Descriptive Statistics Calculator
Before any test, model or argument, a data set deserves a portrait: where is its centre, how wide is its spread, and where do its middle half and its extremes live? This calculator paints the whole portrait from one list of numbers — count, sum, mean, median, mode, range, both standard deviations, variance, quartiles and the interquartile range — using the standard definitions (quartiles by the median-of-halves method), so the summary you quote in a report is the summary every textbook would compute from the same data. Paste your numbers separated by commas or spaces and read the data set’s character in one glance.
The Portrait, Statistic by Statistic
Count (n) and sum — the census and the total; every other line divides by one of them, and n is the first honesty check on any summary (a mean without an n is a rumour). Mean — sum ÷ n, the balance point of the data, pulled by every value including the wildest. Median — the middle value of the sorted list, immune to extremes: half the data lies below it, half above. Mode — the most frequent value (a data set can have several, or none), the only centre that exists for purely categorical data. Range — max minus min, the crudest spread, set entirely by two values. Standard deviation — the typical distance of values from the mean, in the data's own units; its square is the variance. Quartiles Q1 and Q3 and the IQR — the medians of the lower and upper halves, and their difference: the span of the middle 50%, and the backbone of the box plot. Quote mean with SD for well-behaved, symmetric data; median with IQR when the data skew — the calculator gives you both alliances so the choice is yours and your reader's.
Worked Example: Ten Numbers, Fully Read
The default data — 4, 8, 6, 5, 3, 8, 9, 8, 7, 2 — sorts to 2, 3, 4, 5, 6, 7, 8, 8, 8, 9. Count 10, sum 60, mean 6.0. The median sits between the 5th and 6th values: 6.5 — a little above the mean, because the low tail (the 2) drags the mean down while the median ignores it. The mode is 8, appearing three times. Min 2, max 9, range 7. Squared deviations from 6 sum to 52, so variance is 5.2 (population) or 5.78 (sample), giving SD 2.28 or 2.40 — the two versions the output prints side by side. Halves split at the median: the lower half 2, 3, 4, 5, 6 has median Q1 = 4; the upper half 7, 8, 8, 8, 9 gives Q3 = 8; the IQR is 4. One data set, thirteen numbers, and a reader now knows its centre, its typical wobble, its middle half and its fences — before any chart is drawn or any test proposed.
Sample Versus Population: The n − 1 That Matters
The two standard deviations differ by one quiet denominator, and choosing between them is a statement about what your data is. If the list in front of you is the entire population you care about — every sale this March, every student in this class — divide by n: the population SD. If the list is a sample standing in for a larger reality — 200 voters for an electorate, 30 parts for a production run — the deviations are measured from the sample's own mean, which snuggles closer to the data than the true mean does, shrinking the spread; dividing by n − 1 (Bessel's correction) restores an unbiased estimate of the population's variance. The sample figure is always the larger, the difference fading as n grows (at n = 100 the two differ by half a percent). Most scientific and survey work wants the sample version; most "describe exactly these numbers" work wants the population one. This page prints both, labelled — because a startling share of published confusion is nothing but an unlabelled denominator.
When the Mean Lies: Skew and Outliers
Descriptive statistics is also the art of noticing when a summary misbehaves. Incomes, house prices, reaction times and city sizes are right-skewed: a long upper tail hauls the mean above where most data live, and the median becomes the honest centre (which is why official statistics quote median incomes and median house prices). Outliers deserve investigation before deletion — a value ten SDs from the mean is usually a data-entry error (a misplaced decimal is the classic) but is occasionally the entire discovery, from a contaminated assay to a fraud signal. The descriptive routine that protects you is unglamorous and total: look at min and max for impossibilities, compare mean with median for skew, read the IQR for the middle's width and the range for the extremes' reach, and only then proceed to inference with the data's character known. Every later tool in this hub — regression, tests, intervals — assumes you have done this first.
A Second Portrait: Skew You Can See in the Summary
Run a different animal through the same machine — ten household incomes in thousands: 22, 25, 25, 28, 30, 32, 35, 40, 55, 120. The calculator returns n = 10, sum 412, mean 41.2 — and median 31, mode 25, range 98, population SD 27.8, Q1 = 25, Q3 = 40, IQR 15. Read the story the numbers tell together: the mean sits above nine of the ten values, dragged by the single 120, while the median and mode describe the neighbourhood most households actually live in; the SD (27.8) is nearly as large as the median itself, a signature of spread dominated by one tail; and the IQR of 15 gives the middle half's honest width where the range of 98 only reports the extremes. Nothing here required a chart: the descriptive set, quoted whole, diagnosed the skew, priced it, and chose its own headline statistic (the median) — which is why official income statistics are published exactly this way, and why a mean quoted alone about skewed data should always be read as the beginning of a question.
The Other Average: When Data Multiply, Use the Geometric Mean
The mean this calculator headlines is the arithmetic one, right for added quantities — and wrong for multiplied ones. Growth rates, returns and ratios compound, and their honest centre is the geometric mean: multiply the factors, take the n-th root. The canonical trap: an investment rises 10% one year and falls 10% the next. The arithmetic mean of the returns is 0%, suggesting breakeven; the money says 1.10 × 0.90 = 0.99 — a 1% loss, a geometric mean of √0.99 − 1 ≈ −0.50% per year. Volatility itself taxes compounding, and the gap between the two means measures the tax. Rule of the craft: if the data are levels (heights, scores, prices at a moment), summarise arithmetically; if they are changes chained across periods, the geometric mean is the only average that reconciles with the final value — a reconciliation the arithmetic mean fails every time the numbers move at all.
Related Tools
More from the Statistics Calculators hub: the Coefficient of Determination Calculator for how well a line explains paired data, the Residual Calculator for what a regression leaves behind, the RSE Calculator for a survey estimate's relative precision, and the live Correlation Calculator for the strength of a linear relationship.
Frequently Asked Questions
Mean, median or mode — which centre should I report?
Match the centre to the data's shape and type: mean for symmetric numeric data (it uses every value), median for skewed data or data with outliers (incomes, prices), mode for categorical data or "most common" questions. Reporting mean and median together, and noting their gap, is the professional habit — the gap itself describes the skew.
Why are there two standard deviations?
Population SD (divide by n) describes a complete data set; sample SD (divide by n − 1) estimates a population's spread from a sample, correcting the bias caused by measuring deviations from the sample mean rather than the true mean. The calculator prints both, labelled, so you can pick by what your data represents.
How are the quartiles computed?
By the median-of-halves (Tukey) method: split the sorted data at the median, then Q1 and Q3 are the medians of the lower and upper halves. Spreadsheet percentile functions interpolate instead and can differ slightly on small samples — a known, documented divergence, not an error in either.
What if two values tie for the mode?
The data set is bimodal (or multimodal for more ties) and the calculator lists all of them. Two modes often signal two mixed populations — heights of adults of both sexes is the classroom example — worth splitting and summarising separately.
What does the IQR tell me that the range does not?
The range reports the two most extreme values only, so one outlier can stretch it without bound. The IQR spans the middle 50% and ignores the tails entirely: together they separate "the bulk of the data is wide" from "a few wild values are loose," which call for different responses.
How many numbers can I paste?
The calculator handles ordinary pasted lists comfortably — hundreds to thousands of values. For very large data sets, dedicated statistical software adds the histograms and tests this summary is meant to precede; within this hub, the Descriptive output is the standard first page of any analysis.