Statistics formulas

Mean, median, range, and the difference between population and sample variance and standard deviation.

These pages explain formulas for reference. They don't mean Mathover can solve every problem that uses them — interactive calculations are limited to the linked tools.

Arithmetic mean

x̄ = Σx ÷ n

Add all the values and divide by how many there are.

Symbols

x̄: the mean
Σx: sum of all values
n: number of values

When it applies

Numerical data with at least one value. Sensitive to extreme values (outliers).

Worked example

Find the mean of 4, 8, 6, 5, 7.

1.Σx = 4 + 8 + 6 + 5 + 7 = 30
2.n = 5
3.x̄ = 30 ÷ 5 = 6

Answer: 6

Try it in the Average (Mean) tool →

Median

median = value in position (n + 1) ÷ 2 of the sorted list

Sort the data. With an odd count take the middle value; with an even count average the two middle values.

Symbols

n: number of values

When it applies

Numerical data. Less affected by outliers than the mean.

Worked example

Find the median of 3, 9, 1, 7, 5, 11.

1.Sorted: 1, 3, 5, 7, 9, 11
2.n = 6 (even), so position 3.5 — between the 3rd and 4th values
3.(5 + 7) ÷ 2 = 6

Answer: 6

Try it in the Statistics Calculator tool →

Range

range = maximum − minimum

The simplest measure of spread: the gap between the largest and smallest values.

Symbols

maximum: largest value
minimum: smallest value

When it applies

Numerical data. One extreme value can change it a lot.

Worked example

Find the range of 12, 4, 19, 7.

1.Maximum = 19, minimum = 4
2.19 − 4 = 15

Answer: 15

Try it in the Statistics Calculator tool →

Population variance and standard deviation

σ² = Σ(x − μ)² ÷ N, σ = √σ²

Average squared distance from the mean, used when your data is the whole population.

Symbols

μ: population mean
N: population size
σ²: variance
σ: standard deviation

When it applies

Use when the data includes every member of the group you care about. Divide by N.

Worked example

Find the population standard deviation of 2, 4, 4, 4, 5, 5, 7, 9.

1.μ = 40 ÷ 8 = 5
2.Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16 — sum 32
3.σ² = 32 ÷ 8 = 4
4.σ = √4 = 2

Answer: σ² = 4, σ = 2

Try it in the Statistics Calculator tool →

Sample variance and standard deviation

s² = Σ(x − x̄)² ÷ (n − 1), s = √s²

Like population variance but divides by n − 1 (Bessel's correction), which gives a fairer estimate of the population's spread from a sample.

Symbols

x̄: sample mean
n: sample size
s²: sample variance
s: sample standard deviation

When it applies

Use when the data is a sample from a larger group. Needs n ≥ 2.

Worked example

Find the sample variance and standard deviation of 2, 4, 6, 8.

1.x̄ = 20 ÷ 4 = 5
2.Squared deviations: 9, 1, 1, 9 — sum 20
3.s² = 20 ÷ (4 − 1) = 20/3 ≈ 6.667
4.s = √(20/3) ≈ 2.582

Answer: s² = 20/3 ≈ 6.667, s ≈ 2.582

Try it in the Statistics Calculator tool →