Enter a list of numbers to calculate population and sample standard deviation with full step-by-step solution.
| # | Value (x) | x − mean | (x − mean)² |
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Standard deviation is a measure of how spread out numbers are in a dataset. It tells you how much individual values typically differ from the mean. A small standard deviation means data points are close to the mean. A large standard deviation means data is more spread out. It is one of the most widely used statistics in science, finance, and research.
Population standard deviation (σ) is used when your dataset includes every member of a group — for example, the heights of all students in a specific class. Sample standard deviation (s) is used when your data is a random sample from a larger population — for example, surveying 100 people to estimate the average height of an entire country. Sample standard deviation uses N−1 in the denominator (Bessel's correction) to account for the fact that a sample tends to underestimate the true population variance.
Variance is the average of the squared differences from the mean. Standard deviation is simply the square root of variance. Variance is harder to interpret directly because it is in squared units, while standard deviation is in the same units as the original data.
Squaring ensures that negative and positive differences do not cancel each other out. It also gives more weight to values far from the mean, making it more sensitive to outliers than other measures of spread.
The coefficient of variation (CV) is the standard deviation divided by the mean, expressed as a percentage. It allows you to compare variability across datasets with different units or scales.