How to use the standard deviation calculator
- Enter or paste your numbers into the data set box. Separate values with commas, spaces, semicolons or new lines.
- Choose Population if the data contains every member of the group, or Sample if it represents part of a larger group.
- Select the number of decimal places and click Calculate statistics.
- Review the standard deviation, variance, mean, range and individual deviations. Copy the results if needed.
Standard deviation formulas
σ = √[Σ(x − μ)² / N]
Divide the sum of squared deviations by the population count N.s = √[Σ(x − x̄)² / (n − 1)]
Divide by n − 1 to apply the sample correction.Mean = Σx / n
Add all values and divide by the number of values.Variance = (standard deviation)²
Variance is expressed in squared units.What does standard deviation mean?
Standard deviation describes how spread out values are around their mean. A smaller standard deviation means the values tend to cluster closer to the average. A larger standard deviation indicates more variation. Standard deviation is expressed in the same units as the original data, while variance uses squared units.
Sample vs. population standard deviation
Use population standard deviation when your data represents the entire group you want to describe. Use sample standard deviation when your observations are a subset used to estimate a larger population. Population variance divides by N; sample variance divides by n − 1. A sample standard deviation requires at least two values.
Frequently asked questions
Can I enter negative numbers and decimals?
Yes. Negative values and decimal values are supported. Separate each number with a comma, space, semicolon or line break. Avoid using commas as thousands separators; for example, enter one thousand as 1000.
What is the difference between standard deviation and variance?
Variance is the average squared deviation from the mean, using the population or sample denominator. Standard deviation is the square root of variance and is in the same units as the original data.
Why does sample standard deviation divide by n − 1?
The sample formula uses n − 1, often called Bessel's correction, to account for estimating the population mean from a sample.
What if every value is the same?
If all values are identical, every value equals the mean, so the squared deviations and standard deviation are zero.
How many numbers can I calculate at once?
You can enter up to 5,000 numeric values in one calculation. Invalid entries and a sample containing fewer than two values are rejected with an explanatory message.
Are my numbers uploaded or saved?
No. The calculations run in your browser. This calculator does not send your dataset to the server or automatically save it.
This calculator provides descriptive statistics for the values entered. The result does not by itself establish whether a sample is representative of a larger population.