Sample and population standard deviation, variance, mean, median, quartiles and outliers
Runs in your browser. Nothing you type is sent anywhere. Updated on 24 Oct 2026.Standard deviation measures how far a set of numbers spreads around its mean. Paste numbers to get the sample and population standard deviation, variance, mean, median, mode, quartiles and outliers, with every step shown. For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5 and the population standard deviation is 2. Free; nothing is uploaded.
Paste or type your numbers, separated by commas, spaces or new lines, or switch to Value + frequency.
Read the sample and population standard deviation, the variance, mean, median and mode at once.
Open Working to see each deviation from the mean, the sum of squares and the division by n or n − 1.
Check the quartiles, the IQR and any outliers, then copy the summary or share the link.
Find the mean, subtract it from every value, square each difference and add the squares. Divide that sum by n for the population variance, or by n − 1 for the sample variance, then take the square root. For 2, 4, 4, 4, 5, 5, 7, 9 the squared differences add up to 32, so the population variance is 32 ÷ 8 = 4 and the standard deviation is 2.
Use the population standard deviation (σ, divide by n) when the numbers are the whole group, such as every student in one class. Use the sample standard deviation (s, divide by n − 1) when the numbers are a sample drawn from a larger group, such as 50 customers surveyed out of thousands. Dividing by n − 1, called Bessel's correction, makes up for a sample usually lying closer to its own mean than to the true one.
| Measure | Formula | For 2, 4, 4, 4, 5, 5, 7, 9 |
|---|---|---|
| Population SD (σ) | √(Σ(x − x̄)² ÷ n) | 2 |
| Sample SD (s) | √(Σ(x − x̄)² ÷ (n − 1)) | 2.138 |
| Population variance | Σ(x − x̄)² ÷ n | 4 |
| Sample variance | Σ(x − x̄)² ÷ (n − 1) | 4.571 |
By default Q1 and Q3 are the medians of the lower and upper halves of the sorted data, leaving out the middle value when the count is odd, as the TI-84 and most textbooks do. The inclusive method of Excel's QUARTILE.INC can be chosen instead. Values more than 1.5 times the interquartile range below Q1 or above Q3 are listed as outliers.
Yes. Switch to Value + frequency and type one value and its count per line, such as "3 8" for eight threes. Up to 100,000 values are accepted. The sums are computed around the mean in two passes, so large values close together keep their precision. Related tools: Grade Calculator, Scientific Calculator.
Population standard deviation divides the sum of squared deviations by n, the number of values; sample standard deviation divides by n − 1. Use the population figure when the data are the whole group and the sample figure when they are a sample from a larger group. For 2, 4, 4, 4, 5, 5, 7, 9 the two are 2 and 2.138.
Work out the mean, subtract it from each value, square the results and add them up. Divide by n (population) or n − 1 (sample) to get the variance, then take the square root. For 1, 2, 3, 4, 5 the mean is 3, the squares add up to 10, and the sample standard deviation is √(10 ÷ 4) ≈ 1.581.
Standard deviation shows how spread out the values are around the mean, in the same units as the data. A small value means the numbers sit close to the mean; a large one means they are widely scattered. In roughly bell-shaped data about 68% of values lie within one standard deviation of the mean and about 95% within two.
Work out the first and third quartiles, Q1 and Q3, and their difference, the interquartile range. Any value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR is an outlier, a rule from John Tukey. For 1, 2, 3, 4, 5, 6, 100 the fences are −4 and 12, so 100 is an outlier.
Quartiles have several accepted definitions. The calculator uses the median-of-halves method of the TI-84 and most school textbooks by default; Excel's QUARTILE.INC interpolates between positions instead. For 1, 3, 5, 7, 9, 11, 13 the first gives Q1 = 3 and Q3 = 11, Excel gives 4 and 10. Choose Inclusive to match Excel.
Switch the input to Value + frequency and type one value and its count per line, separated by a space, tab, colon or semicolon: 3 8 means the value 3 occurs 8 times. A header line such as Value Frequency is skipped. The results equal those of the expanded list, up to 100,000 values in total.
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