Descriptive statistics turn a column of raw numbers into a short, readable profile: where the data center, how widely they spread, whether they lean to one side and whether anything looks unusual. Paste your data once and this calculator fills in more than two dozen statistics, draws a histogram and explains how the headline figures were computed.
How to use the descriptive statistics calculator
- Paste your numbers into the box. Commas, spaces, tabs and line breaks all work, so you can copy a column straight from a spreadsheet.
- Choose Sample if the data are a subset of a larger group, or Population if they are the whole group.
- Read the mean and standard deviation on the tape, then scroll down for the full summary table and the histogram.
Key formulas
Here m₂ and m₃ are the average squared and cubed deviations from the mean. The excess kurtosis uses the fourth power in the same way and subtracts 3.
Worked example
Fifteen students’ scores on a statistics quiz: 52, 61, 58, 70, 66, 49, 73, 64, 58, 81, 67, 55, 62, 58, 76.
- Center: the scores add to 950, so the mean is 950 ÷ 15 ≈ 63.33. The sorted middle (8th) value gives a median of 62, and 58 is the mode, appearing three times.
- Spread: the squared deviations add to about 1,127.33. Dividing by n − 1 = 14 gives a sample variance of about 80.52, so s ≈ 8.97 points. The range is 81 − 49 = 32.
- Quartiles: Q1 = 58 and Q3 = 70, so the IQR is 12. The outlier fences sit at 40 and 88, and every score falls inside them.
- Shape: sample skewness G₁ ≈ 0.38, a mild lean toward higher scores, and excess kurtosis ≈ −0.38, slightly flatter than a normal curve.
The coefficient of variation, about 14.2%, says a typical score sits within roughly a seventh of the mean.
Reading the summary table
Center
Compare the mean and median first. If they are close, the data are roughly balanced; if the mean is noticeably larger, a long right tail is pulling it up. The midrange is quick to compute but depends entirely on the two most extreme values.
Spread
The standard deviation is in the original units and is the usual headline measure of spread. The IQR covers the middle 50% of the data and ignores extremes. Mean absolute deviation averages the plain distances from the mean and is easier to explain than variance, though less common in formal work.
Precision of the mean
The standard error shrinks as the sample grows. Roughly, the true mean lies within about two standard errors of the sample mean when the data are not too skewed — about 63.33 ± 4.6 points in the example.
Shape
As a rough guide, skewness between −0.5 and 0.5 indicates fairly symmetric data, and values beyond ±1 show strong skew. Both shape measures are unstable for small samples, so treat them as hints when n is under about 30.
The histogram uses Sturges' rule for the number of bins, rounded to a tidy bin width. Each bin includes its lower edge and excludes its upper edge.
To focus on one measure with fuller steps, use the standard deviation calculator, the quartile calculator or the statistics formulas sheet.
Frequently asked questions
Should I choose sample or population?
Choose Sample when your data are a subset used to learn about a larger group, which is the usual case for surveys, experiments and quality checks. Choose Population only when the list contains every member of the group you care about. The choice changes the variance, standard deviation, skewness and kurtosis.
How is skewness calculated here?
For a sample the calculator uses the adjusted Fisher–Pearson coefficient G1, the same formula as Excel's SKEW function. For a population it uses g1 = m3 ÷ m2^1.5, where mk is the average of (x − mean)^k. Values near 0 suggest symmetry; positive values mean a longer right tail.
Why is kurtosis shown as a negative number?
The calculator reports excess kurtosis, which subtracts 3 so that a normal distribution scores 0. Negative values mean lighter tails and a flatter peak than a normal curve; positive values mean heavier tails and more outliers.
Which quartile method does the calculator use?
The median-of-halves method that leaves the median out of both halves when n is odd, as on TI-83 and TI-84 calculators. Software such as Excel's QUARTILE.INC can give slightly different values; the quartile calculator lets you compare methods.
How are outliers identified?
With Tukey's 1.5 × IQR rule: any value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR is flagged. It is a screening rule, not proof that a value is wrong.