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Get the full descriptive summary of any dataset — mean, median, mode, range, variance and standard deviation, all in one place.
One paste gives you the entire descriptive summary of a dataset — no spreadsheet formulas, no manual sorting. Follow these steps to get a complete, correct picture of your numbers.
Make it a habit to glance at mean, median, and SD together before drawing conclusions — those three numbers tell you the center, the skew, and the spread, which is 90% of descriptive statistics.
Descriptive statistics are numbers that summarize a dataset — they describe what the data looks like without trying to prove anything about a larger population. That second job belongs to inferential statistics (hypothesis tests, confidence intervals, p-values), which always starts from a good descriptive summary. You cannot do inference responsibly without first knowing your mean, spread, and shape.
The centers: the mean (Σx/n) is the balance point of the data, sensitive to every value including outliers; the median is the middle value of the sorted list, immune to extremes; the mode is the most frequent value, the only “average” that works for categorical data like favorite colors. A symmetric dataset has mean ≈ median; a right-skewed one (a few huge values) has mean > median; a left-skewed one has mean < median. The spread: the range (max − min) is instant but fragile; the variance averages squared deviations; the standard deviation (σ or s) returns that spread to original units. Shape extras: quartiles split sorted data into quarters, and the five-number summary (min, Q1, median, Q3, max) plus outlier fences give the full box-plot picture.
The field began with John Graunt’s 1662 mortality tables, grew through Quetelet’s “average man,” Galton’s studies of regression, Pearson’s formalization of the SD and correlation, and Fisher’s foundations of inference. What started as counting burials became the language of science, business, and public policy.
Descriptive stats run the modern world: businesses track average order value and revenue spread; medicine reports trial outcomes as means with SDs; sports compare players via averages and consistency; education summarizes classes with means and medians; quality control watches process means and variation; governments publish median incomes (not means, because billionaires skew). Whenever data must become a decision, description comes first.
Every number in the summary comes from a short, exact recipe:
Count n = 10; sum = 55; mean = 55/10 = 5.5. Sorted already; with even n, median = (5+6)/2 = 5.5. Mode: none — each value appears once. Min 1, max 10, range = 9. Deviations from 5.5: ±4.5, ±3.5, ±2.5, ±1.5, ±0.5; squares: 20.25, 12.25, 6.25, 2.25, 0.25 (each twice); total 82.5. Population variance = 82.5/10 = 8.25; σ = √8.25 ≈ 2.8723. Sample variance = 82.5/9 ≈ 9.1667; s ≈ 3.0277. Mean equals median: perfectly symmetric, as expected.
Count 8; sum = 47; mean = 47/8 = 5.875. Sorted: 3, 4, 5, 5, 6, 7, 8, 9; median = (5+6)/2 = 5.5; mode = 5 (appears twice); min 3, max 9, range = 6. Squared deviations from 5.875: 3.515625, 4.515625, 0.015625, 0.765625, 8.265625, 1.265625, 9.765625, 0.765625; total 28.875. Population variance = 28.875/8 = 3.609375; σ ≈ 1.8998. Sample variance = 28.875/7 = 4.125; s ≈ 2.0308. Mean (5.875) > median (5.5): a slight right skew, pulled by the 9.
A café collects 15 ratings (1–5 stars): 5, 4, 5, 3, 5, 4, 2, 5, 4, 5, 3, 4, 5, 1, 4. Sum = 59; mean = 59/15 ≈ 3.93. Sorted: 1, 2, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5; median (8th value) = 4; mode = 5 (six occurrences); min 1, max 5, range 4.
The pattern mean (3.93) < median (4) < mode (5) diagnoses left skew: most customers love the café, but a few harsh reviews (the 1 and 2) drag the mean down. The owner learns two things: the typical experience is 4–5 stars (median/mode), and the priority is finding what upset the 1-star reviewer, not overhauling a menu most people rate 5 stars. Reporting only the mean (“3.9 stars”) would undersell the business; the full summary tells the true story.
Nine homes sell for (in thousands): 280, 310, 295, 320, 305, 290, 2400, 315, 300 — eight ordinary houses and one mansion. Sum = 4815; mean = 4815/9 = 535 (i.e., $535,000). Sorted: 280, 290, 295, 300, 305, 310, 315, 320, 2400; median (5th) = 305 ($305,000).
The mean exceeds the median by 75% — a single $2.4M outlier inflates the “average” far beyond any normal home’s price. This is exactly why real-estate reports quote median prices: the median ($305k) describes the typical house, while the mean ($535k) describes nobody. Rule of thumb: report the median whenever skew is strong (incomes, house prices, wait times); report the mean for symmetric data (heights, test scores, measurement errors). The gap between them is itself a diagnostic — a large mean–median split always deserves an explanation.
