represents a sample from a larger population, the squared deviations are divided by n − 1 instead of n. This adjustment is known as Bessel's correction and helps provide an unbiased estimate of population variance from sample data.

Variance vs. Standard Deviation

Variance and standard deviation both describe data spread, but they use different units. Variance is expressed in squared units because the deviations are squared. Standard deviation is the square root of variance and is expressed in the same units as the original data.

For example, if the variance is 25, the standard deviation is:

√25 = 5

Why Variance Is Useful

Variance helps compare the amount of variability in datasets. It is used in statistical analysis, probability distributions, quality control, financial risk analysis, and many other fields where understanding data dispersion is important.

Common Mistakes to Avoid

Make sure you calculate the correct mean before finding deviations. Also determine whether your data represents an entire population or a sample, because the denominator is different for each calculation.

Remember that variance uses squared deviations, so it should not be confused with the average absolute distance from the mean.

Quick Summary

Variance measures how far values tend to spread from their mean. Population variance divides the sum of squared deviations by N, while sample variance divides it by n − 1. A variance calculator can quickly determine the result and help verify manual calculations.