Descriptive Statistics: Definition, Measures and a Worked Example
Reviewed by Prof. Dr. Alexander Holtermann · Updated 16 October 2026 · 5 min read
Descriptive statistics summarize your data with numbers, tables and charts: sample size, central tendency (mean, median) and spread (SD, IQR).
Key takeaways
- Descriptive statistics describe the data you have; inferential statistics (tests, confidence intervals) draw conclusions about a larger population.
- A complete description has three parts: the sample size n, a measure of central tendency (mean or median) and a measure of variability (standard deviation or interquartile range).
- For roughly symmetric metric data report M and SD; for skewed data or data with outliers add or prefer the median and the interquartile range.
- Shape matters too: skewness tells you whether the tail is longer on the right (positive) or left (negative); a histogram or box plot shows it at a glance.
- In APA style, descriptive statistics come first in the results section, usually in a table with M, SD and n for every variable or group.
What are descriptive statistics?
Descriptive statistics condense a data set into a few numbers, tables and charts so that a reader can understand it quickly. They answer questions such as: How many people took part? What is a typical value? How much do the values differ? Are there unusual cases?
They differ from inferential statistics, which use a sample to make statements about a population, for example with a t-test or a confidence interval. Descriptive statistics make no such claim. They simply describe what you observed. Every quantitative thesis needs them, even if the main analysis is a regression model, because readers and examiners want to see the raw picture before the tests.
| Question | Measure | Examples |
|---|---|---|
| How many? | Frequency | n, counts, percentages |
| What is typical? | Central tendency | Mean, median, mode |
| How different are the values? | Variability (dispersion) | Standard deviation, variance, range, interquartile range |
| What does the distribution look like? | Shape | Skewness, kurtosis, histogram, box plot |
| Where does a value stand? | Position | Percentiles, quartiles, z-scores |
Worked example: weekly study hours
Twelve students report how many hours they studied in the week before an exam:
4, 6, 7, 8, 8, 9, 10, 11, 12, 13, 15, 25
You can paste exactly these numbers into the descriptive statistics tool and get the values in the table. Quartiles are computed with linear interpolation, the default in R and Python and in most statistics software; other methods can give slightly different quartiles for small samples.
| Measure | Calculation | Result |
|---|---|---|
| n | count the values | 12 |
| Mean (M) | 128 / 12 | 10.67 |
| Median (Mdn) | average of 6th and 7th value: (9 + 10) / 2 | 9.50 |
| Mode | most frequent value | 8 |
| Standard deviation (SD) | √(Σ(x − M)² / (n − 1)) | 5.47 |
| Range | 25 − 4 | 21 |
| Quartiles (Q1, Q3) | linear interpolation | 7.75, 12.25 |
| Interquartile range (IQR) | 12.25 − 7.75 | 4.50 |
| Skewness | adjusted Fisher-Pearson coefficient | 1.70 |
Outliers: the 1.5 × IQR rule
A common rule (Tukey’s fences, used in box plots) flags values as potential outliers if they lie more than 1.5 interquartile ranges below Q1 or above Q3:
lower fence = Q1 − 1.5 × IQR, upper fence = Q3 + 1.5 × IQR
In the example: 12.25 + 1.5 × 4.50 = 19.0 and 7.75 − 6.75 = 1.0. The value 25 lies above the upper fence and is flagged.
Do not delete outliers automatically. First check whether the value is an error (typo, impossible answer). If it is a genuine but extreme case, keep it and report how robust your results are. Without the 25, the mean drops to 9.36 and the SD to 3.23, while the median barely moves (9.00). This is exactly why the median is called a robust statistic.
Which measures should you report?
The right summary depends on the level of measurement and the shape of the distribution.
- Always report n, and for every group if you compare groups.
- Report the number of missing values and how you handled them.
