College statistics, lab work and research use the sample standard deviation, which divides by n − 1. This calculator shows the sample and population versions side by side so you always report the right one.
Enter your data values separated by commas.
Read the sample and population standard deviation.
Report s for a sample, σ for a full population.
Sample SD: s = √(Σ(x − x̄)² ÷ (n − 1))
Population SD: σ = √(Σ(x − μ)² ÷ n)
Standard deviation measures how spread out a set of numbers is around its mean. A small value means the readings cluster tightly; a large one means they scatter. This calculator works out both versions side by side: the sample standard deviation s, which divides by n − 1, and the population standard deviation σ, which divides by n. It also shows the mean, every squared deviation and the sample variance.
The two versions exist because most data is a sample. A lab group timing a pendulum, a quality engineer checking 10 packets from a day's production, or a researcher surveying 200 students all want to say something about a larger population they did not fully measure. For that job the n − 1 version is the right one, and it is what university statistics, lab reports and software like Excel's STDEV.S expect.
1. Add all n values and divide by n to get the mean x̄.
2. Subtract the mean from each value to get its deviation, x − x̄.
3. Square each deviation so that negative and positive ones do not cancel.
4. Add the squares to get the sum of squares, Σ(x − x̄)².
5. For a sample, divide by n − 1 to get the variance s². For a whole population, divide by n to get σ².
6. Take the square root to get the standard deviation, which is back in the original units.
The sum of squared deviations is smallest when measured from the sample's own mean. Measured from the true population mean, which you do not know, it would almost always be a little larger. So using x̄ systematically understates the spread. Dividing by n − 1 instead of n inflates the result just enough that the sample variance is, on average, equal to the population variance. This adjustment is called Bessel's correction. The difference is large for tiny samples and negligible for large ones: with 5 values the factor is 5/4, with 500 it is almost 1.
Another way to see n − 1: the deviations from the mean always add up to zero. Once you know n − 1 of them, the last one is fixed. So only n − 1 deviations carry independent information about spread, and statisticians say the estimate has n − 1 degrees of freedom. The same idea returns in t-tests and chi-square tests. It also explains why a single value gives no sample standard deviation at all: with n = 1 there is nothing left to vary, and the calculator shows the sample result as undefined.
Variance is in squared units, such as beats² per minute², which is hard to interpret. Taking the square root returns to the original units, so a standard deviation of 4 beats per minute can sit beside a mean of 73. One subtle point: although s² is an unbiased estimate of σ², s itself still slightly underestimates σ on average, because the square root is a curved function. For practical reporting this is ignored. The shortcut formula Σx² − (Σx)²/n gives the same sum of squares but can lose accuracy through rounding when values are large and close together.
For a biology practical, Neha counted the resting pulse rate of eight classmates in beats per minute: 68, 72, 75, 70, 81, 77, 69 and 74. She treats them as a sample of her whole school.
Mean: (68 + 72 + 75 + 70 + 81 + 77 + 69 + 74) ÷ 8 = 73.25
Sum of squared deviations: 27.5625 + 1.5625 + 3.0625 + 10.5625 + 60.0625 + 14.0625 + 18.0625 + 0.5625 = 135.5
Sample variance (divide by n − 1): 135.5 ÷ 7 = 19.3571 → SD = √ = 4.3997
Population variance (divide by n): 135.5 ÷ 8 = 16.9375 → SD = √ = 4.1155
Answer: Sample standard deviation (s) 4.3997; Population standard deviation (σ) 4.1155; Sample variance (s²) 19.3571
Reporting the population value σ for data that is only a sample, which understates the spread.
Rounding the mean before computing deviations, which adds error to every squared term.
Forgetting the final square root and quoting the variance as the standard deviation.
Using a calculator's σₙ key when the question expects σₙ₋₁, or STDEV.P in Excel instead of STDEV.S.
Comparing standard deviations of data in different units or with very different means; use the coefficient of variation for that.
Reporting the uncertainty of repeated readings in physics and chemistry practicals.
Quality control, where the spread of weights or dimensions in a sample batch is monitored.
Research papers and theses that report mean ± SD for each group.
Measuring the volatility of mutual fund or stock returns from a sample of past months.
Feeding the SD into confidence intervals, t-tests and sample size calculations.
Which one does Excel's STDEV use?
STDEV and STDEV.S give the sample SD; STDEV.P gives the population SD.
Why is s larger than σ?
Because it divides by the smaller number n − 1.