Statistics
Cumulative frequency and the median
Find the median of grouped data using cumulative frequency.
The median is the middle value once the data is in order. Half the values are below it and half above. For n values: if n is odd, it's the ((n + 1)/2)th; if n is even, it's the average of the (n/2)th and (n/2 + 1)th.
Cumulative frequency (cf) is a running total: how many values are up to each point.
Two kinds of cf table:
| Kind | Read it as | Example |
|---|---|---|
| less than | how many are below each upper limit | 'less than 20': 2 + 3 + … |
| more than or equal to | how many are at or above each lower limit | '10 or more': everyone except the 0–10 class |
Think it through with common sense: if 80 pupils scored 0 or more and 77 scored 10 or more, then 3 scored between 0 and 10.
Median of grouped data, step by step:
- find n/2
- go down the cf column to the first cf bigger than n/2: that is the median class
- take cf from the row just above it (the class before)
- put the numbers in: Median = l + (n/2 – cf)/f × h
Here l is the lower limit of the median class, f its frequency and h the class size.
Example: 53 pupils, so n/2 = 26.5. The median class is 60–70 (f = 7), and the cf before it is 22. Median = 60 + (26.5 – 22)/7 × 10 ≈ 66.4.
Careful with brackets: if cf is 27 + x, write n/2 – (27 + x). The minus applies to both parts.
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