Descriptive Statistics Calculator

Descriptive Statistics — The Complete Guide

Descriptive statistics are the tools you reach for first with any new data set: a handful of numbers that summarize hundreds or thousands of individual values into something you can actually reason about. Before you test a hypothesis, build a model, or make a claim about a wider population, you describe what's in front of you. Get this step wrong — miss an outlier, misread a skewed distribution as symmetric — and every downstream analysis inherits the mistake.

Measures of Central Tendency: Mean, Median, and Mode

The mean (arithmetic average) is the sum of all values divided by how many there are: x̄ = (Σxᵢ) / n. It's the most familiar measure of "typical," and it uses every value in the data set — which is both its strength and its weakness. Every value pulls on the mean, including extreme ones.

The median is the middle value when the data is sorted — or the average of the two middle values if n is even. Unlike the mean, the median only cares about position, not magnitude. This makes it resistant to outliers: a single absurdly large or small value barely moves it.

The mode is the most frequently occurring value. It's the only central-tendency measure that works on categorical (non-numeric) data — you can find the mode of a list of eye colours, but not a meaningful mean or median. A data set can have no mode (every value unique), one mode, or several tied modes.

Worked Example — Mean vs. Median with an Outlier

Problem: Eight employees' commute times, in minutes: 12, 14, 13, 15, 12, 90, 14, 13. The 90 is a genuine outlier — a day someone got stuck in unusual traffic.

Mean: (12+14+13+15+12+90+14+13) / 8 = 183 / 8 = 22.9 minutes

Sorted: 12, 12, 13, 13, 14, 14, 15, 90 → Median = (13+13)/2 = 13 minutes

The mean (22.9) is pulled well above what any single employee actually experiences on a typical day; the median (13) reflects the typical commute far better. This is exactly why household income, house prices, and commute times are usually reported as medians in the news — these distributions are routinely skewed by a small number of extreme values.

Measures of Spread: Range, Variance, and Standard Deviation

Two data sets can share the same mean and look completely different. {48, 50, 52} and {10, 50, 90} both average to 50 — but the second is far more spread out. Range (max − min) is the simplest spread measure, but it only uses two values and is extremely sensitive to a single outlier.

Variance measures the average squared distance of each value from the mean. Squaring does two things: it makes every deviation positive (so they don't cancel out to zero), and it penalizes larger deviations disproportionately. Standard deviation is the square root of variance, which brings the units back to the same scale as the original data — this is why SD, not variance, is the number usually reported and interpreted directly.

Note the sample formula divides by n − 1, not n — this is Bessel's correction, and it matters more than it looks. See the FAQ below for exactly why.

Worked Example — Standard Deviation, Step by Step

Problem: A sample of 5 quiz scores: 6, 8, 10, 12, 14. Find the sample standard deviation.

Step 1 — Mean: (6+8+10+12+14)/5 = 50/5 = 10

Step 2 — Squared deviations: (6−10)²=16, (8−10)²=4, (10−10)²=0, (12−10)²=4, (14−10)²=16 → sum = 40

Step 3 — Sample variance: 40 / (5−1) = 40/4 = 10

Step 4 — Sample SD: √10 ≈ 3.16

Quartiles, IQR, and Detecting Outliers

Quartiles split sorted data into four equal parts. Q1 (the 25th percentile) marks the boundary of the lowest quarter; Q3 (the 75th percentile) marks the boundary of the highest quarter; Q2 is just the median. The interquartile range, IQR = Q3 − Q1, measures the spread of the middle 50% of the data — and unlike range or standard deviation, it's almost completely unaffected by outliers, since it ignores the extreme quarters entirely.

The standard rule for flagging outliers is the 1.5×IQR rule: any value below Q1 − 1.5·IQR or above Q3 + 1.5·IQR is flagged as a potential outlier. This is exactly the rule that produces the "whiskers" and individual dots on a box plot — see our percentiles and quartiles guide for the full five-number-summary breakdown.

Skewness: Reading the Shape of a Distribution

When mean and median are close together, a distribution is roughly symmetric. When the mean is noticeably larger than the median, the data is right-skewed (a long tail of high values, like income or house prices). When the mean is noticeably smaller than the median, the data is left-skewed (a long tail of low values, like age at retirement in a workforce with a hard minimum). Comparing mean and median costs nothing and tells you this at a glance — always report both, not just one.

Common Mistakes

Reporting only the mean. Two data sets with the same mean can have wildly different spreads and shapes — always pair a central-tendency measure with a spread measure.

Using population SD (÷n) on sample data. This is the single most common statistics-homework error — it understates the true variability. If your data is a sample (almost always true outside a full census), use ÷(n−1).

Treating the mode as meaningful on continuous data with no repeats. With truly continuous measurements (like height to two decimal places), every value is often unique, so "no mode" is the correct, honest answer — not a failure of the data.

Frequently Asked Questions