Variance Formula: Population, Sample, Calculator and Examples

Quick Answer: What Is the Variance Formula?

Variance measures how far a set of data values spreads out from the mean. A small variance means the values are close to the mean, while a large variance means the values are more spread out.

There are two main variance formulas:

In these formulas:

• xᵢ means each data value
• μ means the population mean
• x̄ means the sample mean
• N means the total number of values in a population
• n means the total number of values in a sample
• Σ means “add them all together”

To calculate variance, find the mean, subtract the mean from each value, square each difference, add the squared differences, and divide by N for population variance or n – 1 for sample variance.

Variance Formula: Population vs Sample

The formula you use depends on whether your data represents an entire population or only a sample.

If you collect the test scores of every student in one class, you can use the population variance formula because you have the full group. If you collect scores from only 20 students to estimate the performance of a much larger grade level, you should use the sample variance formula.

The most important difference is the denominator. Population variance divides by N, while sample variance divides by n – 1. This adjustment helps a sample give a better estimate of the population’s true variance.

Online Variance Calculator

How to Calculate Variance in 5 Steps?

You can calculate variance by following the same basic steps every time.

Step 1: Find the mean

Add all values together and divide by the number of values.

For the data set 8, 12, 10, 14, 16:

Mean = (8 + 12 + 10 + 14 + 16) / 5

Mean = 60 / 5

Mean = 12

Step 2: Subtract the mean from each value

Find how far each value is from the mean.

Step 3: Square each difference

Squaring removes negative signs and gives more weight to values that are farther from the mean.

Step 4: Add the squared differences

16 + 0 + 4 + 4 + 16 = 40

The sum of squared differences is 40.

Step 5: Divide by N or n – 1

For population variance:

σ² = 40 / 5 = 8

For sample variance:

s² = 40 / (5 - 1) = 10

So the population variance is 8, and the sample variance is 10.

Population Variance Formula

The population variance formula is:

σ² = Σ(xᵢ - μ)² / N

Use this formula when your data set includes every member of the population you are studying.

Population Variance Example

Suppose five students took a quiz, and these are all the scores in the class:

6, 8, 10, 12, 14

First, find the mean:

μ = (6 + 8 + 10 + 12 + 14) / 5 μ = 50 / 5 μ = 10

Next, subtract the mean and square each difference:

Add the squared differences:

16 + 4 + 0 + 4 + 16 = 40

Divide by N:

σ² = 40 / 5 = 8

The population variance is 8.

Sample Variance Formula

The sample variance formula is:

s² = Σ(xᵢ - x̄)² / (n - 1)

Use this formula when your data set is only a sample from a larger population.

Sample Variance Example

A teacher randomly selects five students from a larger grade level and records their test scores:

78, 82, 85, 88, 90

Find the sample mean:

x̄ = (78 + 82 + 85 + 88 + 90) / 5 x̄ = 423 / 5 x̄ = 84.6

Now calculate the squared differences:

Add the squared differences:

43.56 + 6.76 + 0.16 + 11.56 + 29.16 = 91.2

Divide by n – 1:

s² = 91.2 / (5 - 1) s² = 91.2 / 4 s² = 22.8

The sample variance is 22.8.

Why Does Sample Variance Use n – 1?

Sample variance uses n – 1 because a sample usually underestimates how spread out the full population is.

When you calculate the sample mean, the values in the sample are already centered around that sample mean. This often makes the sample look slightly less spread out than the full population. Dividing by n – 1 instead of n makes the variance a little larger, which helps correct that underestimation.

This adjustment is called Bessel’s correction.

For students, the simple rule is:

• Use N when you have the whole population.
• Use n – 1 when you only have a sample.

Variance vs Standard Deviation

Variance and standard deviation both measure spread, but they are reported in different units.

Variance is measured in squared units. Standard deviation is the square root of variance, so it uses the same units as the original data.

For example, if exam scores have a variance of 25 points squared, the standard deviation is:

√25 = 5

That means scores typically vary by about 5 points from the mean.

Alternative Variance Formula

Another useful formula for variance is:

Var(X) = E(X²) - [E(X)]²

This means:

Variance = mean of the squares - square of the mean

For a data set, the population version can be written as:

σ² = [Σxᵢ² / N] - μ²

This formula is often faster when you already know the sum of the values and the sum of the squared values.

Alternative Formula Example

For the data set:

2, 4, 6

Find the mean:

μ = (2 + 4 + 6) / 3 = 4

Find the mean of the squares:

(2² + 4² + 6²) / 3 = (4 + 16 + 36) / 3 = 56 / 3 = 18.67

Subtract the square of the mean:

18.67 - 4² = 18.67 - 16 = 2.67

The population variance is about 2.67.

Variance Formula for Grouped Data

When data is organized in a frequency table, use the grouped data variance formula.

For population variance:

σ² = Σf(x - μ)² / N

For sample variance:

s² = Σf(x - x̄)² / (n - 1)

In these formulas:

• f means frequency
• x means the value or class midpoint
• μ or x̄ means the mean
• N or n means the total frequency

Grouped Data Example

Suppose the table shows how many students earned each score:

First, find the mean:

Mean = (70 × 2 + 80 × 3 + 90 × 5) / (2 + 3 + 5)

Mean = (140 + 240 + 450) / 10

Mean = 830 / 10 = 83

Then calculate the squared differences with frequency:

Add the weighted squared differences:

338 + 27 + 245 = 610

Population variance:

σ² = 610 / 10 = 61

The population variance is 61.

How to Find Variance in Excel, Google Sheets, and Python

Many students and teachers calculate variance using spreadsheets or programming tools.

Excel and Google Sheets

Use these functions:

Use VAR.P when your data is the full population. Use VAR.S when your data is a sample.

Python

In Python, you can use NumPy:

import numpy as np data = [8, 12, 10, 14, 16]

population_variance = np.var(data)

sample_variance = np.var(data, ddof=1) print(population_variance) print(sample_variance)

The ddof=1 setting tells Python to divide by n – 1, which gives the sample variance.

Low Variance vs High Variance

Variance helps you understand whether values are consistent or widely spread out.

Low variance often suggests consistency. High variance suggests more variability or less predictability.

For example, two classes may have the same average score, but one class may have much higher variance. That means students in that class performed very differently from one another, even if the average score looks normal.

Common Mistakes When Calculating Variance

Mistake 1: Using n instead of n – 1 for a sample

If your data is a sample, use n – 1 in the denominator. Using n will usually make the variance too small.

Mistake 2: Forgetting to square the differences

If you add the differences without squaring them, positive and negative values can cancel out. Squaring makes every difference positive.

Mistake 3: Confusing variance with standard deviation

Variance is in squared units. Standard deviation is in the original units. If you need an easier real-world interpretation, standard deviation is often more useful.

Mistake 4: Choosing the wrong mean

Use μ for population variance and x̄ for sample variance. The calculation is similar, but the meaning is different.

Nathan | WuKong Math Teacher

Nathan, a graduate of the University of New South Wales, brings over 9 years of expertise in teaching Mathematics and Science across primary and secondary levels. Known for his rigorous yet steady instructional style, Nathan has earned high acclaim from students in grades 1-12. He is widely recognized for his unique ability to blend academic rigor with engaging, interactive lessons, making complex concepts accessible and fun for every student.