Standard deviation measures how spread out a set of numbers is around their average: a small value means the numbers cluster near the mean, a large one means they scatter widely. Paste your numbers and this tool reports the standard deviation and variance in both the sample and population forms, along with the mean, count and sum.
Separate values with spaces, new lines, tabs or semicolons. A comma or dot is a decimal point.
Enter some numbers to see the standard deviation.
How it works
The tool first finds the mean (average) of your values. It then takes each value’s distance from the mean, squares it, and adds those squares up — that total is the sum of squared deviations. The variance is that sum divided by the number of values (population) or by one less (sample), and the standard deviation is the square root of the variance, back in the original units.
The difference between the two forms is the divisor. Population uses N because you have every member of the group. Sample uses N − 1 (Bessel’s correction) because a sample tends to underestimate the true spread, and dividing by the smaller number corrects for that. A single value has no sample standard deviation — there is nothing to vary against — so the tool shows only the population figure and says so.
Practical examples
A textbook data set
Paste 2, 4, 4, 4, 5, 5, 7, 9. The mean is 5 and the population standard deviation is exactly 2. The sample standard deviation is about 2.138 — larger, because it divides by 7 instead of 8.
Two exam groups with the same average
Group A scored 70, 72, 68, 71, 69 and group B scored 50, 90, 60, 95, 55 — both average 70. Group A’s standard deviation is about 1.4, group B’s about 18.7, showing B’s results are far less consistent.
Test measurement repeatability
You weigh the same object five times: 10.1, 10.0, 10.2, 9.9, 10.1. The sample standard deviation (about 0.11) describes the scatter of your measurements — a common way to state instrument precision.
Frequently asked questions
What is standard deviation?
It is the typical distance between the values in a set and their mean. Roughly, it answers "how far off from average is a normal data point?" It is expressed in the same units as the data.
What is the difference between sample and population standard deviation?
Population standard deviation divides the sum of squared deviations by N and is used when your data includes every member of the group. Sample standard deviation divides by N − 1 and is used when your data is a sample drawn from a larger group.
Which one should I use?
If the numbers are the entire group you care about (for example, the marks of all students in one class), use population. If they are a sample meant to represent something bigger (a survey, repeated measurements), use sample. When in doubt in statistics coursework, sample (N − 1) is the usual default.
What is variance?
Variance is the average of the squared deviations from the mean — the standard deviation squared. It is useful in calculations but is in squared units, which is why standard deviation (its square root) is easier to interpret.
Why square the deviations instead of just adding them?
The plain deviations from the mean always sum to zero, because positives and negatives cancel. Squaring makes them all positive and gives larger errors more weight; the square root at the end brings the result back to the original scale.
What does a standard deviation of zero mean?
Every value is identical, so there is no spread at all. As the numbers become more varied, the standard deviation grows.
Can standard deviation be negative?
No. It is a square root of a sum of squares, so it is always zero or positive. If you see a negative sign, something has gone wrong with the input.
How many numbers do I need?
One value gives a mean but no meaningful spread, so there is no sample standard deviation. From two values up, both forms are defined. More values generally give a more reliable estimate.
How do I enter my numbers?
Type or paste them separated by spaces, new lines, tabs or semicolons. A comma or dot inside a number is treated as a decimal point, so 6,5 and 6.5 are read as six-and-a-half.
Does the order of the numbers matter?
No. Standard deviation depends only on the set of values and their mean, not the order you enter them in.
How precise is the answer?
Calculations use full double-precision floating point and results are shown to six significant digits. For typical class and lab data that is well beyond the precision of the measurements themselves.
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