01 · Target only
A cumulative diagram ends at 80 observations. What vertical level locates the median?
Hint
Half of 80.
Worked solution
Cumulative count 40. The median measurement must then be read on the horizontal axis.
Understand · explore · practise
Reading cumulative frequency diagrams
Estimate medians and percentiles, read counts below and between thresholds and compare unequal sample sizes using cumulative percentages.
Before you startCumulative frequency construction and linear interpolation.
01 / Choose which axis you know
Percentiles start with a count; thresholds start with a value.
For percentile p of n observations, target cumulative count = pn/100.
Move across from that count to the curve, then down to the measurement axis.
The measurement you read is an estimate when the graph joins grouped boundary counts. Its accuracy depends on the grouping, interpolation model and plotting precision.
Across to the graph; down to the valueExplore
Twenty journey times are grouped as [0,10), [10,20), [20,40) minutes with counts 4, 8, 8. Straight segments assume uniform spread in each class.
Target cumulative count = 2; estimated 10th percentile = 5 minutes.
Read a measurement from the horizontal axis, not the percentile rank from the vertical axis.
02 / Read the halfway count
The median target is n/2 for grouped interpolation.
For the model, n = 20 and the median target is 10.Worked example
The target lies between cumulative counts 4 and 12
Use the class [10,20).
It is (10 − 4)/(12 − 4) = 3/4 through that count increase
The interpolation uses the fractional target.
Median estimate = 10 + 3/4×10 = 17.5 minutes
Read a measurement, not the vertical count 10.
02 · Read a segment
A straight cumulative segment joins (20,12) to (40,20). Find the horizontal value at cumulative count 16.
Hint
16 is halfway between 12 and 20.
Worked solution
30, halfway between 20 and 40. It is an estimate under the straight-segment model.
03 / Read two values before subtracting
IQR is measured horizontally.
Watch: a percentile count leads to a measurement
Pause, replay or seek freely. The notes explain the same idea and stay in view.
Q₁ uses n/4; Q₃ uses 3n/4. IQR = Q₃ − Q₁.
The vertical rank difference n/2 is not the IQR.
03 · Quartiles from the model
Estimate Q₁ and Q₃ for the model and hence its IQR.
Hint
Targets are 5 and 15.
Worked solution
Q₁ ≈ 10 + (5 − 4)/8×10 = 11.25. Q₃ ≈ 20 + (15 − 12)/8×20 = 27.5. IQR ≈ 16.25 minutes.
04 · A mistaken IQR
A learner subtracts quartile ranks 15 − 5 and reports IQR 10 minutes. Explain the error.
Hint
The vertical axis counts observations.
Worked solution
The difference 10 is a count, not a time interval. Read the two horizontal quartile estimates first, then subtract them.
04 / Generalise the target
Keep fractional ranks for grouped interpolation.
05 · A fractional target
For n = 30, what cumulative count is used to estimate the 25th percentile? Should it be rounded to rank 8?
Hint
Grouped interpolation uses pn/100 directly.
Worked solution
Use 7.5. Do not apply the separate listed-data quartile convention that rounds a non-integer position up.
06 · Upper percentile
Use the model to estimate the 90th percentile.
Hint
Target count 18 lies in the last segment.
Worked solution
Estimate 20 + (18 − 12)/8×20 = 35 minutes.
05 / Start from a measurement
Read up to the curve, then across to a cumulative count.
07 · Below a threshold
Estimate how many model journeys took less than 15 minutes.
Hint
Read halfway along the segment from (10,4) to (20,12).
Worked solution
Estimated count 8. The exact grouped table does not determine the individual count below 15 without a within-class assumption.
08 · At least a threshold
Estimate how many model journeys took at least 25 minutes.
Hint
First estimate the count below 25, then subtract from 20.
Worked solution
Estimated count below 25 = 12 + (25 − 20)/20×8 = 14. Thus about 20 − 14 = 6 took at least 25 minutes. Strict versus inclusive wording matters if exact observations tie at a boundary.
06 / Subtract cumulative counts
Use the same counting convention at both ends.
09 · Between boundaries
In the model, how many observations lie in [10,20)?
Hint
Both endpoints are original class boundaries.
Worked solution
Cumulative count below20 minus count below10 = 12 − 4 = 8, exactly the class frequency.
10 · Between interior values
Estimate the number in [15,30) using the straight graph.
Hint
Estimated cumulative counts are 8 and 16.
Worked solution
About 16 − 8 = 8 observations. This interior-interval result is an estimate, unlike question 9.
07 / Compare unequal sample sizes fairly
Raw cumulative counts are not proportions.
11 · Two groups
At a threshold, group A has cumulative count 30 out of 40; group B has 60 out of 100. Which has the larger proportion below the threshold?
Hint
Divide each count by its own total.
Worked solution
A: 75%; B: 60%. B has more observations below it in absolute number, but A has the larger proportion.
12 · Percentage axis
A cumulative percentage diagram reaches 100%. At what vertical levels should Q₁, median and Q₃ be read?
Hint
The count normalisation is already built into the axis.
Worked solution
25%, 50%, 75%. Read the corresponding horizontal measurements; no further multiplication by n is needed.
08 / Respect what the graph can tell you
Smoothness does not create extra information.
13 · Excessive precision
A hand-drawn graph has horizontal divisions of one minute. Is an estimate of 17.483926 minutes justified by visual reading alone?
Hint
Reading accuracy is limited by scale and drawing.
Worked solution
No. Report precision supported by the graph. Algebraic interpolation can give a more precise model calculation, but the grouped-data assumption still limits its meaning.
14 · Box-plot endpoints
Can a grouped cumulative diagram alone give exact raw minimum and maximum observations for a box plot?
Hint
The outer plotted values may only be class boundaries.
Worked solution
Generally no. Label estimated quartiles and boundary-based endpoints appropriately; do not call the first and last class boundaries exact observed extrema.
09 / Keep counts and measurements distinct
Choose the direction before reading.
For percentiles, start from a cumulative count or percentage. For counts, start from a measurement. Subtract cumulative counts for intervals, normalise when sample sizes differ, and keep interpolation assumptions and precision visible.
Section 1 of 9 · Choose which axis you know