Cumulative frequency diagrams.
Build running totals, draw a smooth curve and estimate quartiles, percentiles and the interquartile range.
Use upper class boundaries · read median, quartiles and percentiles · calculate and interpret the IQR
Keep a running total of the frequencies
An ordinary frequency tells you how many observations belong to one class. A cumulative frequency keeps a running total, answering how many lie below a sequence of boundaries.
Start with the first class frequency and keep adding the next class. The last cumulative value must equal the total number of observations. Subtracting two cumulative totals recovers the count between their boundaries, because the smaller total is included within the larger.
Cumulative frequency tells you how many observations are below each upper class boundary. Add each frequency to all those before it.
| Time t (min) | Frequency | Cumulative frequency |
|---|---|---|
| 0 ≤ t < 10 | 10 | 10 |
| 10 ≤ t < 20 | 22 | 32 |
| 20 ≤ t < 30 | 22 | 54 |
| 30 ≤ t < 40 | 10 | 64 |
Find the number of journeys shorter than 30 minutes, and the number from 10 to under 30 minutes.
Below 30 minutes: 54. From 10 to under 30 minutes: 54 − 10 = 44.
Pause and explainIf 12 observations are below 10 and 35 are below 20, how many lie from 10 to under 20?
35 − 12 = 23. Remove the earlier total to isolate that interval.
Plot upper boundaries, then join smoothly
A cumulative point belongs at the upper boundary of its class because its total includes that class and all earlier classes. A midpoint would place the whole-class total too early on the horizontal axis.
Begin at the lower boundary with cumulative count zero where appropriate, plot the running totals at their upper boundaries, and join with a smooth non-decreasing curve. The smoothness supports approximate readings between known boundaries; it does not reveal the exact original observations.
Plot (0, 0), (10, 10), (20, 32), (30, 54) and (40, 64). Use the upper boundaries, not class midpoints. Mark each point clearly with a small cross, then join with a smooth increasing curve.
Read t ≈ 20.0 minutes
This is one smooth curve through the grouped totals. Readings are estimates; the original times within each class are unknown.
Pause and explainWhy should a cumulative frequency curve not slope downward?
Adding later observations cannot reduce the number already counted. Its total can rise or stay level, but cannot fall.
Read quartiles, percentiles and the IQR
Quartiles divide an ordered population into quarters. The median is halfway through the observations, the lower quartile is one quarter through, and the upper quartile is three quarters through.
On a cumulative graph, those positions are vertical-axis counts. Find the required fraction of the total, move horizontally to the curve and then down to the measurement axis. The interquartile range subtracts the two quartile values and describes the spread of the middle half of the data.
Median: cumulative frequency N2
Q₃: cumulative frequency 3N4
Interquartile range = Q₃ − Q₁
Here N = 64. Read horizontally from the required cumulative frequency to the curve, then down to the time axis.
Estimate the median, quartiles and interquartile range for the 64 journeys.
At cumulative frequencies 16, 32 and 48, the illustrated curve gives Q₁ ≈ 13.1, median = 20, and Q₃ ≈ 26.9 minutes.
The middle 50% of journey times lie between about 13.1 and 26.9 minutes. Readings and their difference depend on the precision of the graph.
Pause and explainIf N = 80, is the lower quartile read at a measured value of 20?
Not necessarily. Read across from cumulative frequency 20; the corresponding measurement comes from where that level meets the curve.
Read counts from a value, or values from a percentage
A cumulative curve can be read in either direction. Starting with a measurement gives an estimated count below it; starting with a count or percentage gives an estimated measurement.
For a percentile, convert the percentage into a count using the total N before reading across. For a count above a measurement, read the count below and subtract from N. Keep count units and measurement units separate in the final interpretation.
Estimate the 90th percentile of the journey times.
Reading from 57.6 to the curve and then down gives about 32.2 minutes. About 90% of journeys are shorter than this time.
How many of the journeys lasted at least 30 minutes?
For a value between plotted boundaries, read the curve to estimate the cumulative count. Subtract from N to estimate the count above that value.
Pause and explainWhy multiply a percentile percentage by N before using the vertical axis?
The axis is a cumulative count, not a percentage, so the requested proportion must be converted to a number of observations.
Your turn. Choose the correct reading level.
Build and check the cumulative totals before drawing. Plot upper boundaries with clear crosses and a smooth curve on consistent numerical scales.
For every reading, identify which axis you start from and what unit the answer will have. Label the median, quartiles or percentile as an estimate when read from grouped data.
Class frequencies are 5, 9, 12 and 4. Give the cumulative frequencies.
Reveal worked answer
A cumulative frequency curve represents 80 observations. Give the reading levels for Q₁, the median and Q₃.
Reveal worked answer
From a curve, Q₁ ≈ 18 and Q₃ ≈ 31. Estimate the IQR.
Reveal worked answer
A total of 120 observations has cumulative frequency 87 below 50. How many are at least 50?
Reveal worked answer
At what cumulative frequency do you read the 60th percentile for 150 observations?