Standard Deviation Calculator

Calculate the sample and population standard deviation, variance, and mean for your dataset.

Enter your dataset values (separated by commas, spaces, or lines)

Sample / Population

Statistics

What Is Standard Deviation?

Standard deviation is a number that shows how much values differ from the mean. A small standard deviation means the numbers are close to the average, while a large standard deviation means the numbers are more spread out.

A Standard Deviation Calculator helps measure spread in statistics, finance, science, research, education, sports, business, and data analysis.

How to Use

How to Use the Standard Deviation Calculator

Enter numbers separated by commas, spaces, semicolons, or new lines. Then choose whether the values represent a sample or a full population.

Count: The number of values in the data set.
Mean: The average of the values.
Variance: The average squared distance from the mean.
Standard deviation: The square root of variance, shown in the original unit of the data.
Step-by-step table: A breakdown of each value, its difference from the mean, and the squared difference.

Sample vs. Population

Population vs. Sample Standard Deviation

The formula changes depending on whether your data is the full group or only a sample from a larger group.

Type	Use when	Denominator
Population standard deviation	Use when the data includes every value in the full group	Divide by N
Sample standard deviation	Use when the data is only part of a larger group	Divide by n - 1

Use population standard deviation when you have every value in the group, such as every student score in one class. Use sample standard deviation when you only have part of a larger group, such as a survey of 100 people from a city.

The Math

Standard Deviation Formulas

Standard deviation starts with the mean, then measures how far each value is from that mean.

Mean

\bar{x}=\frac{\sum x}{n}

The mean is the average of the values. For 2, 4, 4, 4, 5, 5, 7, 9, the sum is 40 and the count is 8, so the mean is 5.

Population standard deviation

\sigma=\sqrt{\frac{\sum (x-\mu)^2}{N}}

Use this formula when the data represents the full population.

Sample standard deviation

s=\sqrt{\frac{\sum (x-\bar{x})^2}{n-1}}

Use this formula when the data is a sample from a larger population.

Variance

\text{Variance}=(\text{Standard Deviation})^2

Standard deviation is usually easier to interpret because it is in the same unit as the original data.

Example

Step-by-Step Standard Deviation Example

Suppose the data set is 2, 4, 4, 4, 5, 5, 7, 9. The mean is 5, and the squared differences add up to 32.

Value	Value - Mean	Squared Difference
2	-3	9
4	-1	1
4	-1	1
4	-1	1
5	0	0
5	0	0
7	2	4
9	4	16

Calculation	Formula	Result
Population variance	$\frac{32}{8}$	4
Population standard deviation	$\sqrt{4}$	2
Sample variance	$\frac{32}{7}$	4.5714
Sample standard deviation	$\sqrt{4.5714}$	2.1381

Interpretation

Variance vs. Standard Deviation

Variance and standard deviation are related, but they are not the same.

Term	Meaning
Variance	Average of squared differences from the mean
Standard deviation	Square root of variance
Range	Maximum value minus minimum value

Range only uses the smallest and largest values. Standard deviation uses every value in the data set, giving a better overall measure of spread.

Use Cases

Why Standard Deviation Is Useful

Standard deviation helps answer whether values are close to the average, spread out, consistent, unusual, risky, or variable.

Area	Use
Finance	Measures investment volatility and risk
Education	Shows how spread out test scores are in a classroom
Business	Measures variation in sales, performance, or manufacturing quality
Science	Shows how consistent experimental measurements are
Sports	Compares consistency across players, teams, or events

Low vs. High

Low and High Standard Deviation

A low standard deviation means values are clustered near the mean. A high standard deviation means values are more spread out.

Low standard deviation: Example data: 48, 49, 50, 51, 52. The values are close to the mean.
High standard deviation: Example data: 10, 25, 50, 80, 100. The values are spread farther from the mean.

Standard Deviation Calculator FAQ

Frequently Asked Questions

It finds the mean, variance, standard deviation, range, count, and useful step-by-step statistics from a set of numbers.

It shows how spread out numbers are from the average. A small standard deviation means the numbers are close to the average, while a large standard deviation means they are more spread out.

Population standard deviation is used when you have data for the entire group. Sample standard deviation is used when you only have part of the group and want to estimate the spread of the larger group.

Sample standard deviation uses n - 1, also called Bessel's correction, to better estimate population variation when only a sample is available.

No. Variance is the average squared difference from the mean. Standard deviation is the square root of variance, which brings the value back to the original unit of measure.

No. Standard deviation can never be negative. The smallest possible standard deviation is 0, which means every value is exactly the same.

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