Compute population σ or sample s, variance, mean, median, mode, range, quartiles, IQR and CV — with a live histogram and step-by-step deviation table.
Paste numbers separated by commas, spaces, or newlines — results update live.
Standard deviation
Mean
5
μ or x̄
Median
4.5
50th percentile
Variance
4
σ² or s²
Range
7
max − min
Min
2
smallest value
Max
9
largest value
Sum
40
Σx
CV
40%
σ / μ
Q1
4
25th percentile
Q3
5.5
75th percentile
IQR
1.5
Q3 − Q1
Mode
4
most frequent
| # | x | x − μ | (x − μ)² |
|---|
The Method
Standard deviation measures the typical distance between each value and the mean. The procedure is the same in every textbook: subtract the mean from each value (the deviation), square each deviation, average them (this is the variance), and take the square root. The only quirk is the divisor: population σ uses N; sample s uses n−1 (Bessel's correction) to give an unbiased estimate when working from a sample.
Working for current dataset
About This Tool
A standard deviation calculator measures how spread out a set of values is around the mean. Paste your data and the calculator returns the standard deviation (σ or s), the variance (σ² or s²), the mean, median, mode, range, quartiles, the interquartile range (IQR), and the coefficient of variation (CV) — every common single-variable statistic in one view.
Two flavours of standard deviation matter. Population σ divides by N and applies when you have every value in the population. Sample s divides by n − 1 and applies when your data is a sample drawn from a larger population — the subtraction (Bessel's correction) compensates for the bias that arises when the mean is itself estimated from the sample.
Standard deviation is the natural unit of spread for normally distributed data: about 68% of values lie within ±1σ of the mean, about 95% within ±2σ, and about 99.7% within ±3σ — the famous 68-95-99.7 rule. Even for non-normal data, Chebyshev's inequality guarantees that at least 1 − 1/k² of values fall within k standard deviations of the mean.
Use this free standard deviation calculator for science labs, quality control, finance (volatility), exam analysis, and any time you need a quick read on how variable your data is. All calculation is local — no sign-up, no tracking, no data sent to any server.
σ and s in One Tool
Toggle between population and sample formulas with one click.
Live Histogram
Auto-binned via the √n rule; the mean's bin is highlighted in green.
Deviation Table
Row-by-row x, x − μ and (x − μ)² so you can verify by hand.
Quartiles & IQR
Q1, Q3 and interquartile range alongside the spread metrics.
100% Free & Private
No account, no tracking — every calculation runs locally in your browser.
Flexible Input
Paste comma-, space-, tab- or newline-separated values from any source.
From paste to full statistics in seconds.
Drop in your numbers — any mix of commas, spaces, tabs, or newlines works. The aux label shows the value count as you type.
Toggle Population if you have every value, Sample if your data is a subset of a larger group. The formula's divisor changes between N and n − 1.
The big number is your standard deviation. Below, twelve stat tiles cover mean, median, mode, variance, range, min, max, sum, CV, Q1, Q3, and IQR.
The auto-binned histogram shows the shape of your data; the bin containing the mean is highlighted in green. Useful for spotting skew or outliers.
The sortable table lists x, x − μ, and (x − μ)² for every value — perfect for verifying a hand calculation in school.
The formula card shows the full substitution and step-by-step reasoning — handy for assignment write-ups.
Everything you need to know about σ, variance, and statistical spread.
Standard deviation measures how spread out values are from the mean. A small standard deviation means values cluster near the average; a large one means they vary widely. It is the most-used single number summary of spread in statistics.
Use population σ (divide by N) when your data contains every value in the population. Use sample s (divide by n − 1) when your data is a sample drawn from a larger population — the subtraction is Bessel's correction, which removes bias when the mean is estimated from the same sample.
Variance is the average of squared deviations from the mean — the standard deviation squared. Because it has squared units (e.g. cm²) it's harder to interpret directly, which is why we take the square root to get the standard deviation back into the same units as the data.
For a roughly normal distribution: about 68% of values fall within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ. So a value more than three standard deviations from the mean is a 1-in-370 event — usually flagged as an outlier.
CV = σ / μ, often expressed as a percentage. It measures spread relative to the mean, so you can compare variability between datasets with very different scales (e.g. small daily temperatures versus large annual budgets).
The interquartile range (IQR) is the spread of the middle 50% of values: Q3 − Q1. It is robust to outliers (unlike σ), which is why box-and-whisker plots and outlier detection rules use it.
Squaring gives all deviations a positive sign (so they don't cancel) and emphasises larger differences. It also gives variance the nice mathematical property of being additive for independent random variables — Var(X + Y) = Var(X) + Var(Y) — which is why σ² is the natural measure even though σ is the more intuitive one.
This calculator uses the square-root rule: number of bins ≈ √n, capped between 4 and 20. It is a simple, fast heuristic that works well for n between 10 and 1000 and avoids the need for distribution assumptions.