Standard Deviation Calculator
Paste a list of numbers to get the mean, variance, and both population and sample standard deviation — with every step shown.
Standard deviation calculator
Standard deviation measures how spread out a set of numbers is around its mean. A small standard deviation means the values cluster tightly near the average; a large one means they are scattered widely. Paste or type your data set above and this calculator returns the count, sum, mean, variance, and both the population and sample standard deviation, along with the full step-by-step work so you can follow exactly how the answer is built.
The formula
Standard deviation is the square root of the variance. Variance is the average of the squared distances from the mean:
Population: σ = √[ Σ(x − x̄)² / N ]
Sample: s = √[ Σ(x − x̄)² / (n − 1) ]
Here x̄ is the mean, x is each value, and the Σ symbol means "add up over all values." The only difference between the two formulas is the divisor: the population formula divides by N (the full count) and the sample formula divides by n − 1.
Population vs. sample — which do I use?
This is the single most common point of confusion. Use the population standard deviation (σ) when your numbers represent the entire group you care about — for example, the test scores of every student in one specific class when that class is all you are describing. Use the sample standard deviation (s) when your numbers are a sample drawn from a larger population that you are trying to estimate — for example, 30 surveyed customers standing in for all your customers.
The n − 1 divisor in the sample formula is called Bessel's correction. Dividing by n − 1 instead of n makes the result a little larger, which corrects for the fact that a sample tends to underestimate the true spread of the population it came from. When you are unsure which to use, the sample standard deviation is the safer default — almost all real-world statistics work with samples, not entire populations.
Worked example
For the data set 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32. Dividing by N = 8 gives a population variance of 4, so the population standard deviation is √4 = 2. Dividing instead by n − 1 = 7 gives a sample variance of about 4.571 and a sample standard deviation of about 2.138.
Standard deviation is everywhere in statistics: it defines the width of the normal distribution, underlies z-scores and confidence intervals, and quantifies risk in finance. Everything is computed locally in your browser — your data is never uploaded.
Standard deviation calculator
Standard deviation measures how spread out a set of numbers is around its mean. A small standard deviation means the values cluster tightly near the average; a large one means they are scattered widely. Paste or type your data set above and this calculator returns the count, sum, mean, variance, and both the population and sample standard deviation, along with the full step-by-step work so you can follow exactly how the answer is built.
The formula
Standard deviation is the square root of the variance. Variance is the average of the squared distances from the mean:
Population: σ = √[ Σ(x − x̄)² / N ]
Sample: s = √[ Σ(x − x̄)² / (n − 1) ]
Here x̄ is the mean, x is each value, and the Σ symbol means "add up over all values." The only difference between the two formulas is the divisor: the population formula divides by N (the full count) and the sample formula divides by n − 1.
Population vs. sample — which do I use?
This is the single most common point of confusion. Use the population standard deviation (σ) when your numbers represent the entire group you care about — for example, the test scores of every student in one specific class when that class is all you are describing. Use the sample standard deviation (s) when your numbers are a sample drawn from a larger population that you are trying to estimate — for example, 30 surveyed customers standing in for all your customers.
The n − 1 divisor in the sample formula is called Bessel's correction. Dividing by n − 1 instead of n makes the result a little larger, which corrects for the fact that a sample tends to underestimate the true spread of the population it came from. When you are unsure which to use, the sample standard deviation is the safer default — almost all real-world statistics work with samples, not entire populations.
Worked example
For the data set 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32. Dividing by N = 8 gives a population variance of 4, so the population standard deviation is √4 = 2. Dividing instead by n − 1 = 7 gives a sample variance of about 4.571 and a sample standard deviation of about 2.138.
Standard deviation is everywhere in statistics: it defines the width of the normal distribution, underlies z-scores and confidence intervals, and quantifies risk in finance. Everything is computed locally in your browser — your data is never uploaded.
Need help shipping something?
Productized MVP development for founders. 9 SaaS apps shipped — yours could be next, in 6 weeks.
Frequently Asked Questions
Common questions about the Standard Deviation Calculator
They use different divisors. Population standard deviation divides the sum of squared deviations by N (the total count) and is used when your data is the entire group you care about. Sample standard deviation divides by n − 1 (Bessel's correction) and is used when your data is a sample meant to estimate a larger population. The n − 1 version is slightly larger and is the more common choice in real-world statistics.
Five steps: (1) find the mean of your values; (2) subtract the mean from each value and square the result; (3) add up those squared deviations; (4) divide by N for the population or by n − 1 for a sample to get the variance; (5) take the square root of the variance. This calculator shows all five steps with your actual numbers.
It tells you how spread out the data is around the average. A low standard deviation means most values are close to the mean; a high one means they are widely scattered. For data that follows a normal distribution, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three.
Standard deviation is simply the square root of the variance. Variance is the average squared distance from the mean and is expressed in squared units, which is hard to interpret. Taking the square root returns the spread to the original units of your data, which is why standard deviation is usually reported instead of variance.
It can be zero — that happens only when every value in the data set is identical, so there is no spread at all. It can never be negative, because it is a square root of an average of squared (non-negative) numbers. The smallest possible standard deviation is zero.
ℹ️ Disclaimer
This tool is provided for informational and educational purposes only. All processing happens entirely in your browser - no data is sent to or stored on our servers. While we strive for accuracy, we make no warranties about the completeness or reliability of results. Use at your own discretion.