# Standard Deviation Calculator — Mean, Variance & Stats with Steps

> Calculate mean, median, mode, range, population and sample standard deviation, variance, quartiles, IQR and outliers from a data set, with a histogram and ±σ bands.

- **Category:** Math
- **Interactive calculator:** https://youcalc.com/en/math/statistics-standard-deviation/
- **Price:** Free, no sign-up required

## Overview

This calculator analyses any list of numbers and returns a full summary: mean, median, mode, range, population and sample standard deviation, variance, quartiles, IQR, and outliers. Use it whenever you need to understand how spread out a data set is, whether for a school assignment, a scientific study, or a quality-control check.

## How to read your result

The headline figure is the standard deviation — either population (σ, divides by n) or sample (s, divides by n−1), switchable with the toggle. Below it you see variance, median, and range at a glance. The full statistics table lists every measure including Q1, Q3, IQR, and any flagged outliers. The histogram plots the frequency of each bin and shades the ±1σ band around the mean, with a dashed line marking the mean itself.

## Method

The mean is the arithmetic average (sum ÷ count). Variance is the average of squared deviations from the mean: divide by n for population, by n−1 for sample. Standard deviation is the square root of variance. Median is the middle value of the sorted list; for even counts it is the average of the two central values. Quartiles use the median-of-halves method, excluding the median itself for odd-length lists. Outliers are flagged using the 1.5 · IQR rule applied to the resulting fences. Sources: Wolfram MathWorld and Khan Academy (see references).

## Example

- **Setup:** Enter the classic dataset 2, 4, 4, 4, 5, 5, 7, 9 (eight values used in most statistics textbooks) and select the population mode.
- **Result:** The mean is 5, the population standard deviation is exactly 2, and the variance is 4. Switching to sample mode gives s ≈ 2.1381 and variance ≈ 4.5714, because the denominator becomes n−1 = 7.

## Frequently asked questions

### What is the difference between population and sample standard deviation?

Population standard deviation (σ) divides the sum of squared deviations by n, the total count. Sample standard deviation (s) divides by n−1 — a correction called Bessel's correction — which removes a small downward bias when you are estimating from a subset of a larger group. Use population mode when your list is the entire group; use sample mode when it is a subset drawn from a larger population.

### How are outliers detected?

The calculator uses the interquartile range (IQR) fence rule: any value below Q1 − 1.5 · IQR or above Q3 + 1.5 · IQR is flagged as a potential outlier. This method works well for roughly symmetric distributions; very skewed data may need a different approach.

### What does a higher standard deviation tell me?

A high standard deviation means the values are spread widely around the mean; a low one means they cluster closely. Two datasets with the same mean can behave very differently in practice — the one with the larger standard deviation carries more variability and, in many fields, more risk.

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- [Z-Score & Normal Distribution Calculator](https://youcalc.com/en/math/z-score-normal-distribution/)
- [Percentage Calculator](https://youcalc.com/en/math/percentage/)
- [Permutations & Combinations Calculator](https://youcalc.com/en/math/permutations-combinations/)

## Sources

- https://mathworld.wolfram.com/StandardDeviation.html
- https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/variance-standard-deviation-population/a/calculating-standard-deviation-step-by-step

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Interactive version: https://youcalc.com/en/math/statistics-standard-deviation/ · From YouCalc — https://youcalc.com