Measures of centre — mean, median, mode
Three different answers to the same question 'what is typical?'.
Mean. Arithmetic average.
For grouped data, is the midpoint of each class.
- Strengths: uses every value, mathematically tractable, links to variance.
- Weaknesses: pulled by outliers; can be misleading when data is skewed.
Median. Middle value of the ordered data.
- Raw data of size : median = -th value (interpolating between two values when is even).
- Grouped data: locate position in the cumulative frequency, then linear-interpolate within the median class.
Strengths: robust to outliers; meaningful for skewed data. Weakness: ignores most of the data.
Mode. Most frequent value (or modal class for grouped data).
- Strength: works for categorical/qualitative data.
- Weakness: may not exist or may be multiple. Least informative for continuous data.
Choosing. Symmetric, no outliers mean. Skewed or outliers present median. Categorical mode.
- Mean uses all data — sensitive to outliers.
- Median = middle — robust.
- Mode = most frequent — best for categorical data.
- For grouped data, mean uses midpoints; median uses linear interpolation.