Measures of Central Tendency and Dispersion
What you must understand
Know the three averages: mean (sum ÷ count), median (middle value of ordered data), and mode (most frequent value, or modal class for grouped data). Understand when each is best: the mean uses all values but is distorted by outliers; the median resists outliers and suits skewed data; the mode suits categorical data and identifies the most common item.
For dispersion, know the range, interquartile range (IQR = upper quartile − lower quartile), semi-interquartile range, variance and standard deviation. The IQR describes the middle 50% and ignores extreme values; the standard deviation uses every value and is expressed in the original units.
For grouped data, calculate the mean using class midpoints: mean = Σ(f × x) ÷ Σf, where x is the midpoint. Locate the median class by cumulative frequency.
Traps that appear in questions
- Forgetting to order the data before finding the median.
- With an even number of values, forgetting to average the two middle values.
- Using class limits instead of midpoints in grouped mean calculations.
- Confusing variance (units squared) with standard deviation (original units).
- Assuming a distribution is symmetric; if mean > median the data are usually positively skewed.
- A data set can have no mode or several modes.
Worked example
Find the mean, median and IQR of 3, 7, 8, 9, 13.
Answer: Mean = 40 ÷ 5 = 8. Median = 8 (third of five ordered values). Lower quartile = 7, upper quartile = 9, so IQR = 9 − 7 = 2. The data are tightly clustered in the middle despite a range of 10.
Rule of thumb: compare mean and median to judge skew; a large gap signals outliers pulling the mean.