Descriptive statistics

Descriptive Statistics Calculator

Every summary statistic for one column of numbers, with the distribution drawn beneath it.

Updated August 2026Runs in your browser — nothing is uploadedVerified against R

How to read this result

Read centre, spread and shape in that order. The mean and median say where the data sit; the standard deviation and IQR say how spread out they are; skewness and kurtosis say what shape they make.

When the mean and median disagree, believe the gap. A mean well above the median means a long right tail — a few large values pulling the average up. That is the situation where quoting the mean alone misleads, and it is why incomes are reported as medians.

The standard error is not the standard deviation. The SD describes how much individual values vary and does not shrink as you collect more data. The SE describes how much the mean would vary between samples and falls as 1/√n. Confusing them is the single most common error in reporting.

Skewness beyond about ±1 and excess kurtosis beyond about ±2 are worth acting on: they suggest the mean and standard deviation are not summarising the data well, and that a median-based description or a rank-based test would serve better.

The mean chases the tail and the median does not. That is the whole reason to report both, and the reason median income is quoted rather than mean income.

The formula

\bar{x} = \frac{\sum x_{i}}{n} s^{2} = \frac{\sum (x_{i} - \bar{x})^{2}}{n - 1} S E = \frac{s}{\sqrt{n}}

$\bar{x}$: the sample mean
$s^{2}$: the sample variance — divided by n − 1, not n
$S E$: the standard error of the mean

Computed with Welford's online algorithm rather than Σx² − (Σx)²/n, which loses catastrophic precision when the mean is large relative to the spread.

Worked example

Twelve exam scores

A class of twelve sits an exam. The scores are 72, 85, 78, 90, 68, 95, 88, 76, 82, 91, 79 and 84. What is there to say about them?

Add them up and divide by twelve.
988 / 12 = 82.33
Sort them and take the middle. With an even count, average the sixth and seventh.
68 72 76 78 79 82 | 84 85 88 90 91 95 → (82 + 84)/2 = 83
Sum the squared deviations from the mean, then divide by n − 1.
718.67 / 11 = 65.33
The standard deviation is its square root.
s = 8.08
The standard error of the mean is s divided by √12.
8.08 / 3.464 = 2.33
Quartiles by linear interpolation.
Q1 = 77.5, Q3 = 88.5, IQR = 11

mean 82.33median 83SD 8.08IQR 11SE 2.33

Mean 82.3 and median 83 sit almost on top of each other, which says the scores are roughly symmetric — one of the few cases where either summary would do.

Checked against R's mean, median, sd, var, IQR and quantile(type = 7).

The calculator above is loaded with these numbers by the Load the worked example button.

Assumptions, and when to use something else

Descriptive statistics assume nothing about the data, which is exactly why they come first. They do assume the numbers are what you think they are: a mistyped value, a missing-data code such as −99 or 9999, or two different units mixed in one column will pass silently into every statistic here.

Two habits catch most of that: look at the minimum and maximum first, and look at the histogram before the numbers.

You want just the centreMean, median and modeThe three measures of centre with the differences between them explained.
You want just the spreadStandard deviationSample and population versions with the sum of squares shown.
Something looks extremeOutlier detectionThree rules for flagging unusual values, and what to do about them.
You want to compare two groupsT-testDescriptive statistics describe; a test decides whether a difference is real.

Questions people ask

What is the difference between descriptive and inferential statistics?

Descriptive statistics summarise the data you have. Inferential statistics use that sample to say something about a population you did not measure — with a confidence interval or a hypothesis test quantifying the uncertainty in doing so.

Which statistics should I report?

For roughly symmetric data: n, mean, standard deviation, and the range. For skewed data: n, median, quartiles, and the range. Reporting a mean and SD for badly skewed data is technically correct and practically misleading.

What does skewness of 1.5 mean?

A long right tail — the mean is being pulled above the median by a few large values. As a rough guide, |skew| under 0.5 is near-symmetric, 0.5 to 1 is moderate, and above 1 is substantial enough that median-based summaries and rank-based tests are worth preferring.