Standard Deviation & Variance Calculator
Computes population and sample standard deviation, variance, mean, and sum of squared deviations with full step-by-step arithmetic proofs.
How This Calculation Formula is Formulated
Standard deviation measures the dispersion or spread of numerical data points around their arithmetic mean. For a sample drawn from a larger population, Bessel's correction divides by (n − 1) instead of n to produce an unbiased estimator of true population variance.
Variable Definitions & Measurement Units
Sample Dataset: {10, 12, 23, 23, 16, 23, 21, 16}
Why Bessel's Correction Uses (n − 1)
When calculating the variance of a sample, the observations tend to be closer to the sample mean than to the true, unknown population mean.
Dividing by n systematically underestimates population variance. Dividing by (n − 1) mathematically corrects this negative bias, ensuring the expected value of sample variance equals true population variance.
Frequently Asked Questions About Standard Deviation & Variance Calculator
When should I use Sample vs Population standard deviation?
Use Population (N) if your dataset includes every single member of the group you are studying (e.g., all 30 students in a specific classroom). Use Sample (n − 1) if your data is a random sample drawn from a larger population (e.g., 500 voters surveyed out of millions).
What does a high standard deviation indicate?
A high standard deviation means the data points are spread out widely across a large range of values. A low standard deviation means the data cluster tightly around the average.
What is the 68-95-99.7 empirical rule?
In a normal (bell-shaped) Gaussian distribution, ~68.2% of data falls within ±1σ of the mean, ~95.4% within ±2σ, and ~99.7% within ±3σ.
Can standard deviation be negative?
No. Standard deviation is the positive square root of variance (squared deviations), so it is always greater than or equal to zero.
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