Standard Deviation Calculator

Standard deviation (SD) measures how spread out a set of numbers is around their average. A low SD means the values cluster tightly near the mean; a high SD means they are scattered widely. Because it is expressed in the same units as the data, it is one of the most useful measures of consistency — comparing two students with the same average marks but different SDs tells you who is more reliable, and the idea drives risk in mutual funds, tolerance in manufacturing, and grading on a curve. This tool reports both the sample and population SD, the variance, the full five-number summary, and a deviation table so you can see each step.

Who uses it, and why

Standard deviation turns up wherever consistency matters. A teacher uses it to see whether a class scored uniformly or split into strong and weak groups around the same average. An investor reads a fund's SD of returns as a measure of volatility — two funds can share an average return while one swings far more wildly. A manufacturer tracks the SD of a part's dimensions to keep output inside tolerance, and a researcher reports the SD alongside the mean so readers can judge how reliable an average really is. In every case, the mean alone hides the spread; the SD reveals it.

Variance first, then standard deviation

Variance is the average of the squared distances of each value from the mean. Squaring removes negative signs (so deviations cannot cancel out) and penalises large gaps more heavily. The drawback is that variance comes out in squared units — squared marks, squared rupees — which has no intuitive meaning. Taking the square root brings the figure back to the original unit, and that square root is the standard deviation. In short, standard deviation = √variance and variance = (standard deviation)²; the two always travel together.

Formula

Sample SD (s): s = √[Σ(x − x̄)² / (n − 1)] — use this for a sample drawn from a larger population.

Population SD (σ): σ = √[Σ(x − μ)² / n] — use this when your data is the entire population.

The only difference is the divisor: sample uses n − 1 (the Bessel correction), population uses n. Dividing by n−1 makes the sample SD an unbiased estimator of the true population SD.

When to use each — and why n−1

Choose population SD (÷ n) only when your numbers are the entire group you care about: the marks of every student in one specific class, or the heights of all eleven players on a single team. Choose sample SD (÷ n−1) when your numbers are a subset used to estimate a wider whole — a survey of 500 voters, a batch pulled off a production line, or readings from a few trial runs. Because a sample rarely captures the population's most extreme values, its raw spread tends to underestimate the true spread. Bessel's correction — dividing by n−1 rather than n — nudges the variance up just enough to remove that bias on average. The smaller the dataset, the bigger the correction: with 5 values, n−1 raises the variance by 25%; with 500 values the adjustment is well under 1% and the two formulas nearly coincide.

How to calculate it (worked example)

Take the 10 values 4, 8, 6, 5, 3, 2, 8, 9, 2, 5.

• Step 1 — mean: sum = 52, so x̄ = 52 ÷ 10 = 5.2.
• Step 2 — squared deviations: subtract the mean from each value, square it, and add them all up: Σ(x − x̄)² = 57.6.
• Step 3 — variance: sample variance = 57.6 ÷ (10 − 1) = 6.4; population variance = 57.6 ÷ 10 = 5.76.
• Step 4 — standard deviation: take the square root: sample SD = √6.4 ≈ 2.53; population SD = √5.76 = 2.4.

Notice that the sample SD (2.53) is slightly larger than the population SD (2.4) on identical data — that gap is Bessel's correction at work.

Tips

• If in doubt, choose sample (n−1) — most real-world data is a sample, not a complete population.
• SD is sensitive to outliers because deviations are squared; one extreme value can inflate it sharply.
• To compare spread across different scales, use the coefficient of variation (CV) shown above the result.
• For roughly bell-shaped data, the 68-95-99.7 rule lets you sanity-check the SD: about two-thirds of values should fall within one SD of the mean.

Common mistakes

• Picking the wrong divisor. Using population SD on sample data understates the true spread; this is the single most frequent error, and it matters most on small datasets.
• Confusing variance with standard deviation. They answer the same question in different units — report the SD when you want a figure readers can interpret directly.
• Comparing SDs across different units or scales. An SD of 50 is not automatically "bigger" spread than an SD of 5; convert to the coefficient of variation first.
• Letting a single outlier drive the result. Because deviations are squared, one mistyped value can dominate the SD — check the deviation table for any row with an unusually large (x − x̄)².