E9.6 Cumulative frequency diagrams
- Syllabus
- 0580–2028–2029
- Topic
- E9.6
- Level
- Extended
Learning objectives
E9.6.1Draw and interpret cumulative frequency tables and diagrams.• Draw and interpret cumulative frequency tables and diagrams.E9.6.2Estimate and interpret the median, percentiles, quartiles and• Estimate and interpret the median, percentiles, quartiles and interquartile range from cumulative frequency diagrams. Plotted points on a cumulative frequency diagram should be clearly marked, for example as small crosses (×), and be joined with a smooth curve.
Build and read a cumulative frequency diagram
Cumulative frequency is a running total: at each boundary it counts every observation up to and including that boundary.
| Class | Frequency | Upper boundary | Cumulative frequency |
|---|---|---|---|
| 0<x≤10 | 3 | 10 | 3 |
| 10<x≤20 | 5 | 20 | 8 |
| 20<x≤30 | 4 | 30 | 12 |
Add frequencies successively, then plot each upper class boundary against its cumulative frequency as a small clear cross. Include the lower boundary with cumulative frequency 0 when it is known. Label both axes and join the points with a smooth increasing curve; cumulative frequency must never decrease.
number in a<x≤b≈CF(b)−CF(a),number greater than a≈N−CF(a)
Read values from the curve using the axis scales, not from the class frequencies in isolation. Plot at upper boundaries, not class midpoints. Graph readings are estimates, and the final cumulative frequency must equal the total N.
Estimate percentiles, quartiles and IQR from a cumulative curve
A percentile is the data value below which a stated percentage of the observations lie. On a cumulative curve, convert the percentage into a cumulative-frequency position first.
| Measure | Cumulative-frequency position |
|---|---|
| lower quartile Q1 | 0.25N |
| median Q2 | 0.50N |
| upper quartile Q3 | 0.75N |
| pth percentile | 100pN |
Mark the required position on the cumulative-frequency axis, move horizontally to the curve, then move vertically to the data axis. For N=200, the median is read at cumulative frequency 100 and the 80th percentile at 160.
IQR=Q3−Q1
The median divides the ordered observations into two halves. The IQR measures the spread of the middle 50%, so a smaller IQR means the central half is more tightly clustered. Keep graph readings consistent with the scale and report them as estimates.
Do not read the 80th percentile at CF=80 unless N=100. Do not subtract cumulative-frequency positions to obtain the IQR: first read the two data values Q1 and Q3, then subtract them.