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Statistics Calculator
Paste a list of numbers — test scores, measurements, survey results — and get every standard descriptive statistic at once: mean, median, mode, range, variance and standard deviation (both sample and population), quartiles and the interquartile range, plus the sorted list so you can check the work. Numbers can be separated by commas, spaces or new lines.
Quartiles use Tukey’s hinges: Q1 is the median of the lower half of the sorted data and Q3 the median of the upper half (the median itself is excluded when the count is odd). The coefficient of variation is the population standard deviation as a percentage of the mean. All figures are estimates of your data only — they describe the numbers you entered, nothing more.
How it works
All statistics come from the sorted list of your numbers. Mean = sum ÷ count. Median = the middle value (or the average of the two middle values when the count is even). Mode = the value that appears most often; if every value appears once, there is no mode. Range = maximum − minimum. Quartiles use Tukey’s hinges: split the sorted list into a lower half and an upper half (excluding the median when the count is odd); Q1 is the median of the lower half, Q3 the median of the upper half, and the interquartile range is Q3 − Q1. Variance is the average squared distance from the mean: population variance divides by n, sample variance divides by n−1 (Bessel’s correction — use the sample figures when your numbers are a sample of a larger group). Standard deviation is the square root of the variance, back in the original units. Worked example 1: 2, 4, 4, 4, 5, 5, 7, 9 — sorted already, sum = 40, n = 8, so the mean is 40 ÷ 8 = 5. The median is (4 + 5) ÷ 2 = 4.5, the mode is 4 (three times), the range is 9 − 2 = 7. Squared deviations from 5 are 9, 1, 1, 1, 0, 0, 4, 16, totalling 32: population variance = 32 ÷ 8 = 4, population SD = 2; sample variance = 32 ÷ 7 = 4.5714, sample SD = 2.1381. Q1 = 4 (median of 2, 4, 4, 4), Q3 = 6 (median of 5, 5, 7, 9), IQR = 2. Worked example 2: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 — mean 5.5, median 5.5, no mode, population variance 8.25 (SD 2.8723), sample variance 9.1667 (SD 3.0277), Q1 = 3, Q3 = 8, IQR = 5, coefficient of variation 52.22%. Worked example 3: a single value, 7 — mean, median and mode are all 7, both deviations are 0, and the sample figures are blank because a sample of one has no spread to measure.
Frequently asked questions
What is the difference between sample and population standard deviation?
Population standard deviation divides the squared deviations by n (the whole group); sample standard deviation divides by n−1. Use the population figures when your numbers are the entire group you care about (a whole class, a full year of sales). Use the sample figures when your numbers are a sample drawn from a larger group (a survey of 200 voters, a batch of 30 parts) — dividing by n−1 corrects the small downward bias of a sample. With large n the two are nearly identical.
How are the quartiles calculated?
This calculator uses Tukey’s hinges, the most common textbook method: sort the data, split it into a lower half and an upper half (leaving the median out when the count is odd), then take the median of each half. For 2, 4, 4, 4, 5, 5, 7, 9 the halves are 2, 4, 4, 4 and 5, 5, 7, 9, so Q1 = 4 and Q3 = 6. Other methods (Excel’s inclusive/exclusive quartiles, for example) can differ by a little on small data sets — that is normal, not an error.
What does it mean if there is no mode?
It means every value in your data appears exactly once, so no value is more frequent than any other. That is common with continuous measurements (heights, times, prices). A data set can also have two or more modes — for example 1, 2, 2, 3, 3 has modes 2 and 3 — and this calculator lists them all.
How do I enter my data?
Paste or type numbers separated by commas, spaces, semicolons or new lines — any mix works. Decimals and negative numbers are fine. Anything that is not a number is ignored and reported (for example a stray word or a currency symbol), so check the note under the results if the count looks wrong.
What is the coefficient of variation?
It is the standard deviation expressed as a percentage of the mean: SD ÷ mean × 100. It lets you compare spread across data sets with different scales — a SD of 2 means something different for test scores averaging 90 than for incomes averaging 50,000. It is only meaningful when the mean is not near zero.