Go deeper with the specialist tools. The Standard Deviation Calculator focuses on spread alone; the Mean / Median / Mode Calculator zooms in on centers; the Probability Calculator powers the inference step; the Z-Score Calculator standardizes any value; and the Percentage Calculator handles rate summaries. The full collection lives in Math Calculators.
This page provides a descriptive statistics calculator for a numeric list. Method: count n, sum Σx, mean Σx/n, median of sorted values (average of two middle for even n), mode (most frequent; none if all unique), min, max, range max−min, population variance Σ(x−μ)²/N, population SD σ, sample variance Σ(x−x̄)²/(N−1), sample SD s. Verified numeric facts: 1–10 gives n=10, sum 55, mean 5.5, median 5.5, mode none, range 9, σ²=8.25, σ≈2.8723, s²≈9.1667, s≈3.0277; 4,8,6,5,3,7,9,5 gives mean 5.875, median 5.5, mode 5, range 6, σ≈1.8998, s≈2.0308; café ratings (n=15) give mean ≈3.93, median 4, mode 5; houses (280–2400k) give mean 535 vs median 305. Inputs accept comma/space/newline-separated decimals and negatives.
Sources: OpenStax, Introductory Statistics (descriptive statistics); NIST/SEMATECH e-Handbook of Statistical Methods; Ronald Fisher, Statistical Methods for Research Workers (1925). All numerical examples were computed by hand and cross-checked against the calculator.
The standard deviation is the single most informative number in the summary, yet also the least intuitive. Read it as the typical miss: for the numbers 1 through 10, σ ≈ 2.87 means a typical value sits about three units away from the mean of 5.5. Judge the SD against the size of the values — an SD of 3 around a mean of 82 (exam scores) signals remarkable consistency, while an SD of 35 around a mean of 40 (monthly sales in thousands) signals near-total unpredictability. A standard deviation close to zero means the values barely differ; a standard deviation larger than the mean means the data are all over the place.
For roughly bell-shaped data, the empirical rule turns the SD into plain English: about 68% of values fall within one SD of the mean, 95% within two, and 99.7% within three. With mean 50 and SD 10, expect two-thirds of observations between 40 and 60, and nearly all of them between 20 and 80. A value more than three SDs from the mean is either extraordinary or an error — either way, it deserves an investigation, not silent inclusion.
When means differ, compare spreads with the coefficient of variation: CV = (s ÷ mean) × 100%. Portfolio X (mean return 8%, SD 4%) has CV 50%; Portfolio Y (mean 12%, SD 9%) has CV 75% — relatively more volatile even though every number is bigger. Because the CV is unit-free, it compares the consistency of exam scores, delivery times, and stock returns on a single scale. As a rule of thumb: CV below 15% is tight, 15–35% is moderate, and above 35% is volatile.
An investor compares two portfolios’ annual returns over 8 years (percent). Portfolio A: 6, 7, 8, 7, 9, 6, 8, 7. Portfolio B: −4, 12, 3, 15, 0, 10, 8, 14. Both sum to 58 — paste either list into the calculator and the mean is 7.25% for both. The difference is entirely in the spread: A never strays far from its mean (population SD ≈ 0.97, CV ≈ 13%), while B swings from −4 to 15 (population SD ≈ 6.46, CV ≈ 89%).
The summary turns an abstract choice into a concrete one: A is the steady earner, B the roller coaster, and the identical means hid two opposite risk profiles. This is exactly why comparing centers without comparing spreads is one of the most expensive mistakes in statistics. Whenever two datasets share a mean, the standard deviation becomes the whole story — and the CV tells you which one to trust with your money.
Numbers that summarize a dataset. For 1–10: mean 5.5, SD ≈ 2.87.
Mean for symmetric data, median for skewed data, mode for categorical questions.
Minimum, first quartile, median, third quartile, maximum.
Real inference usually wants n ≥ 30.
Skew: a few huge values pull the mean up but barely move the median.
Sort (already sorted), take the middle value. With 7 values, the 4th is the median: 12. For an even count, average the two middle values.
Range = max − min = 20 − 5 = 15. It is the simplest measure of spread, though one outlier can inflate it.
Yes. Two modes make it bimodal (e.g., 2, 2, 5, 5, 9 has modes 2 and 5); more than two is multimodal. If every value appears once, there is no mode — like 1 through 10.