Descriptive Statistics
Descriptive Statistics
Descriptive statistics summarise and describe the key features of a dataset — they do not make inferences about the population, but provide a clear, concise picture of the data collected. The two most important aspects of any dataset to summarise are where the data are centred (measures of central tendency) and how spread out they are (measures of dispersion).
Measures of Central Tendency
A measure of central tendency identifies the 'typical' or 'average' value in a dataset.
The mean is calculated by summing all scores and dividing by the number of scores. It uses all values and is the most mathematically sensitive measure — essential for parametric statistical tests. Its key weakness is sensitivity to outliers: a single extreme value can pull the mean far from the typical score (e.g. in {2, 3, 3, 4, 48} the mean is 12 — unrepresentative of the cluster at 2–4).
The median is the middle value when scores are arranged in order (or the mean of the two middle values if n is even). It is unaffected by outliers and is preferred for skewed distributions and ordinal data. In the example above, the median is 3 — far more representative.
The mode is the most frequently occurring value. It is the only appropriate measure for nominal (categorical) data. A dataset may have one mode (unimodal), two modes (bimodal), or none.
| Measure | Calculation | Best used when | Outlier sensitivity |
|---|---|---|---|
| Mean | Sum ÷ n | Interval/ratio data; symmetrical distribution | High |
| Median | Middle value in ordered set | Ordinal data; skewed distributions; outliers present | None |
| Mode | Most frequent value | Nominal/categorical data; bimodal distributions | None |
Measures of Dispersion
A measure of dispersion describes how spread out scores are around the central value.
The range is the simplest measure: maximum value minus minimum value. It is quick to calculate but severely affected by outliers — one extreme score dramatically inflates the range and gives a misleading impression of typical spread.
The standard deviation (SD) measures the average distance of each score from the mean. A small SD means scores cluster tightly around the mean; a large SD means they are widely spread. It uses all data values, making it the most precise and informative measure of dispersion and essential for parametric statistical tests. It is also sensitive to outliers (because the mean it is calculated around is itself distorted by outliers).
In research reports, the mean and SD are always reported together (e.g. M = 14.2, SD = 3.1) to give a complete picture of a distribution's centre and spread.
Key Takeaways
- Descriptive statistics summarise data — measures of central tendency (where data centre) and dispersion (how spread out data are).
- Mean: sum ÷ n — sensitive to outliers; required for parametric tests. Median: middle value — robust to outliers; use for skewed data or ordinal data. Mode: most frequent — only valid for nominal/categorical data.
- Range: max − min — quick but severely distorted by outliers. Standard deviation: average distance of scores from mean — uses all values, most precise, essential for parametric tests.
- A large SD relative to the mean indicates high variability; scores are widely spread. A small SD indicates scores cluster tightly around the mean.
- Mean and SD are always reported together in psychology research reports: e.g. M = 14.2, SD = 3.1.
- Choose the right measure: mean for symmetrical interval/ratio data; median for skewed data or outliers; mode for nominal data.