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Standard Deviation Calculator

Free standard deviation calculator. Find the sample or population standard deviation, variance, mean, median, range, and sum of a set of numbers.

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Deep dive guides

A LITTLE CONTEXT · 5 MIN READHow Standard Deviation Works
All 8 related guides
  • Mean vs Median vs Mode: How Averages Work — The three kinds of average, what each one hides, and how to choose: outliers drag the mean, the median resists, and the mode is for categories.
  • How Standard Deviation Works — What standard deviation actually measures, the step-by-step variance recipe, why sample mode divides by n−1, and the 68-95-99.7 bell-curve rule.
  • Z-Scores Explained: Standardising the Unusual — What a z-score really measures, how it becomes a percentile on the normal curve, and when the reading breaks down.
  • How Weighted Averages Work — Why some numbers count more than others: the weighted average formula, weights that don't sum to 1, and the grade and group-size traps.
  • How Percentile Rank Works — Position, not points: the counting formula behind percentile ranks and the below / at-or-below / midpoint conventions that ties create.
  • The 68-95-99.7 Rule Explained — The empirical rule: why bell-shaped data puts 68%, 95% and 99.7% within one, two and three standard deviations of the mean.
  • Geometric Mean vs Arithmetic Mean — One average adds, one multiplies: the geometric mean formula, volatility drag on returns, the AM–GM inequality, and when each is correct.
  • Grade Curving: Flat, Linear and Top-Score Methods — The three standard ways to curve a set of scores, what each does to the class distribution, and how to defend the choice.

How standard deviation calculator works

Standard deviation measures how spread out a set of numbers is around the mean. A low standard deviation means the values cluster near the average; a high one means they are spread widely. It is the square root of the variance, which is the average of the squared differences from the mean.

There are two flavors. The sample standard deviation divides by n−1 (one fewer than the count) and is used when your numbers are a sample of a larger population — the n−1 correction makes the estimate unbiased. The population standard deviation divides by n and is used when you have every member of the population. This calculator reports both the chosen deviation and the variance, plus the mean, median, sum, min, max, and range.

Enter your numbers separated by commas or spaces and choose sample or population. The calculator parses the list, validates that every entry is a number, and computes the statistics. Sample mode requires at least two numbers because you cannot measure spread from a single value.

Frequently asked questions

What is the difference between sample and population standard deviation?
Sample (n−1) is used when your data is a sample of a larger group — it is an unbiased estimate of the population spread. Population (n) is used when you have the entire population. This calculator defaults to sample mode and reports which one it used.
How is standard deviation calculated?
Take the mean, square the difference of each value from the mean, average those squares to get the variance, then take the square root. For the dataset 2,4,4,4,5,5,7,9 the sample standard deviation is about 2.14.
Why does sample mode need at least two numbers?
With one number the spread is undefined — dividing by n−1 = 0 is impossible. You need at least two values to measure variation, so sample mode requires a minimum of two numbers.
Can I paste numbers separated by spaces or newlines?
Yes. The calculator splits on any whitespace and commas, so "2, 4, 4" and a column of newline-separated numbers both work. Every entry must parse as a number, or it will report an error.

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