Grouped Data Standard Deviation Calculator
Mean and standard deviation from a frequency table — when you have counts per class rather than the raw observations.
With grouped data you no longer know the individual values, only how many fell in each class.
How the grouped data standard deviation calculator works
With grouped data you no longer know the individual values, only how many fell in each class. The standard approach treats every observation in a class as sitting at its midpoint, then computes a weighted mean and variance from those midpoints.
That introduces grouping error, which is small when the classes are narrow relative to the spread and can be material when they are wide. The result is an estimate, and should be quoted as one.
Formula: x̄ = Σfx / Σf; s² = Σf(x − x̄)² / (Σf − 1)
Worked examples
| Inputs | Standard deviation (sample) | Note |
|---|---|---|
| Five classes, 32 observations | 10.582815 | mean 24.4, SD 10.3 |
| One class only | 0 | SD = 0 |
| Two equal classes | 5.270463 | mean 5 |
FAQFrequently asked questions
What is grouped data?
Data recorded as counts per class — "12 people aged 20 to 30" — rather than as individual values.
How accurate is the estimate?
Good when classes are narrow relative to the spread. The midpoint assumption introduces grouping error that grows with class width.
Why use the midpoint?
Because it is the best single guess for a class when you know nothing about the distribution inside it. If the data is heavily skewed within a class, it will be slightly off.
Can I find the exact median?
Not from grouped data — only the class it falls in, or an interpolated estimate. This calculator reports the containing class.
Should I divide by n or n − 1?
By Σf − 1 if the table is a sample, by Σf if it is the whole population. Both are shown.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.