Descriptive Statistics Calculator

The full summary of a dataset, with its denominators and quartiles stated

Paste or type your numbers and this calculator returns the complete descriptive picture: count, sum, mean, median, mode, quartiles, range, interquartile range, variance and standard deviation, mean absolute deviation, standard error and coefficient of variation, plus the sorted data, a histogram and a box plot. Two conventions that pages usually hide are printed here instead: variance is shown with BOTH denominators (n for a population, n − 1 for a sample, labelled, since the choice changes the answer), and quartiles use the median-of-halves method, named in the method so the figures can be reproduced by hand. Frequencies are supported for grouped tallies: enter value:count pairs when one number stands for many.

Disclaimer: This calculator gives estimates for general information only. It is not financial, medical or other professional advice. Quartiles follow the median-of-halves (Tukey) method stated in the text; other conventions can differ slightly on small datasets.

What this calculator does

Descriptive statistics is the discipline of summarising without yet inferring: where the centre is (mean, median, mode, each honest about different things), how far the data spread (range, IQR, variance, SD, MAD), and how the shape looks (histogram and box plot, computed from the same numbers rather than sketched). The page computes on the expanded dataset (frequencies multiply their values out), uses the numerically stable two-pass route for variance (centre first, then squared deviations, so large values with small spreads do not lose their precision to floating-point cancellation), and prints its conventions because “the standard deviation” of the same six numbers is legitimately two different figures until someone says which denominator they meant.

How to use it

  • Any separators work in plain format. Commas, spaces, tabs and line breaks are all read as boundaries; a stray letter or an empty gap is reported with its position rather than silently dropped, since a silently dropped value is a changed dataset.
  • Sample versus population changes the headline, not the data. Both variance rows always print; the selector only chooses which one headlines, and the right choice is about what the data ARE: everything that exists (population) or a draw from something larger (sample).
  • Frequencies are for tallies. Exam marks counted per mark, defects counted per day: value:count lines expand exactly, and the count n in the results is the expanded count, which the results show so the expansion can be checked.
  • Read the pictures with the numbers. The histogram’s bin count follows the square-root rule and is stated; the box plot marks points beyond 1.5 × IQR as individual dots, the standard convention, named so the dots are interpreted rather than admired.

The method

  • Mean = sum / n. Median = the middle of the sorted data (mean of the middle pair for even n). Mode = the most frequent value or values; all tied modes are listed, and “no mode” is said when every value occurs once.
  • Quartiles (Tukey hinges): split the sorted data at the median; Q1 is the median of the lower half, Q3 of the upper half; IQR = Q3 − Q1. Box-plot fences at 1.5 × IQR; points outside are plotted individually and the whiskers end at the farthest points inside the fences.
  • Variance two-pass: mean first, then the sum of squared deviations; population variance divides by n, sample variance by n − 1 (undefined for n = 1, and reported so). SD is the root of each. MAD = the mean of the absolute deviations from the mean. Standard error = sample SD / √n. CV = SD / mean as a percentage.
  • Histogram bins = the ceiling of √n bins across the range, counts by bin, drawn to scale.

Worked examples

4, 8, 15, 16, 23, 42. n = 6, sum = 108, mean = 18, median = 15.50 (the middle pair 15 and 16 averaged), no mode, range 38. Quartiles: lower half 4, 8, 15 gives Q1 = 8; upper half 16, 23, 42 gives Q3 = 23; IQR = 15. Squared deviations sum to 910: population variance 151.67 (SD 12.32), sample variance 182.00 (SD 13.49), the denominator’s whole effect visible in one dataset. MAD = 58/6 = 9.67. The mean sitting above the median is the 42 pulling, and the box plot shows it as a long upper whisker rather than an outlier (the fence is at 23 + 22.5 = 45.5, just beyond it).

A frequency example. Marks tallied as 10:3, 12:5 expand to eight values (three 10s, five 12s): mean 11.25, median 12, mode 12. Grouped data entered honestly beats eight lines typed hopefully, and the expanded count printed in the results is the check that the tally was read as intended.

Reading your results

Compare mean and median first: their gap is the skew’s signature, and the honest centre for lopsided data is usually the median, which is why both print. Read SD against the mean (the CV does it as a percentage) rather than alone: an SD of 12 is tight for incomes and wild for temperatures. Read the IQR as the middle half’s width, the robust spread that outliers cannot inflate, and when SD and IQR tell different stories, believe both: the data have a calm centre and wild edges, which is a finding, not a contradiction.

Limitations

  • Quartile conventions genuinely differ. Tukey hinges are used and named; percentile-interpolation methods (the spreadsheet default) give slightly different quartiles on small datasets. Neither is wrong; they are different definitions, and exam boards specify theirs.
  • The mode of continuous data is fragile. Measurements rounded differently have different modes or none; the page reports what the entered values support and does not bin silently to manufacture one.
  • Nothing here infers. Confidence intervals, tests and regression live on their own pages (linked below); descriptive figures describe THIS dataset, and generalising from them is a step this page deliberately does not take.
  • Very large datasets are capped by the browser. Tens of thousands of values summarise happily; beyond that, statistical software is the right instrument and the page’s error will suggest it if parsing fails.

Frequently asked questions

When do I use n and when n − 1?

Divide by n when your data are the whole population you care about (every employee’s salary, every day of last year) and by n − 1 when they are a sample standing in for more. The n − 1 (Bessel’s correction) compensates for the sample mean being fitted to the sample, which otherwise understates the spread on average. This page prints both with labels and lets the selector choose the headline, because the figures differ exactly when the sample is small, which is when the choice matters most.

Why are there so many ways to compute quartiles?

Because “the value a quarter of the way through” is ambiguous for finite data: include the median in both halves or neither, interpolate between neighbours or take the nearest. Tukey’s hinges (used here) split at the median and take the halves’ medians; spreadsheets typically interpolate by rank percentile. On large datasets they converge; on six numbers they visibly differ, and the professional habit is to name your method, as this page does in its method section and its disclaimer.

What is the difference between SD and standard error?

The standard deviation describes the spread of the DATA (how far individual values stray from the mean); the standard error describes the precision of the MEAN (how far sample means would stray from the true mean across repeated samples), computed as sample SD over √n. Confusing them flatters results: error bars drawn with the SE look tighter because they are answering an easier question. Both print here with their names in full.

What does the box plot show?

A box from Q1 to Q3 with the median lined inside it (the middle half of the data, boxed), whiskers to the farthest points within 1.5 × IQR of the box, and any points beyond drawn as dots, conventionally called outliers. It is the five-number summary (minimum inside fence, Q1, median, Q3, maximum inside fence) as a picture, and beside the histogram it gives the dataset’s shape at two resolutions: the box for the robust skeleton, the histogram for the flesh.

Related calculators

Method: two-pass variance with both denominators printed, Tukey-hinge quartiles, 1.5 × IQR box-plot fences, square-root-rule histogram bins, and frequency expansion for tallied data. Last reviewed: 15 October 2026.