Standard Deviation Calculator
Enter up to five data points to instantly compute the arithmetic mean, population variance, and population standard deviation. Standard deviation is the most widely used measure of spread in statistics.
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Formula
σ = √[ Σ(xᵢ − μ)² / N ]
First compute the mean (μ) by summing all values and dividing by N. Then subtract the mean from each value and square the result. Average those squared differences to get the variance (σ²). Finally, take the square root of the variance to get the standard deviation (σ). This calculator uses the population formula (dividing by N), not the sample formula (dividing by N−1).
How to use the Standard Deviation Calculator
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Results update in real time as you type.
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What standard deviation tells you
Standard deviation measures how spread out the values in a dataset are around the mean. A small standard deviation means the values cluster tightly near the average. A large standard deviation means the values are spread widely. For the default dataset {10, 20, 30, 40, 50}, the mean is 30 and the standard deviation is about 14.14 — each value is, on average, about 14 units away from the center.
In practice, standard deviation is used in finance to measure investment volatility, in manufacturing to track process consistency, in education to understand score distributions, and in science to report measurement uncertainty. Whenever you see error bars on a chart, they almost always represent one standard deviation above and below the mean.
Population vs. sample standard deviation
There are two versions of standard deviation. The population standard deviation (σ) divides the sum of squared deviations by N — the total number of data points. This is correct when you have data for every member of the group you care about.
The sample standard deviation (s) divides by N−1 instead. This correction, called Bessel's correction, accounts for the fact that a sample tends to underestimate the true spread of the population. If your five values are drawn from a larger group and you want to estimate that group's spread, use the sample formula. This calculator computes the population version; for the sample version, see the Variance Calculator, which shows both.
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The empirical rule (68-95-99.7)
When data follows a normal (bell-curve) distribution, roughly 68% of values fall within one standard deviation of the mean, about 95% fall within two standard deviations, and about 99.7% fall within three. This is called the empirical rule or the 68-95-99.7 rule.
For example, if a class of students scores a mean of 75 with a standard deviation of 8, then about 68% of students scored between 67 and 83, about 95% scored between 59 and 91, and nearly all students scored between 51 and 99. This rule only applies to roughly normal distributions — highly skewed data does not follow this pattern.
Tips & Insights
Variance is standard deviation squared
If you already have the standard deviation, variance is just σ². If you have variance, take the square root for standard deviation. They measure the same thing in different units — variance is in squared units of the original data.
Outliers inflate standard deviation dramatically
A single extreme value can pull the standard deviation far above what is typical for the rest of the data. If your dataset has outliers, consider reporting median and IQR instead of mean and standard deviation.
Use sample std dev for survey data
If your five values are a sample drawn from a larger population (e.g., five survey respondents out of thousands), divide by 4 (N−1) instead of 5 (N). The Variance Calculator on this site shows both versions side by side.
Worked Examples
Test scores
Mean: 82. Standard deviation: ≈ 9.87. This tells you scores are fairly spread — a typical student is about 10 points from the class average.
Daily temperatures (°F)
Mean: 71°F. Standard deviation: ≈ 2.19°F. The week was very consistent — temperatures stayed within about 2°F of the average each day.
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Frequently Asked Questions
What is standard deviation?
Standard deviation is a measure of how spread out a set of values is. A low standard deviation means values are close to the mean; a high standard deviation means they are far from the mean.
What is the difference between population and sample standard deviation?
Population standard deviation divides by N (all data points) and is used when you have complete data. Sample standard deviation divides by N−1 and is used when your data is a sample from a larger population.
What does a standard deviation of 0 mean?
A standard deviation of 0 means every value in the dataset is identical. There is no variation at all.
Is a higher or lower standard deviation better?
It depends on context. In quality control or finance, a low standard deviation often means consistency, which is desirable. In biology or social science, it simply describes natural variation without a value judgment.
How does standard deviation relate to variance?
Variance is the average of the squared differences from the mean. Standard deviation is the square root of variance. Standard deviation is preferred because it is in the same units as the original data.
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