Standard Deviation Calculator (Sample and Population, with Steps)
Paste a list of numbers and get the sample or population standard deviation, variance, mean, standard error, confidence interval, quartiles, z-scores, a frequency table and a histogram, with every step worked out.
How it works
mean = Σx ÷ n · σ = √(Σ(x − mean)² ÷ n) · s = √(Σ(x − mean)² ÷ (n − 1)) · SE = s ÷ √n · CI = mean ± t × SE · z = (x − mean) ÷ SD · CV = SD ÷ |mean| × 100%
Standard deviation measures how spread out the values are around the mean: square each value’s distance from the mean, add the squares up (the sum of squares), divide to get the variance, and take the square root. The population version divides by n; the sample version divides by n − 1 (Bessel’s correction), which compensates for a sample underestimating the true spread. Pick the one that fits your data with the Sample/Population switch: the main result follows it, and the other is shown alongside. The calculator also gives the standard error of the mean (s ÷ √n) and a confidence interval for the mean built from the t distribution, a z-score for every value, the median, quartiles and IQR with 1.5 × IQR outlier fences, the mean absolute deviation, skewness, excess kurtosis and the coefficient of variation, plus a frequency table and histogram. Under More options you can enter a frequency table instead of a raw list, choose the confidence level and quartile method, round the results, or switch to a decimal comma.
Worked example
For 2, 4, 4, 4, 5, 5, 7, 9: the mean is 40 ÷ 8 = 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4 and 16, which sum to 32. Population variance = 32 ÷ 8 = 4, so σ = √4 = 2. The sample version divides by 7: s² = 32 ÷ 7 ≈ 4.5714 and s ≈ 2.1381. The standard error is 2.1381 ÷ √8 ≈ 0.7559, and with t = 2.3646 for 7 degrees of freedom the 95% confidence interval for the mean is 5 ± 1.7875, or about 3.21 to 6.79. The value 9 has a z-score of (9 − 5) ÷ 2.1381 ≈ 1.87.
Sources: NIST/SEMATECH e-Handbook of Statistical Methods — measures of scale · NIST/SEMATECH e-Handbook — confidence limits for the mean
Frequently asked questions
Sample or population — which one do I need?
If your data is every member of the group you care about (all 30 students in one class), use population. If it is a sample standing in for something larger (30 students representing all students), use sample — that’s the more common case, and it gives a slightly larger answer because it divides by n − 1.
What does one standard deviation actually mean?
For roughly bell-shaped data, about 68% of values fall within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD — the empirical rule. The chart under the results draws the mean and the ±1 SD band over your values.
What is the difference between standard deviation and standard error?
Standard deviation describes how spread out the individual values are. Standard error (s ÷ √n) describes how precisely the sample mean estimates the true mean, so it shrinks as you collect more data while the standard deviation does not.
Why square the deviations instead of just averaging distances?
Squaring keeps everything positive so spread doesn’t cancel out, and it weights big deviations more heavily. The plain average of absolute distances is the mean absolute deviation, which is listed in the statistics table for comparison.
How do I enter a frequency table?
Open More options, choose “Values + frequencies”, type each distinct value once in the data box and the matching counts in the frequency box — for example values 2, 4, 5, 7, 9 with counts 1, 3, 2, 1, 1 is the same as the list 2, 4, 4, 4, 5, 5, 7, 9.
How do I calculate standard deviation in Excel or Google Sheets?
Use STDEV.S for the sample standard deviation and STDEV.P for the population one; VAR.S and VAR.P give the variances. For the default data they return 2.1381 and 2, the same as this calculator.
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This calculator is for educational purposes. Double-check important results before acting on them.