Standard Deviation Calculator — Sample and Population
Paste a list of numbers — commas, spaces, or new lines. Get the sample standard deviation, population standard deviation, variance, mean, count, and sum in one panel.
Use sample stddev when your data is part of a larger group; use population stddev when you have every value.
What is standard deviation calculator?
Standard deviation is the standard way to describe how spread out a list of numbers is. The mean tells you the center; the standard deviation tells you how far typical values sit from that center. Two datasets can share the same mean and look very different: scores 78, 80, 82 cluster tightly around 80, while scores 60, 80, 100 straddle the same average with a much wider spread. Standard deviation captures that distinction in a single number — a smaller value means the data clusters near the mean, and a larger value means the data is more dispersed.
There are two formulas worth knowing. The sample standard deviation divides the sum of squared deviations by n − 1; the population standard deviation divides by n. The difference matters most for small datasets and disappears as n grows. Use the sample formula when your numbers are a sample drawn from a larger group (which is what almost every stats class assumes by default). Use the population formula when your numbers genuinely represent every value in the group of interest. The calculator shows both side-by-side so you don’t have to guess which one your problem wants — pick the one that matches your context and the formula your textbook is asking for.
Variance is the squared standard deviation: same information, different unit. If your data is in centimeters, the standard deviation is in centimeters but the variance is in cm². Statisticians often work with variance internally because it has nicer algebraic properties (independent samples’ variances add), but reports usually cite the standard deviation because it shares units with the original data and is easier to interpret. Both numbers appear in the result panel so you can match whichever form your assignment or spec asks for.
Privacy: every calculation runs locally in your browser. The numbers you paste never leave your device — there is no server, no API, no analytics on input values. The page carries a third-party display ad slot (which is how the site stays free), but the ad has no access to the calculation inputs.
When to use a standard deviation calculator
- Stats homework — sample vs population stddev — The calculator shows both formulas side-by-side: sample (divides by n−1) and population (divides by n). When your problem says 'sample from a population', read the sample value; when it says 'all the data', read the population value.
- Quality control — spread of measurements — Drop in a list of measurements (lengths off a milling line, weights off a fill machine) and the sample stddev quantifies how consistent the process is. A small stddev relative to the mean indicates tight tolerance; a large one means the process is drifting.
- Lab data — uncertainty of repeated readings — Three repeated readings of the same sensor produce three slightly different numbers. The sample stddev is the standard estimate of measurement uncertainty — paste the readings to read it, then report mean ± stddev.
- Comparing two datasets' spread — Two classes both have mean test scores of 80, but one stddev is 5 and the other is 15. Same average, very different consistency. Compute each separately and compare the stddevs to see which class has more uniform performance.
- Verifying a spreadsheet formula — Spreadsheet stddev functions vary: <code>STDEV</code> / <code>STDEV.S</code> compute sample stddev; <code>STDEVP</code> / <code>STDEV.P</code> compute population stddev. If you're unsure which your formula picked, paste the same data here and compare both values against your spreadsheet result.
How to use the Standard Deviation Calculator — Sample and Population
- Paste or type your numbers — Type the numbers into the textarea or paste them from anywhere — a spreadsheet column, a CSV, a lab notebook. Lists of any size work; there is no upper limit.
- Use any common separator — Numbers can be separated by commas, spaces, tabs, semicolons, or new lines, and you can mix them freely. The parser ignores empty entries and shows you any non-numeric tokens it had to skip.
- Read both standard deviations — The headline shows the sample stddev (the most common form taught in stats classes). The stats grid below shows the population stddev alongside the variance, mean, count, and sum. A line beneath explains when to use which.
Worked examples
Five numbers, sample stddev
Input: 2, 4, 4, 4, 5
Output: Sample stddev: 1.0954, Population stddev: 0.9798, Mean: 3.8Deviations from the mean: -1.8, 0.2, 0.2, 0.2, 1.2. Squared: 3.24, 0.04, 0.04, 0.04, 1.44 = 4.8. Sample variance = 4.8/4 = 1.2; sample stddev = √1.2 ≈ 1.0954.
Two-value pair
Input: 1, 5
Output: Sample stddev: 2.8284, Population stddev: 2Mean 3, deviations of 4 + 4 = 8 squared total. Sample variance 8/1 = 8 → stddev √8 ≈ 2.83. Population variance 8/2 = 4 → stddev 2.
All identical — zero spread
Input: 7, 7, 7, 7
Output: Sample stddev: 0, Population stddev: 0When every value equals the mean, both stddevs are zero. The calculator shows 0, not NaN.
Frequently asked questions
What is the difference between sample and population standard deviation?
n − 1; population stddev divides by n. Sample is the default in stats classes because data is usually a sample drawn from a larger group; population is correct when you literally have every value in the group.Why divide by n−1 instead of n?
n systematically underestimates the population variance when you're working with a sample. The n − 1 correction (Bessel's correction) compensates for the fact that the sample mean is slightly closer to your sample's values than the true population mean is. With small samples the difference between the two formulas is noticeable; with large samples (n ≥ 30) the two values are nearly identical.How is the standard deviation calculated?
n (population) or n − 1 (sample). The result is the variance. The standard deviation is the square root of the variance. The calculator shows all four numbers (sample variance, sample stddev, population variance, population stddev) so you can see the relationship.Does the calculator handle decimals and negative numbers?
-3.5, 0, 1e2 is valid input. Very long lists with mixed magnitudes can occasionally show tiny rounding artifacts in the last decimal place because the math uses floating-point arithmetic.Why is the sample stddev shown as a dash when I have only one number?
n − 1, which is undefined for n = 1. With only one value there is no spread to measure as a sample. The population stddev is shown as 0 in that case (the value equals its own mean, so the deviation is zero).