Measures of Dispersion for Grouped Data
How to Read the median from an ogive
Use an ogive (cumulative frequency curve) to find the median and quartiles of grouped data.
Before you start
- Reading values off a graph's scale accurately
- Finding the upper boundary of a class (e.g. 30.5 for 21–30)
- Building cumulative frequency as a running total
- Knowing the median is the middle value of the data
When to use it
Use an ogive (cumulative frequency curve) to find the median and quartiles of grouped data.
The steps
- Build a cumulative frequency table by adding each frequency to the running total.
- Plot cumulative frequency against the upper boundary of each class.
- Join the points with a smooth curve.
- For the median, find half the total frequency on the vertical axis.
- Draw across to the curve and down to the horizontal axis to read the value.
Worked example
The marks of 30 students are grouped as 1–10 (3 students), 11–20 (6), 21–30 (9), 31–40 (8), 41–50 (4). Use an ogive to estimate the median mark.
- Build the cumulative frequency by adding each frequency to the running total: 3, 3+6=9, 9+9=18, 18+8=26, 26+4=30.
- Plot cumulative frequency against the upper boundary of each class: (10.5, 3), (20.5, 9), (30.5, 18), (40.5, 26), (50.5, 30).
- Join the points with a smooth curve (the ogive).
- For the median, find half the total frequency on the vertical axis: 30 ÷ 2 = 15.
- Draw across from 15 to the curve, then down to the horizontal axis; it meets at about 27, so the median is about 27 marks.
A second example, with a twist
Instead of the median you read two values, the lower and upper quartiles at ¼ and ¾ of the total, then subtract them to get the interquartile range. The times (in seconds) of 40 runners are grouped as 0–5 (4), 5–10 (10), 10–15 (14), 15–20 (8), 20–25 (4).
Use an ogive to estimate the interquartile range.
- Build the cumulative frequency: 4, 4+10=14, 14+14=28, 28+8=36, 36+4=40.
- Plot cumulative frequency against the upper boundary of each class: (5, 4), (10, 14), (15, 28), (20, 36), (25, 40), and join with a smooth ogive.
- For the lower quartile, use ¼ of the total on the vertical axis: 40 ÷ 4 = 10; read across and down to get about 8 s.
- For the upper quartile, use ¾ of the total: 3 × 40 ÷ 4 = 30; read across and down to get about 16 s.
- Interquartile range = upper quartile − lower quartile ≈ 16 − 8 = 8 s.
Formulae you may need
Formula pages
Practise this in a KBAT problem
Frequently asked questions
Why plot against the upper boundary and not the midpoint?
Cumulative frequency counts everything up to and including a class, and that count is only complete at the class's upper boundary. Plotting there lines the curve up correctly; using the midpoint would shift every point and give a wrong median.
My median off the ogive is a bit different from my friend's, is one of us wrong?
Not necessarily. An ogive gives an estimate read by eye, so small differences are normal and answers within a reasonable range are accepted.
Draw a large, smooth curve and use a ruler for the lines to keep your reading as accurate as possible.
What's the difference between the upper limit and the upper boundary?
For the class 21–30 the upper limit is 30, but the upper boundary is 30.5, halfway between 30 and the next class's 31. Cumulative frequency graphs use boundaries, so plot 30.5, not 30, to keep the classes touching with no gaps.