Standard deviation measures how spread out a dataset is around its mean. Low standard deviation means data points cluster near the average; high standard deviation means they're scattered. This calculator returns both population and sample standard deviation along with full descriptive statistics.
Sample SD (s)
5.237229
Divide by n − 1
Population SD (σ)
4.898979
Divide by N
Sample variance (s²)
27.428571
Population variance (σ²)
24
Mean
18
Count
8
Sum
144
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Standard deviation is the single most important measure of dispersion in statistics - it quantifies how much the values in a dataset typically deviate from the mean. A low standard deviation indicates the data points cluster tightly around the mean; a high standard deviation indicates wider spread. In finance, SD of returns is the standard measure of volatility and risk. In quality control, SD measures consistency. In academic grading, SD identifies whether a class scored uniformly or showed wide variation. Wherever you summarise data with the mean, you should report standard deviation alongside it.
This calculator computes both flavors of standard deviation: population SD (divides by N - use when you have data on every member of the group) and sample SD (divides by N−1 using Bessel's correction - use when your data is a sample from a larger population, which is the typical case in research and surveys). The output includes the variance, mean, and a full set of descriptive statistics so you get the complete picture from one calculation.
Paste numbers (separated by commas, spaces or newlines).
See mean, sample SD, population SD, variance and quartiles.
Variance is the average squared deviation from the mean. Standard deviation is its square root, which restores the original units. Sample SD divides by (n−1) instead of n - Bessel's correction - to give an unbiased estimate of the population SD when working from a sample.
Population: σ = √( Σ(xᵢ − μ)² / N ) Sample: s = √( Σ(xᵢ − x̄)² / (n−1) )
Variance is computed in four steps: (1) compute the mean of the data, (2) subtract the mean from each data point to get deviations, (3) square each deviation, (4) average the squared deviations. Squaring serves two purposes - it makes all values positive (so positives and negatives don't cancel) and it amplifies the influence of large deviations. Standard deviation is the square root of variance, restoring the original units (squared units in variance become regular units in SD).
Bessel's correction (dividing by N−1 instead of N for sample variance) compensates for the fact that the sample mean tends to be 'closer' to the data than the true population mean, which underestimates variance. The N−1 correction provides an unbiased estimate of population variance from sample data. For large N, the difference between N and N−1 is negligible; for small N, it matters significantly. Always use sample SD unless you have population data.
The empirical rule (68-95-99.7 rule) gives intuition for what standard deviation means in a normal distribution. Approximately 68% of data lies within ±1 SD of the mean. About 95% lies within ±2 SD. About 99.7% within ±3 SD. So if exam scores are normally distributed with mean 65 and SD 10, you'd expect 68% of students between 55-75, 95% between 45-85, and 99.7% between 35-95. Used heavily in process control (Six Sigma uses 6 SD = roughly 1 in a billion defects), academic grading, and risk management.
Mean: 70. Deviations: -10, -5, 0, 5, 10. Squared deviations: 100, 25, 0, 25, 100 → sum 250. Population variance: 250/5 = 50. Population SD: √50 ≈ 7.07. Sample variance: 250/4 = 62.5. Sample SD: √62.5 ≈ 7.91. Different results depending on whether you treat this as the entire class (population) or a sample.
Mean return: 11%. Sample SD: 10.7%. The fund averages 11% per year but with significant year-to-year variation. Returns outside ±21% (mean ± 2 SD) are unlikely in any given year if the volatility pattern persists.
Sample SD when your data is a subset of a larger population (the usual case). Population SD when you have data on every member of the group.
Dividing by (n−1) instead of n. Compensates for the fact that the sample mean is a biased estimator of the true mean - using n underestimates variance slightly.
SD = √variance. Variance has squared units (e.g. cm²) while SD has the same units as the data (cm). SD is more interpretable; variance has nicer mathematical properties.
In a normal distribution: ~68% of data lies within 1 SD of the mean, ~95% within 2 SD, ~99.7% within 3 SD. Known as the empirical rule or 68-95-99.7 rule.
No. SD is a square root of a non-negative quantity, so it's always ≥ 0. SD = 0 only when every value equals the mean.
Standard deviation of returns is the standard measure of risk / volatility. Higher SD means more variable returns - both upside and downside.
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