Statistics
Statistics of grouped data
Find the mean, median and mode of grouped data, and read a cumulative frequency curve.
Once data is grouped into classes, the individual values are gone. A table saying that four people scored between 10 and 20 does not say whether they scored 11 or 19. So grouped statistics works with a stand-in: the class mark, the midpoint of the class, used as though every value in that class sat exactly there. It is an approximation, and it is a good one as long as the values are spread reasonably evenly.
The mean of grouped data
Mean = Σfx ÷ Σf, where x is each class mark and f its frequency. In words: multiply each midpoint by how many values it stands for, add all of those, and divide by how many values there were altogether.
Worked example
Find the mean of: 0–10 with frequency 3, 10–20 with frequency 5, 20–30 with frequency 2.
Class marks: 5, 15 and 25.
Midpoint of each class. Write these above the table before anything else — every column that follows depends on them.
The median needs running totals instead. Add the frequencies down the table to get the cumulative frequency, and find the class where the running total first passes half the data — that is the median class. The mode needs the fullest class and the two classes either side of it, because what the mode formula really measures is how much the peak leans towards one neighbour.
The other two formulas
Median = l + ((n/2 - cf) ÷ f) × h, where l is the median class's lower limit, cf the running total before it, f its own frequency and h its width. Mode = l + ((f₁ - f₀) ÷ (2f₁ - f₀ - f₂)) × h, with f₁ the modal class's frequency and f₀ and f₂ its neighbours'. Both start at the lower limit of a class and step part of the way into it.
Try it together
Find the mean of: 0–20 with frequency 4, 20–40 with frequency 6, 40–60 with frequency 10.
The classes are 20 wide this time, so the class marks are not the small numbers you may be expecting.
1.What is the class mark of the class 20–40?
Have a go
Have a go on your own: what is the class mark of the class 15–35?
Hint: Add the ends together and halve the result.
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.