Descriptive statistics summarise what a data set looks like before any test or model is applied. They answer three questions: where is the centre (mean, median, mode, midrange), how spread out are the values (range, variance, standard deviation, IQR) and what is the shape of the distribution (skewness, kurtosis, outliers)? Looking at all three together prevents misleading conclusions – a mean alone says nothing about whether a few extreme values dominate.
This calculator produces the full report in one go. Choose whether your numbers are a sample, in which case variance and standard deviation divide by n − 1 and skewness and kurtosis use the adjusted sample formulas, or a complete population, which divides by n. The five-number summary uses the quartile method of your choice, and the 1.5 × IQR rule lists any outliers.
Two charts complete the picture: a histogram shows the shape of the distribution and a box plot shows the quartiles, whiskers and outliers. The summary table can be copied straight into a report.
How to use the Descriptive Statistics Calculator
- Data set
- Data are a
- Quartile method
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The data are sorted once. Sums and moments give the mean, variance, standard deviation, standard error, skewness and kurtosis; positions in the sorted list give the median, quartiles and five-number summary; a frequency count gives the mode(s).
Worked example
Ten ages: 23, 25, 28, 31, 34, 36, 41, 45, 52, 67 (a sample). n = 10, sum = 382, mean = 38.2, median = (34 + 36)/2 = 35, no mode, range = 67 − 23 = 44, midrange = 45. Σ(x − x̄)² = 1,657.6, so s² = 1,657.6 ÷ 9 = 184.18 and s = 13.57; SE = 4.29 and CV = 35.5%. Q1 = 28, Q3 = 45, IQR = 17, fences 2.5 and 70.5 (no outliers). Sample skewness = 1.08 (right-skewed) and excess kurtosis = 0.96.
How to interpret the report
| Look at | What it tells you |
|---|---|
| Mean vs median | If they differ a lot, the data are skewed; report the median. |
| SD vs IQR | SD much larger than IQR ÷ 1.35 suggests heavy tails or outliers. |
| Skewness | |G₁| < 0.5 roughly symmetric, 0.5–1 moderate, > 1 strong skew. |
| Excess kurtosis | Near 0 is normal-like; above 1 means heavy tails. |
| Outliers | Check each one before deciding what to do with it. |
Sample and population formulas
| Statistic | Sample | Population |
|---|---|---|
| Variance | Σ(x − x̄)² ÷ (n − 1) | Σ(x − μ)² ÷ n |
| Standard deviation | √s² | √σ² |
| Skewness | G₁ = g₁·√(n(n − 1))/(n − 2) | g₁ = m₃ ÷ m₂^1.5 |
| Excess kurtosis | G₂ = ((n + 1)g₂ + 6)(n − 1)/((n − 2)(n − 3)) | g₂ = m₄ ÷ m₂² − 3 |
Formula
x̄ = Σx ÷ n · s² = Σ(x − x̄)² ÷ (n − 1) · SE = s ÷ √n · skewness G₁ = √(n(n−1))/(n−2) · m₃/m₂^1.5 · excess kurtosis from m₄/m₂² − 3
Worked example
Inputs — Data set: 23, 25, 28, 31, 34, 36, 41, 45, 52, 67; Data are a: Sample (divide by n − 1); Quartile method: Median of halves, median excluded (TI-84, most textbooks).
Results — Mean (x̄) = 38.2; Count (n) = 10; Sum (Σx) = 382; Median = 35; Mode = No mode – every value appears once; Midrange = 45.
Key terms
- Central tendency
- The typical or middle value: mean, median, mode.
- Dispersion
- How spread out the values are: range, variance, standard deviation, IQR.
- Skewness
- Asymmetry of the distribution.
- Kurtosis
- Heaviness of the tails relative to a normal distribution.
- Standard error
- Standard deviation of the sample mean, s ÷ √n.
Frequently asked questions
What are descriptive statistics?
Numbers that summarise a data set: measures of centre (mean, median, mode), spread (range, standard deviation, IQR) and shape (skewness, kurtosis).
What is the difference between descriptive and inferential statistics?
Descriptive statistics describe the data you have; inferential statistics use a sample to draw conclusions about a larger population, e.g. with confidence intervals and tests.
Should I choose sample or population?
Sample if your data are a subset of a larger group you care about (most research data); population if you have every member of the group.
What do skewness values mean?
About 0 is symmetric; positive means a long right tail, negative a long left tail. Values beyond ±1 indicate strong skew.
What is excess kurtosis?
Kurtosis minus 3, so a normal distribution scores 0. Positive values mean heavier tails and more outliers than normal; negative values mean lighter tails.
What is the standard error?
s ÷ √n, the typical error of the sample mean as an estimate of the population mean. For the example it is 13.57 ÷ √10 = 4.29.
Why are the mean and median different?
Skewed data pull the mean towards the long tail. In the example the mean (38.2) is above the median (35) because of the value 67.
How many values do I need?
At least two; skewness needs three and kurtosis four values.