Sample and population
Standard Deviation Calculator: Sample and Population
Paste a list of numbers to get the sample and population standard deviation and variance, the mean, and a step-by-step table of squared deviations.
| x | x − mean | (x − mean)² |
|---|---|---|
| 2 | -3 | 9 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 5 | 0 | 0 |
| 5 | 0 | 0 |
| 7 | 2 | 4 |
| 9 | 4 | 16 |
| Sum of squares | 32 | |
Every value is calculated in your browser from the standard normal distribution (Marsaglia’s series for Φ) and rounded to four decimals; the tables were checked cell by cell against an independent calculation. These tables give the area to the left of z. Some textbooks, including the NIST handbook, print the area from 0 to z instead, which is the left area minus 0.5 for positive z.
Sample or population?
Use the sample standard deviation (dividing by n − 1) when your numbers are a sample from a larger group, which is the usual case in class. Use the population standard deviation (dividing by n) only when you have every member of the group. For 2, 4, 4, 4, 5, 5, 7, 9 the population value is 2 and the sample value is 2.1381.
Common questions
Why divide by n − 1?
A sample tends to sit closer to its own mean than to the true population mean, so dividing by n − 1 corrects the variance estimate upward.
How is variance related to standard deviation?
The standard deviation is the square root of the variance. Variance is in squared units; standard deviation is in the original units.
The studio notebook
A useful place to begin.
Clear steps, considered details, and ideas you can put to use.
How to Read a Z Table (Left-Tail Areas, Step by Step)
Learn to read a standard normal z table: find the row and column, get left-tail, right-tail and between-two-values areas, and handle other table layouts.
Z-Score Formula and Worked Examples
Use the z-score formula z = (x − μ) / σ, read what the sign and size mean, convert a z-score to a percentile, and handle sample means with σ/√n.
Sample vs Population Standard Deviation (n vs n − 1)
Know when to divide by n and when to divide by n − 1, with the formulas, a worked data set, the link to variance, and the mistakes that cost points.
Mean, Median and Mode: Definitions, Examples and Which to Use
Find the mean, median and mode of a data set, see how one outlier moves each one, and decide which measure of center to report for your data.
T Table vs Z Table: Which One to Use and How to Read It
Decide between the t table and z table, find degrees of freedom, read one-tailed and two-tailed columns, and see why t values approach z as df grows.
How to Read a Chi-Square Table (df, Critical Values, p-Values)
Find degrees of freedom for goodness-of-fit and independence tests, read right-tail critical values, and compare them with your chi-square statistic.