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Calculate the mean, variance, and both sample standard deviation (s, with Bessel's N−1 correction) and population standard deviation (σ) for any data set — used in statistics, finance, and quality control.
Math
Generated on August 22, 2026
Calculate the mean, variance, and both sample standard deviation (s, with Bessel's N−1 correction) and population standard deviation (σ) for any data set — used in statistics, finance, and quality control.
Standard deviation measures how spread out the values in a data set are from their mean. This calculator returns the mean, the variance, and both the population standard deviation (σ) and the sample standard deviation (s), along with the count of values. Standard deviation is one of the most important tools in statistics, science, quality control, and finance — wherever you need to quantify variability, uncertainty, or risk.
Formula
Population SD: σ = √[Σ(x − μ)² / N]. Sample SD: s = √[Σ(x − x̄)² / (N − 1)]. Variance = SD².Two classes both have an average of 75 on the same test. Are they equivalent? No idea — not without standard deviation. One class might have everyone scoring between 72 and 78 (tight cluster, consistent teaching). The other might have half the class at 95 and half at 55 (bimodal disaster, urgent intervention needed). SD is the single number that separates those two stories. This calculator returns both versions — population SD (divide by N, used when your data is the entire group) and sample SD (divide by N − 1, Bessel's correction, used when you're estimating from a sample). Here's the part most students get wrong on exams: knowing which one to use. Census data, every part in a production batch, all 30 students in your section — that's a population, divide by N. A survey of 500 voters representing millions, a quality-check sample from a much larger run — that's a sample, divide by N − 1. The 68-95-99.7 rule is the practical superpower SD gives you: roughly 68 percent of bell-curve data falls within 1 SD of the mean, 95 percent within 2, 99.7 percent within 3. So if a stock has a mean monthly return of 1 percent with an SD of 5 percent, a month with −9 percent return is two SDs below the mean — uncommon but not unheard of. Useful for grading curves, quality control, and any time you need to flag something as 'unusual.'
Standard deviation measures how spread out the values in a data set are from their mean. This calculator returns the mean, the variance, and both the population standard deviation (σ) and the sample standard deviation (s), along with the count of values. Standard deviation is one of the most important tools in statistics, science, quality control, and finance — wherever you need to quantify variability, uncertainty, or risk. Two data sets with the same mean can look completely different; standard deviation captures that difference in a single number.
Each data point's deviation from the mean is squared (to convert negatives to positives without canceling) and averaged. Taking the square root returns the result to the original units of measurement, which is why standard deviation — not variance — is normally reported. The critical difference between population and sample SD is the divisor: population uses N (when you have all possible values), while sample uses N − 1 (Bessel's correction) to provide an unbiased estimator when your data is a sample of a larger population.
68-95-99.7 rule: in normal (bell-curve) data, 68% fall within 1 SD, 95% within 2 SD, 99.7% within 3 SD.
Z-score = (x − mean) / SD — a standardized 'how far from average' metric, used in grading curves and statistical tests.
SAT/ACT scores use SD-based scaling: mean 1000, SD 200, so 1400 is +2 SD (top 2.5% of test takers).
Financial returns with high SD = volatile = risky; low SD = stable = typically lower return.
Data [2, 4, 4, 4, 5, 5, 7, 9]: Mean 5, Population SD 2.0, Sample SD 2.14 — slightly higher sample SD reflects Bessel's correction.
Test scores [70, 75, 80, 85, 90]: Mean 80, Sample SD ≈ 7.91 — a tight cluster around the mean, typical of a well-taught class.
Quality control: if a process produces parts with SD near zero, it is very consistent; a large SD indicates variability and potential defects.
Finance: the standard deviation of a stock's historical daily returns is the standard measure of its volatility and risk.
Weather: the SD of daily temperatures over a month tells you whether the month was climatically stable or volatile.
Sports: batting averages in cricket can hide variability — two players can share an average while one is consistent and the other swings wildly; SD reveals it.
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