Standard Deviation Calculator
Calculate sample or population standard deviation and variance, with every intermediate step shown and the choice of divisor explained.
How to use this calculator
- 1Paste your numbers, separated however you like.
- 2Choose sample or population. If you are estimating something about a wider group from a subset, it is a sample.
- 3Open the working table to see each deviation and its square.
How the calculation works
s = √( Σ(x − x̄)² / (n − 1) ) σ = √( Σ(x − μ)² / N )- x̄ / μ
- The mean of the sample / of the population
- n / N
- Number of values in the sample / population
- s / σ
- Sample / population standard deviation
The only difference is the divisor: n − 1 for a sample, n for a population.
Standard deviation is in the same units as the data, which is why it is usually preferred to variance for reporting.
The sum of the deviations themselves is always zero, which is why they are squared before being added.
Worked example
Sample standard deviation of 4, 8, 15, 16, 23, 42
- 1.Mean = 108 ÷ 6 = 18.
- 2.Deviations: −14, −10, −3, −2, 5, 24.
- 3.Squared: 196, 100, 9, 4, 25, 576. These sum to 910.
- 4.Sample variance = 910 ÷ 5 = 182.
- 5.Standard deviation = √182 = 13.4907.
Result: s = 13.4907, variance = 182
Population standard deviation of the same 6 values
- 1.The mean and squared deviations are identical to the sample case above — they still sum to 910, since only the final divisor changes.
- 2.Population variance = 910 ÷ 6 = 151.6667 (divide by n, not n − 1).
- 3.Population standard deviation = √151.6667 = 12.3153 — about 9.5% smaller than the sample figure, which is exactly the size of Bessel's correction at n = 6.
Result: σ = 12.3153, variance = 151.6667
What standard deviation actually measures
Standard deviation answers a simple question: on average, how far does each value in a dataset sit from the mean? Two datasets can share an identical average and still tell very different stories — one clustered tightly around it, the other scattered widely on either side. Standard deviation is the single number that captures that difference, expressed in the same units as the original data.
It is built by squaring every deviation from the mean before averaging them — that average is the variance — then taking the square root to bring the units back to normal. Squaring first matters because the deviations above and below the mean always cancel out to zero if left alone; squaring forces every deviation to count regardless of direction, and the final square root undoes the distortion that squaring introduced.
Sample or population — why the divisor matters
Every standard deviation is computed one of two ways, and the difference comes down to a single number in the denominator — which one applies depends on whether your data is the entire group you care about, or only a sample of it.
- Population standard deviation (σ) — used when the data already covers every member of the group in question — every employee, every finished item in a batch. It divides by N, the full count, since there is no estimation involved.
- Sample standard deviation (s) — used when the data is a subset standing in for a larger group. It divides by n − 1 rather than n — a small correction (Bessel’s correction) that compensates for a sample’s own mean sitting slightly closer to its data than the true population mean does, which would otherwise make the estimate run systematically too low.
Where standard deviation shows up
The idea shows up wherever consistency matters as much as the average itself.
- Finance and investing — used as a stand-in for volatility — two funds can post the same average return while one swings far harder along the way.
- Manufacturing and quality control — tracks whether individual parts stay within tolerance, not just whether the average part is on spec.
- Standardized testing — describes how tightly scores cluster around the mean, which is the basis percentile rankings are built on.
- Weather and climate — distinguishes a place with a stable climate from one with wild swings around an identical average temperature.
- Sports analytics — separates a consistently reliable performer from a streaky one with the same season average.
Reading a standard deviation without being misled
A standard deviation is only easy to interpret against something else — the mean, another dataset, or a known distribution shape. On its own, a figure like 12 means nothing without knowing what units the data is in and roughly how large the values themselves are.
The commonly quoted rule that about 68% of values fall within one standard deviation of the mean, and about 95% within two, holds only for roughly normal, bell-shaped data. Skewed or heavy-tailed distributions can miss that pattern by a wide margin, and because deviations are squared before being averaged, a single extreme outlier can inflate the whole figure on its own.
What this assumes, and where it stops
Assumptions
- Every value is weighted equally.
- The data is measured on an interval or ratio scale, so differences are meaningful.
Limitations
- Highly sensitive to outliers, because deviations are squared. A single extreme value can dominate the result.
- The 68–95–99.7 rule only applies to roughly normal distributions. For skewed data it can be badly misleading.
- For grouped or weighted data, a weighted variance formula is needed instead.
Common questions
Should I use sample or population standard deviation?
Sample, almost always. Use the population formula only when your data genuinely covers every member of the group you care about — every employee in a company, every item in a finished batch. If you are generalising from a subset, use the sample formula. With small datasets the difference is substantial: at n = 6 the sample SD is about 10% larger.
Why divide by n − 1 rather than n?
Because a sample’s mean is calculated from the sample itself, the values sit slightly closer to it than they would to the true population mean. Dividing by n would systematically underestimate the spread. Subtracting one from the divisor — Bessel’s correction — removes that bias.
What is a high standard deviation?
There is no universal threshold; it depends on the scale of the data. The coefficient of variation — standard deviation as a percentage of the mean — is the scale-free version, and lets you compare spread between datasets measured in different units.
What is standard deviation actually used for?
Anywhere consistency matters as much as the average. A factory tracks the standard deviation of a part's dimensions because a low average defect rate is meaningless if individual parts swing wildly outside tolerance. Investors use it as a stand-in for risk: two funds can post the same average return while the one with the higher standard deviation swings harder in both directions along the way. Meteorologists use it to describe climate variability — two cities can share an identical average temperature while one has a much wider standard deviation because its seasons swing harder. In every case, the mean says where the data is centred and the standard deviation says how much it wanders.
Formula and content last reviewed on .
Results are estimates for information only, not professional advice.
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