Descriptive Statistics Calculator
Calculate every descriptive statistic for a data set at once: mean, median, mode, variance, standard deviation, skewness, kurtosis, and quartiles.
One data set, every statistic
This is the all-in-one descriptive statistics page: paste in a data set once, and get every commonly used summary statistic back, grouped into four families:
| Family | What it answers | Measures on this page |
|---|---|---|
| Center | Where is the "middle" of the data? | Sum, mean, median, mode, midrange |
| Dispersion | How spread out is the data? | Range, variance (pop/sample), standard deviation (pop/sample), MAD, RMS, SEM, coefficient of variation |
| Position | Where does a value rank within the sorted data? | Min, max, Q1, Q2 (median), Q3, IQR |
| Shape | Is the distribution symmetric, and how heavy are its tails? | Skewness, kurtosis |
For a full step-by-step breakdown of any single family, this calculator links out to the focused calculators below the results — this page trades step-by-step depth for breadth, showing everything at once.
The deviation-to-a-power thread
Four of the measures above — mean absolute deviation, variance, skewness, and kurtosis — are really the same idea repeated at increasing powers. Each one starts from the same building block, the deviation of a value from the mean (xᵢ − x̄), and differs only in what power that deviation is raised to before averaging:
| Measure | Power of (xᵢ − x̄) | Why that power |
|---|---|---|
| Mean absolute deviation (MAD) | 1st (as |xᵢ − x̄|) | Absolute value avoids positive/negative cancellation without changing units |
| Variance | 2nd | Squaring also avoids cancellation, and weights larger deviations more heavily |
| Skewness | 3rd | Cubing preserves sign, so it can detect which side of the mean is more stretched out |
| Kurtosis | 4th | A 4th power grows fastest of all, so kurtosis is dominated by the most extreme values (the tails) |
The advanced measures, one at a time
For the example data set 42, 54, 65, 47, 59, 40, 53 (7 values, mean ≈ 51.4286):
Midrange — the average of the minimum and maximum: (40 + 65) ÷ 2 = 52.5. It's the fastest possible "center" estimate, but only uses two of the seven values, so a single outlier can swing it a lot.
Mean absolute deviation (MAD) — the average distance from the mean, without squaring: 7.2245. It answers "on average, how far off is a typical value?" in the data's own original units, and it's less sensitive to extreme outliers than variance or standard deviation since it doesn't square the deviations.
Root mean square (RMS) — the square root of the average of the squared values themselves (not squared deviations from the mean): 52.1043. RMS shows up constantly in physics and engineering (RMS voltage, RMS speed) because squaring before averaging correctly handles quantities that alternate sign.
Standard error of the mean (SEM) — the sample standard deviation divided by √n: 9.034 ÷ √7 ≈ 3.4147. SEM estimates how much the sample mean itself would likely vary if you repeated the sampling — a smaller SEM means the sample mean is a more precise estimate of the true population mean.
Coefficient of variation (CV) — the standard deviation expressed as a percentage of the mean: (8.364 ÷ 51.4286) × 100 ≈ 16.26%. Because CV is unit-free, it's the right tool for comparing the relative spread of two data sets measured in different units or wildly different scales — standard deviation alone can't do that fairly.
Skewness — how asymmetric the distribution is: 0.1040 here, close to 0, meaning this data set is nearly symmetric (a longer right tail would push skewness positive; a longer left tail, negative).
Kurtosis — how heavy the distribution's tails are, relative to a normal (bell-curve) distribution, which has a kurtosis of exactly 3: 1.3426 here, well below 3, meaning this small data set has noticeably thinner tails and a flatter peak than a normal distribution.
Where a full summary is useful
Running every measure at once is handy for a first look at unfamiliar data — quality-control samples, survey results, exam scores, sensor readings — before deciding which single measure actually matters for the question at hand. Once you know which family you need, the dedicated calculators go deeper on that one: mean, median & mode for central tendency, standard deviation for population-vs-sample variance and spread, quartile & IQR for the exclusive-method quartiles and outlier fences, percentile for linear-interpolation percentiles and percentile rank, and z-score for how many standard deviations a single value sits from the mean.
Frequently asked questions
- What does "descriptive statistics" mean, and what does this calculator report?
- Descriptive statistics summarize a data set with a handful of numbers instead of listing every value. This calculator reports all four families at once — center (sum, mean, median, mode, midrange), dispersion (range, variance, standard deviation, MAD, RMS, SEM, coefficient of variation), position (min, max, quartiles, IQR), and shape (skewness, kurtosis) — so you get the full picture in a single pass instead of running several calculators separately.
- Why does this page show both population and sample variance/standard deviation?
- Every data set has two valid formulas for variance and standard deviation, and which one applies depends on whether your data IS the entire group you care about (population, divide by n) or is a sample drawn from a larger group (sample, divide by n − 1, Bessel's correction). Since this calculator can't know which applies to your data, it reports both side by side.
- Why are skewness, kurtosis, or the coefficient of variation sometimes shown as "—" instead of a number?
- All three divide by a standard deviation (skewness and kurtosis by the sample standard deviation cubed and to the fourth power; CV by the mean). When every value in your data set is identical, the standard deviation is exactly 0, and dividing by 0 is undefined — mathematically a "0/0" indeterminate form, not a large or small number. This calculator shows "—" for that case instead of a nonsensical result. Variance and standard deviation themselves are still reported correctly as 0 in that case, since "no spread" is a perfectly meaningful answer for those two.
- What is the "deviation raised to a power" idea behind MAD, variance, skewness, and kurtosis?
- Four different measures all start from the same building block — how far each value sits from the mean (xᵢ − x̄) — and differ only in what power they raise that deviation to before averaging. Mean absolute deviation uses the 1st power (as an absolute value, to avoid cancellation). Variance uses the 2nd power (squaring also avoids cancellation, and penalizes big deviations more). Skewness uses the 3rd power, which is sensitive to the direction of asymmetry since cubing preserves sign. Kurtosis uses the 4th power, which is dominated by the most extreme values since a 4th power grows very fast.
- Which quartile method does this calculator use — exclusive or linear interpolation?
- The exclusive method — the same one used by the dedicated quartile & IQR calculator. It finds the overall median, then the median of the lower half and the median of the upper half (excluding the overall median from both halves when the count is odd). The dedicated percentile calculator instead uses linear interpolation between ranked values, a different, equally valid convention — the two can disagree slightly on small data sets, which is expected, not a bug.
- Why do I need at least two numbers?
- Several of the measures on this page — sample variance, sample standard deviation, the standard error of the mean, skewness, and kurtosis — divide by (n − 1) or by a sample-based standard deviation, so at least two values are required for every measure on the page to be defined.