- Round sensibly: usually two decimals for M and SD; percentages to whole numbers or one decimal.
| Data | Central tendency | Spread | Chart |
|---|---|---|---|
| Nominal (e.g., major) | Mode | Frequencies, percentages | Bar chart |
| Ordinal (e.g., single Likert item) | Median | IQR, min–max | Bar chart of frequencies |
| Metric, roughly symmetric | Mean | Standard deviation | Histogram |
| Metric, skewed or with outliers | Median (plus mean) | IQR (plus SD) | Box plot |
Charts that describe distributions
A good chart often says more than a table. Choose the chart that fits the variable.
- Histogram: bars for value ranges of a metric variable; shows shape, skewness and gaps.
- Box plot: median, quartiles, whiskers and outliers in one compact figure; ideal for comparing groups.
- Bar chart: frequencies of categories (nominal or ordinal). Do not use pie charts with more than a few categories.
- Scatter plot: relationship between two metric variables, the starting point for correlation and regression.
How to report descriptive statistics in APA style
APA 7 recommends a table when you have more than a few values. In the text, report the key values with the statistic symbol, which is italicized in the final document: M, SD, Mdn, n. Use a leading zero for values that can exceed 1 (M = 0.85) and no leading zero for values that cannot (r = .45, p = .032).
Doing this in SPSS, R, JASP or Excel
Note that programs use slightly different formulas for quartiles in small samples, so the IQR can differ a little between them.
| Program | How to run it |
|---|---|
| SPSS | Analyze → Descriptive Statistics → Frequencies, Descriptives or Explore (by group) |
| R | summary(df$x), sd(df$x), psych::describe(df) |
| JASP / jamovi | Descriptives; tick the statistics and plots you need, split by a grouping variable |
| Excel | Functions such as AVERAGE, MEDIAN, STDEV.S, or Analysis ToolPak → Descriptive Statistics |
Frequently asked questions
What is the difference between descriptive and inferential statistics?
Descriptive statistics summarize the data you collected (mean, SD, percentages). Inferential statistics use the sample to draw conclusions about a population, for example with significance tests and confidence intervals.
Which descriptive statistics do I need in my thesis?
At least n, a measure of central tendency and a measure of spread for every main variable, usually M and SD, and for skewed data also the median and IQR. Add frequencies and percentages for categorical variables such as gender or major.
Should I report the mean or the median?
Report the mean for symmetric metric data. For skewed data, ordinal data or data with outliers, the median describes the typical value better. Many researchers report both.
What does a skewness of 1.7 mean?
A positive skewness means a long tail to the right: most values are low, a few are very high. Values beyond about ±1 indicate clear skewness; around 0 means roughly symmetric.
Can I compute descriptive statistics without statistics software?
Yes. For small data sets you can compute them by hand or in a spreadsheet, or paste your numbers into the free descriptive statistics tool, which shows n, M, Mdn, SD, quartiles, skewness and a histogram.
Sources and further reading
- Gravetter, F. J., & Wallnau, L. B. (2017). Statistics for the Behavioral Sciences (10th ed.). Cengage.
- Agresti, A., & Finlay, B. (2018). Statistical Methods for the Social Sciences (5th ed.). Pearson.
- American Psychological Association. (2020). Publication Manual of the American Psychological Association (7th ed.).
Your university’s examination regulations and style guide always take precedence.
Related articles
- Mean, median, modeThe mean is the arithmetic average, the median the middle value, the mode the most frequent value. Use the median for skewed data or outliers.
- Standard deviation and varianceThe standard deviation is the typical distance of values from the mean. Variance is its square. Sample formula: SD = √(Σ(x − M)² / (n − 1)).
- Levels of measurementVariables are nominal, ordinal, interval or ratio. The level decides which averages, charts and statistical tests you are allowed to use.
- Testing for normalityCheck normality with a histogram or Q-Q plot, skewness and kurtosis, and the Shapiro-Wilk test. With large samples, plots matter more than p-values.
- Tables and figuresIn APA 7, label tables and figures in bold (Table 1), add an italic title in title case above, put notes below, and cite any source in a copyright note.
- Statistics software comparedSPSS and Stata are menu- or command-based commercial packages, R and Python are free code, JASP and jamovi are free point-and-click. Choose by field and task.
