What Is Descriptive Statistics?
Descriptive statistics describe a data set without drawing conclusions beyond it. They cover the centre (mean and median), the spread (range, variance, standard deviation, interquartile range) and the shape (skewness and kurtosis). This calculator reports all of them together so you do not have to switch between tools.
The results include both sample and population versions of the variance and standard deviation, the five-number summary, the standard error of the mean, the coefficient of variation and an outlier check based on the 1.5 × IQR rule. Anyone searching for a summary statistics calculator, five number summary calculator or IQR calculator will find those values here.
The sections below explain each statistic, give the formulas, work through a small example and answer common questions such as when to report the median instead of the mean.
How to Use the Descriptive Statistics Calculator
- Paste or type your numbers, separated by commas, spaces or line breaks.
- Read the primary results at the top, then scroll through the full list for spread, shape and outliers.
- Check the box plot and histogram before trusting any single number. A mean can hide a bimodal shape or an extreme value.
- Use the worked steps to verify the main calculations by hand.
Descriptive Statistics Formula
Mean
Sample variance and standard deviation
Interquartile range and outlier fences
Skewness (moment coefficient g₁)
Excess kurtosis
Descriptive Statistics Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| x̄ | Mean | Σx ÷ n |
| Q1, Q3 | First and third quartiles | 25th and 75th percentiles |
| IQR | Interquartile range | Q3 − Q1 |
| s, σ | Sample and population standard deviation | n − 1 versus n |
| g₁ | Skewness | Asymmetry of the distribution |
Descriptive Statistics Worked Example
Using the example values (press “Load example” above to try them):
- Sort the data: 4, 8, 8, 9, 12, 14, 15, 15, 16, 19, 21, 23, 27, 33, 42.
- Mean = Σx ÷ n = 266 ÷ 15 = 17.7333. Median = 15.
- Sum of squared deviations Σ(x − x̄)² = 1466.93; sample variance = 1466.93 ÷ 14 = 104.781.
- Quartiles by the median-of-halves method: Q1 = 9, Q3 = 23, IQR = 14. Outlier fences = Q1 − 1.5·IQR = -12 and Q3 + 1.5·IQR = 44.
Result: Count (n) = 15; Mean = 17.7333; Median = 15; Sample standard deviation (s) = 10.2363.
Which measure of centre and spread should you use?
The right summary depends on the shape of your data and on what you want to say. The mean uses every value and is the natural partner of the standard deviation, but it is pulled by outliers. The median and IQR resist outliers and are the better pair for skewed data such as incomes, waiting times or house prices.
| Data situation | Centre | Spread |
|---|---|---|
| Roughly symmetric, no outliers | Mean | Standard deviation |
| Skewed or with outliers | Median | IQR |
| Categorical or most-common value matters | Mode | Number of categories |
| Comparing variability across different scales | Mean | Coefficient of variation |
Reading skewness and kurtosis
Skewness measures asymmetry. A value near zero means the data are balanced around the mean; a positive value means a long right tail, so the mean sits above the median; a negative value means a long left tail. As a rough guide, absolute skewness below 0.5 is approximately symmetric, 0.5 to 1 is moderate, and above 1 is strong.
Excess kurtosis compares the tails with a normal distribution (which has excess kurtosis 0). Positive values mean heavier tails and more extreme values than a normal curve would produce; negative values mean lighter tails. Both statistics are unstable for small samples, so treat them as descriptions, not proof of non-normality.
| Statistic | Value | Interpretation |
|---|---|---|
| Skewness | −0.5 to 0.5 | Approximately symmetric |
| Skewness | 0.5 to 1 (or −1 to −0.5) | Moderately skewed |
| Skewness | beyond ±1 | Strongly skewed |
| Excess kurtosis | near 0 | Normal-like tails |
| Excess kurtosis | above 0 | Heavy tails, more outliers |
| Excess kurtosis | below 0 | Light tails, fewer outliers |
How the outlier check works
The calculator flags a value as a potential outlier if it lies more than 1.5 interquartile ranges below Q1 or above Q3. This is Tukey’s rule, the same one used to draw box plot whiskers. A flagged value is a prompt to investigate, not an instruction to delete it. It may be a data entry error, a different population, or a genuine extreme observation that carries real information.
Why quartiles can differ between tools
There is more than one accepted way to compute quartiles. This calculator uses the median-of-halves method: split the sorted data at the median (excluding the median itself when n is odd) and take the median of each half. Excel’s QUARTILE.INC, R’s default type 7 and some textbooks use interpolation and can differ slightly, particularly for small samples. The differences are small and shrink as n grows.
Where Descriptive Statistics Is Used
First look at a new data set
Getting centre, spread and shape before choosing a method.
Reports and assignments
Producing the summary table that sits at the top of an analysis.
Outlier screening
Flagging values beyond 1.5 × IQR from the quartiles.
Choosing mean or median
Comparing the two to see whether skew or outliers are pulling the mean.
Common Descriptive Statistics Mistakes to Avoid
- Always look at the histogram: two very different data sets can share the same mean and standard deviation.
- Report the sample standard deviation (n − 1) for samples and the population one only when you have every member of the population.
- Do not remove outliers just to improve the numbers. Report results with and without them if their status is unclear.
- Skewness and kurtosis need dozens of observations to be reliable.
Descriptive Statistics Terms and Definitions
- Quartile
- A value that splits sorted data into four equal-count groups.
- IQR
- Interquartile range, Q3 minus Q1: the spread of the middle half of the data.
- Mode
- The most frequently occurring value.
- Outlier
- An observation unusually far from the rest of the data.
Descriptive Statistics FAQs
What does a descriptive statistics calculator do?
It summarises a data set with measures of centre (mean, median, mode), spread (range, IQR, variance, standard deviation) and shape (skewness, kurtosis), and can flag outliers.
What is the difference between sample and population standard deviation?
The sample version divides by n − 1 to correct for estimating the mean from the same data; the population version divides by n. Use the sample version for sample data.
What is a five-number summary?
The minimum, Q1, median, Q3 and maximum. It is exactly what a box plot draws, and it is shown here through the quartile and range results.
Why are my quartiles different from Excel?
Different software uses different quartile methods. This tool uses the median-of-halves method; Excel’s QUARTILE.INC interpolates. Results usually differ only slightly.
What counts as an outlier here?
Any value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR.
How many values can I paste?
Up to 5,000 numbers at once, separated by commas, spaces or new lines.
When is the mean a poor summary?
When the data are strongly skewed or contain extreme values. In those cases the median and IQR describe the typical case better.