Statistics
Using the median; choosing a measure
Use the median in problems and choose the best average for a situation.
From a 'less than' table back to frequencies: subtract each cf from the next one. If 4 girls are under 140 cm and 11 under 145 cm, then 7 are in 140–145.
The classes must touch before you use the median or mode formula, because both use the lower limit l. Classes like 118–126, 127–135 have a gap of 1, so move each limit by half the gap: 117.5–126.5, 126.5–135.5, and so on. (For the mean this isn't needed: the class marks don't change.)
Two missing frequencies? You need two facts:
- the total gives one equation (5 + x + 20 + 15 + y + 5 = 60)
- the given median (or mean) gives the other
Which measure for which job? Each tells a different story:
| Measure | Best for | Weakness |
|---|---|---|
| mean | comparing groups; it uses every value | a few very large or very small values pull it a long way |
| median | a 'typical' value when there are extremes: wages, how much a worker produces | ignores how far out the extremes are |
| mode | the most popular: the favourite TV programme, the most-bought car colour | there may be no clear winner |
Example: in a factory where a few managers earn a lot, the median wage says more about a typical worker than the mean does.
A rough rule: 3 Median ≈ Mode + 2 Mean. It is only approximate (the book calls it 'empirical'). Use it to estimate, not to replace working out each measure when a question asks for it.
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