Introduction to Descriptive Statistics
Key concepts, tools, and practical steps for summarizing data
1. What Is Descriptive Statistics?
- Definition – A set of techniques used to summarize, organize, and present data in a meaningful way.
- Purpose – Provide a quick overview of the main features of a dataset without drawing conclusions about a larger population.
- Contrast with Inferential Statistics – Descriptive statistics describe the sample itself; inferential statistics use the sample to make predictions or generalizations about a population.
2. Types of Data
| Data Type |
Characteristics |
Typical Measures |
| Nominal |
Categories without order (e.g., gender, eye color) |
Frequency, mode, proportion |
| Ordinal |
Ordered categories (e.g., rating scales) |
Median, percentiles, mode |
| Interval |
Numerical values with equal intervals, no true zero (e.g., temperature in °C) |
Mean, standard deviation, range |
| Ratio |
Numerical values with a meaningful zero (e.g., height, weight) |
All interval measures + coefficient of variation |
3. Summarizing a Single Variable
3.1. Frequency Distribution
- Absolute frequency (count) – Number of observations in each category or class.
- Relative frequency (proportion) – Count divided by total observations.
- Cumulative frequency – Running total of frequencies; useful for percentiles.
Example Table
| Class / Category |
Frequency (f) |
Relative Frequency (f/N) |
Cumulative Frequency |
| 0‑9 |
5 |
0.10 |
5 |
| 10‑19 |
12 |
0.24 |
17 |
| … |
… |
… |
… |
3.2. Measures of Central Tendency
| Measure |
Formula (for a sample) |
When to Use |
| Mean |
(\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}) |
Symmetric, interval/ratio data, no extreme outliers |
| Median |
Middle value after sorting (or average of two middle values if n is even) |
Skewed distributions, ordinal data |
| Mode |
Most frequently occurring value |
Nominal data, multimodal distributions |
3.3. Measures of Dispersion (Spread)
| Measure |
Formula |
Interpretation |
| Range |
( \max(x) - \min(x) ) |
Total spread; sensitive to outliers |
| Inter‑Quartile Range (IQR) |
( Q_3 - Q_1 ) |
Spread of the middle 50 %; robust to outliers |
| Variance |
( s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} ) |
Average squared deviation from the mean |
| Standard Deviation (SD) |
( s = \sqrt{s^2} ) |
Typical distance of observations from the mean |
| Coefficient of Variation (CV) |
( \frac{s}{\bar{x}} \times 100% ) |
Relative variability; useful for comparing different units |
3.4. Shape of Distribution
- Skewness – Direction and degree of asymmetry.