Introduction
Descriptive statistics summarises and presents data clearly before any deeper analysis is attempted — every later chapter in this subject (probability, inference, regression) assumes the data has first been organised and understood using these basic tools.
1. Types and sources of data
Data is broadly qualitative (categorical, describing a quality — e.g., colour, gender) or quantitative (numerical), and quantitative data is further discrete (countable, distinct values, like number of children) or continuous (any value in a range, like height). Data also comes from different sources: primary data is collected directly for the specific purpose at hand (surveys, experiments), while secondary data is obtained from existing records collected for another purpose — the distinction matters because secondary data may not perfectly match the current question’s needs and its collection methodology may be unknown or unsuitable.
2. Frequency distributions and graphical presentation
A frequency distribution groups data into classes and records how many observations fall into each — the essential first step in making a large raw dataset comprehensible. Suitable graphical presentations include histograms (for continuous grouped data, with bar area proportional to frequency), bar charts (for categorical or discrete data, with bar height proportional to frequency), and frequency polygons or ogives (for showing distribution shape or cumulative frequency respectively) — choosing the graph type appropriate to the data type is itself an examinable skill.
3. Measures of central tendency: mean, median, and mode
The mean (arithmetic average, sum divided by count) uses every data value but is sensitive to extreme outliers, which can pull it away from where most data actually lies. The median (the middle value when data is ordered) is robust to outliers, making it a better summary for skewed data (such as income, where a few very high values would distort the mean). The mode (the most frequently occurring value) is the only measure of central tendency meaningful for purely categorical data, and is useful for identifying the most “typical” or common category or value.
4. Measures of dispersion: range, variance, and standard deviation
Central tendency alone does not describe a dataset fully — two datasets can share the same mean while being very differently spread out. Range (maximum minus minimum) is simple but uses only two data points, making it highly sensitive to a single outlier. Variance measures the average squared deviation of each value from the mean, and standard deviation (its square root) returns this measure to the original units of the data, making it directly interpretable and comparable to the mean itself — a larger standard deviation indicates data more spread out around the mean.
5. Shape measures: skewness and kurtosis
Skewness measures asymmetry in a distribution’s shape: positive (right) skew has a longer tail toward higher values (often pulling the mean above the median), negative (left) skew the opposite. Kurtosis measures the “tailedness” of a distribution relative to a normal distribution — higher kurtosis indicates heavier tails and a sharper peak, meaning extreme values are relatively more common than a normal distribution would predict.
6. Index numbers and time-series basics
An index number expresses a variable’s value relative to a fixed base period (conventionally set to 100), making it easy to track relative change over time (e.g., a price index of 115 means prices have risen 15% relative to the base period). Time-series data (values recorded at successive time points) is often decomposed conceptually into trend (long-term direction), seasonal variation (regular within-year fluctuation), and irregular/random components — recognising which component dominates a given series’ behaviour is the starting point for further time-series analysis.
Key Takeaways
- Data is classified as qualitative or quantitative (discrete/continuous), and by source as primary or secondary; each classification shapes which further techniques are appropriate.
- Choosing the correct central-tendency measure (mean, median, or mode) depends on data type and the presence of outliers or skew, not habit.
- Standard deviation, being in the original units and using every data point, is generally the most informative single measure of spread, though range and variance each have their own uses.
- Skewness and kurtosis describe distribution shape beyond central tendency and spread, indicating asymmetry and tail behaviour respectively.
- Index numbers and basic trend/seasonal/irregular decomposition allow tracking and interpreting change in a variable over time.