01 · Running totals
Class frequencies are 3, 7, 5, 2. Find the cumulative frequencies.
Hint
Add each new count to the preceding total.
Worked solution
3, 10, 15, 17. The total sample size is 17.
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Cumulative frequency diagrams
Build running totals, plot at true upper class boundaries and distinguish known cumulative points from assumptions used between them. Original graph model and worked practice.
Before you startFrequency tables, class boundaries and coordinate plotting.
01 / Accumulate the counts
Each point includes everything in the earlier classes.
Cumulative frequency = running total of frequencies.
For a less-than diagram using [lower, upper) classes, plot cumulative counts at the upper boundaries.
The horizontal axis shows the measured variable. The vertical axis counts observations below the given boundary. The final cumulative frequency is the total number of observations.
Build a running totalExplore
Classes [0,10), [10,20), [20,40) have frequencies 4, 8, 8. Plot cumulative counts at upper boundaries.
At the lower starting boundary: cumulative count 0.
Any straight segment between known boundary counts is a within-class interpolation model, not a record of exact individual values.
02 / Build a cumulative table
Add each class once.
Classes [0,10), [10,20), [20,40) have frequencies 4, 8, 8.Worked example
Below 10: 4
Only the first class contributes.
Below 20: 4 + 8 = 12
Include both first and second classes.
Below 40: 4 + 8 + 8 = 20
The final cumulative frequency is n = 20.
02 · Recover frequencies
Cumulative frequencies at consecutive upper boundaries are 5, 14, 18, 30. Recover the class frequencies.
Hint
Subtract consecutive cumulative counts, taking the initial count as zero.
Worked solution
5, 9, 4, 12. These sum to 30.
03 / Plot boundaries, not midpoints
The cumulative point refers to a threshold.
Watch: frequencies become cumulative boundary points
Pause, replay or seek freely. The notes explain the same idea and stay in view.
Use points (upper boundary, cumulative frequency).
Midpoints are useful for grouped means; they are not the horizontal coordinates here.
03 · Choose points
Classes [5,15), [15,25), [25,45) have frequencies 2, 6, 4. List the three upper-boundary points.
Hint
Cumulative frequencies are 2, 8, 12.
Worked solution
(15,2), (25,8), (45,12). Do not replace the boundaries by midpoints 10, 20, 35.
04 · Rounded readings
Masses rounded to the nearest gram are grouped as 10–14, 15–19 and 20–24 grams, with frequencies 3, 5, 2. Which horizontal coordinates go with cumulative counts 3, 8, 10?
Hint
Convert the rounded integer labels to true class boundaries.
Worked solution
14.5, 19.5 and 24.5 grams. The initial lower boundary is 9.5 grams, subject to the stated rounding convention.
04 / Include the starting zero
Do not force the graph through the origin.
05 · Initial point
For question3, where should the initial zero-frequency point be?
Hint
The lowest class starts at 5.
Worked solution
(5,0), assuming all observations lie in the listed classes. There is no reason to start at (0,0).
06 · What zero means
Does an initial point (5,0) prove that one observation equals 5?
Hint
The vertical coordinate is a count below a threshold.
Worked solution
No. It says there are no observations below 5 under the table’s coverage. The actual minimum may be 5 or greater.
05 / Join without inventing exact values
Interpolation needs an assumption.
A smooth cumulative curve is often drawn through the points. It must remain nondecreasing and inside the known cumulative limits. Straight line segments give a piecewise linear model, corresponding to uniform spread within each class. Follow the requested graph convention and treat readings inside a class as estimates.
07 · A point inside a class
Using the main table and straight interpolation, estimate the cumulative count below 15.
Hint
15 is halfway from 10 to 20.
Worked solution
Estimate 4 + ½×8 = 8. The boundary counts are known, but this interior count assumes uniform spread within the class.
08 · Curve goes backwards
A smoothed curve falls slightly between two boundary points. Why is that impossible for a cumulative count?
Hint
Raising a threshold cannot remove observations already counted.
Worked solution
Cumulative frequency must be nondecreasing. Use a monotone curve or the stated straight interpolation; do not allow smoothing to introduce a decrease.
06 / Deal with empty and unequal classes
A zero class gives a flat section.
09 · Empty class
Classes [0,5), [5,10), [10,20) have frequencies 4, 0, 6. Give the cumulative points and describe the middle section.
Hint
The second class adds zero.
Worked solution
Starting at (0,0), plot (5,4), (10,4), (20,10). The middle section is horizontal because no observations lie in [5,10).
10 · Unequal widths
In the main example, the last two frequencies both equal 8, but widths are 10 and 20. Are their straight cumulative segments equally steep?
Hint
Slope is change in cumulative count divided by horizontal width.
Worked solution
No. Slopes are 8/10 = 0.8 and 8/20 = 0.4 observations per unit. The wider class rises by the same amount over twice the horizontal distance.
07 / Check totals and scales
The graph should agree with its table.
11 · Too high
A table contains 24 observations, but the last plotted cumulative point has height 31. Is that consistent?
Hint
The final cumulative count is the sum of all listed frequencies.
Worked solution
No, if the diagram covers exactly that dataset. Recheck additions, transcription and the axis scale.
12 · Unbounded class
The highest occupied class is “40 or more”. Can a final finite upper-boundary point be plotted without more information?
Hint
There is no given finite upper boundary.
Worked solution
No. You can plot earlier known boundary counts, but a finite final endpoint requires additional information or an explicitly justified assumption.
08 / Interpret inequalities precisely
Boundary conventions determine what has been counted.
13 · At a shared boundary
With [0,10) and [10,20), does an observation exactly equal to 10 contribute to the cumulative count below 10?
Hint
The first interval excludes 10.
Worked solution
No. It belongs to the second class and is not strictly below 10. State conventions if exact ties at boundaries matter.
14 · Individual data
Does connecting the cumulative points reveal the exact order and value of every observation?
Hint
Grouping has already removed within-class detail.
Worked solution
No. The boundary counts are retained, but the distribution within each class is unknown. A line or curve between points is a model for estimation.
09 / Totals at boundaries
Keep data and interpolation separate.
Add frequencies cumulatively, plot true upper boundaries, include the correct lower starting point and use labelled scales. Check the graph never decreases and finishes at the sample size; label interior readings as estimates.
Section 1 of 9 · Accumulate the counts