01 · Midpoints
Find midpoints of [4,10), [10,16), [16,28).
Hint
Average both boundaries in each class.
Worked solution
7, 13 and 22.
Understand · explore · practise
Frequency polygons
Plot class-midpoint polygons, label frequency or density axes correctly and explain the effect of unequal class widths and endpoint conventions.
Before you startClass midpoints, frequencies and histogram density.
01 / Place points at class midpoints
A polygon joins summaries with straight lines.
Class midpoint = (lower boundary + upper boundary)/2.
Pair each midpoint with the vertical quantity specified by the question.
A conventional frequency polygon plots frequencies at midpoints. Joining the centres of histogram tops instead plots frequency densities when the histogram’s vertical axis is density. State which construction you are using; they are not interchangeable for unequal classes.
Choose the vertical quantityExplore
Three classes have frequencies 4, 8 and 8. Change the widths and the quantity plotted at each midpoint.
Frequency points: (5,4), (15,8), (25,8).
The segments connect class summaries; they are not measurements at every intermediate value. This model ends at the first and last midpoint.
02 / Use the centre of each interval
Class boundaries determine horizontal positions.
Classes [0,10), [10,20), [20,40) have frequencies 4, 8, 8.Worked example
Midpoints = 5, 15, 30
The final interval is twice as wide.
Frequency points = (5,4), (15,8), (30,8)
These plot counts per whole class.
Density points = (5,0.4), (15,0.8), (30,0.4)
These join histogram-top centres.
02 · Coordinates
The classes in question 1 have frequencies 6, 12 and 12. Give the frequency-polygon points.
Hint
Use midpoint then frequency.
Worked solution
(7,6), (13,12), (22,12).
03 / Understand equal widths
Counts and densities differ by one constant factor.
Watch: move from bar tops to midpoint joins
Pause, replay or seek freely. The notes explain the same idea and stay in view.
For common class width w, density = frequency/w.
The frequency and density polygons have the same shape after vertical rescaling.
03 · Common width
Classes each have width 5 and frequencies 10, 20, 15. Find their densities.
Hint
Divide every frequency by 5.
Worked solution
2, 4 and 3. Multiplying density heights by 5 gives frequency heights.
04 · Axis label
A polygon joins histogram-top centres with heights 2, 4, 3 measured as density. Should its vertical axis be labelled frequency?
Hint
The values have not been rescaled.
Worked solution
No. Label frequency density, with appropriate units. Shape similarity does not change the meaning of the numbers.
04 / Check unequal widths
Equal class counts need not mean equal concentration.
05 · Two meanings
Classes [0,10) and [10,30) each contain 20 observations. Compare their frequency and density polygon heights.
Hint
Densities use widths 10 and 20.
Worked solution
Frequency heights are both 20. Density heights are 2 and 1: the narrower class has twice the frequency per unit.
06 · Histogram tops
Why does a frequency-height polygon fail to join histogram tops for the model’s unequal bins?
Hint
The last bin’s width is 20 rather than 10.
Worked solution
The last count is 8 but its density is 0.4. A single common vertical scale factor cannot turn all count heights into their histogram-top heights.
05 / State your endpoint convention
Do not invent observed zero counts.
07 · Open ends
Must every polygon be closed to the horizontal axis?
Hint
The task may specify a plotting convention.
Worked solution
No universal endpoint extension should be silently assumed. Follow the stated convention. Ending at the first and last midpoint is valid when no closure is required.
08 · Equal-width closure
A question requires zero-frequency neighbouring classes of width 10 to close a polygon with midpoint positions 5, 15 and 25. Where are the extra points?
Hint
Extend the midpoint sequence by one class on either side.
Worked solution
(−5,0) and (35,0). These are construction points under the given convention, not evidence about observed values beyond the dataset.
06 / Use polygons for supported comparisons
Check axes, bins and denominators.
09 · Different sample sizes
Sample A has 20 of 100 observations in a class; B has 30 of 300. Which has the larger proportion?
Hint
Compare relative frequencies.
Worked solution
A: 20%; B: 10%. The taller count point for B does not imply a larger proportion.
10 · Values between points
A segment passes through (12,6). Does this prove exactly six observations have value 12?
Hint
The point was created by joining grouped summaries.
Worked solution
No. A polygon does not recover individual frequencies inside classes. Its line is a graphical connection, not raw-data evidence.
07 / Audit the construction
Keep it distinct from cumulative frequency.
11 · Midpoint or upper boundary
Which uses midpoint positions: a frequency polygon or a less-than cumulative-frequency diagram?
Hint
Think of a class representative versus a completed running total.
Worked solution
A frequency polygon uses midpoints. A less-than cumulative diagram uses upper class boundaries, with its starting lower boundary at count zero.
12 · Can it decrease?
Can a correct frequency polygon slope down as x increases?
Hint
Class frequencies can decrease.
Worked solution
Yes. Frequencies or densities can rise or fall. A cumulative-frequency diagram, in contrast, must not decrease.
08 / Name both coordinates
Midpoint first; the specified height second.
Calculate midpoint positions, choose the required frequency or density heights and join neighbouring points. Label axes, follow any endpoint convention and avoid treating connecting segments as recovered individual observations.
Section 1 of 8 · Place points at class midpoints