Standard Deviation Calculator — How Spread Out Is Your Data?
Paste any list of numbers and get the full spread story — population and sample variance, both standard deviations, and how much of your data sits within one and two standard deviations of the mean.
Your numbers
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Your spread
See your spread
Every value you entered, sorted low to high. The solid line is the mean; the shaded band is one sample standard deviation on each side.
- The classic example is built in: for 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the population SD is 2 and the sample SD is 2.13809.
- Sample SD divides by n − 1 and population SD by n, so on the same data the sample figure is always the larger of the two.
- At the defaults, 75% of the values sit within one sample SD of the mean and 100% within two — the page reports all three shares for any data you paste.
Keep exploring
Three companion tools for working with numbers.
How it works
Three steps from a messy list to a precise spread.
Paste your numbers
Commas, spaces, or line breaks — any mix works. Non-numbers are skipped and counted, never silently dropped.
We measure the spread
The mean is found first, then every value’s squared distance from it — the average of those is the variance.
Read both SDs
Population SD divides by n; sample SD divides by n − 1. The chart shades the ±1 SD band around the mean.
The math behind your standard deviation
Standard deviation measures spread: how far values typically wander from the average. First find the mean. Then square each value’s distance from the mean — squaring stops negatives from canceling positives and punishes big misses. Average those squared distances to get the variance; the square root brings it back to the original units as the standard deviation.
Worked example — 2, 4, 4, 4, 5, 5, 7, 9:
| Step | Calculation | Result |
|---|---|---|
| Mean | 40 ÷ 8 | 5 |
| Squared deviations | 9, 1, 1, 1, 0, 0, 4, 16 | sum = 32 |
| Population variance | 32 ÷ 8 | 4 |
| Population SD (σ) | √4 | 2 |
| Sample variance (s²) | 32 ÷ 7 | 4.571429 |
| Sample SD (s) | √(32 ÷ 7) | 2.13809 |
How sure are we of the mean? The standard error (s ÷ √n) estimates how much a sample mean would wobble from sample to sample. Multiply it by a critical value to get the margin of error, then add and subtract that from the mean for a confidence interval — the range the true population mean plausibly sits in. For samples under 30 the critical value comes from the t-distribution with n − 1 degrees of freedom; for samples of 30 or more the normal (z) value is used instead. The coefficient of variation (s ÷ mean, shown as a percentage) expresses spread relative to the size of the mean, so two data sets in different units can still be compared.
Go deeper
The details that separate a rough guess from a trustworthy number.
A sample’s spread systematically underestimates the true spread of the population it came from. The reason is subtle: the sample mean is computed from the sample itself, so it sits as close as possible to the sample’s values — the deviations are measured against the friendliest possible center.
- Dividing by n − 1 instead of n inflates the variance just enough to correct this bias (named after Friedrich Bessel).
- With large data sets the difference is tiny; with small ones it matters — for n = 3, sample variance is 50% larger than population variance.
- Rule: whole group in hand → population SD (÷ n); estimating a bigger group from a sample → sample SD (÷ n − 1).
This calculator shows both, so coursework and research always use the right one.
For roughly bell-shaped data, the standard deviation predicts where values land:
- About 68% of values fall within 1 SD of the mean.
- About 95% fall within 2 SD of the mean.
- About 99.7% fall within 3 SD of the mean.
The calculator reports your actual within-1-SD, within-2-SD and within-3-SD percentages so you can see how closely your data follows the rule. A value more than 3 SD from the mean is a genuine outlier worth investigating — or a typo.
Reading the chart on this page: the ±1 SD band
The chart on this page shades the band from one standard deviation below the mean to one above it. For roughly bell-shaped data, about 68% of values fall inside that band — the empirical rule, which the existing article on this site explains in full and which is linked from the related guides below. The chart’s purpose is comparative: two datasets with the same mean but different spreads show visibly different bands, and the width of the band is the story. A narrow band means the values cluster tightly; a wide band means they scatter.
Use the chart to check the calculation, not just to admire it. If the band looks far wider than the data’s visible spread, check for an outlier or a data-entry error — a single mistyped value (1000 instead of 100) inflates the standard deviation dramatically, because the calculation squares distances from the mean. The page counts non-numbers it skips; if that count is not zero, find the offending entries before trusting the result. And remember the band is a description of this dataset, not a prediction: it says where these values fell, not where the next value will fall.
Population or sample: which setting this page should use
This page reports both the population standard deviation (dividing by n) and the sample standard deviation (dividing by n − 1). Use the population figure when your numbers are the entire group you care about — every student in a class, every measurement in a completed experiment. Use the sample figure when your numbers are a sample drawn from a larger group and you want to estimate that group’s spread — a survey sample, a quality-control sample, a sample of patients. The sample figure is always a little larger, because dividing by n − 1 compensates for the fact that a sample’s spread understates the population’s.
The distinction is explained in depth in the companion article on sample versus population linked below, including why the correction exists and how the gap shrinks as samples grow. For most practical work, the choice matters less than consistency: report which one you used, alongside the mean and the count, so anyone reading your result can reproduce it. A standard deviation reported without its n, and without saying population or sample, is an incomplete result.
Frequently asked questions
Quick answers about this standard deviation calculator.
How do you calculate standard deviation step by step?
First find the mean. Subtract the mean from each value and square each difference. Average the squared differences — that is the variance. Take the square root — that is the standard deviation. Divide by n for the population SD, or by n − 1 for the sample SD.
When should I use sample standard deviation instead of population standard deviation?
Use population SD (÷ n) when your numbers are the entire group you care about — every employee, every sale this month. Use sample SD (÷ n − 1) when your numbers are a sample estimating a larger group — a poll, an experiment, a class survey. For school and research work, sample SD is the safe default.
What does a high standard deviation mean?
It means values are widely spread around the mean — the data is inconsistent or volatile. A low standard deviation means values cluster tightly near the mean. Two classes can share the same average test score while one has double the standard deviation: that class has far more very high and very low scores.
Can standard deviation be negative?
No — never. Standard deviation is the square root of a sum of squared differences, and both squaring and square roots of non-negative numbers are non-negative. The smallest possible value is 0, which happens when every value is identical. If you ever compute a negative SD, there is an arithmetic error.
What is the difference between variance and standard deviation?
Variance is the average of the squared distances from the mean, so its units are squared (square dollars, square inches) — hard to interpret. Standard deviation is the square root of variance, which returns the spread to the original units of your data. Report variance for the math; quote standard deviation for humans.
Why do we square the differences when calculating variance?
Two reasons. First, plain differences cancel out — values above and below the mean would sum to zero, always. Second, squaring punishes large deviations more than small ones, so variance reflects how much extreme values dominate. The square root at the end (the SD) then restores the original units.
Should I use population or sample standard deviation?
Population (÷ n) when your data is the whole group of interest; sample (÷ n − 1) when it is a sample estimating a larger group’s spread. This page shows both — report which one you used, with the mean and the count, so the result is reproducible.
Why does one outlier change the standard deviation so much?
Because the calculation squares each value’s distance from the mean. A value far from the mean contributes a squared distance that dominates the average. Check for data-entry errors and outliers before trusting a surprisingly large standard deviation, and consider reporting the median and interquartile range alongside it for skewed data.
Related calculators
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Disclaimer
This calculator provides descriptive statistics for information only. Population and sample formulas follow standard textbook definitions (sample variance uses Bessel’s correction, n − 1). Results depend entirely on the data you enter — check for typos and outliers, and confirm whether your data is a full population or a sample, before quoting a number. For academic, medical, or legal decisions, verify with appropriate statistical guidance.