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Lesson overview

Statistics and data — distributions and diagrams

A frequency polygon is a line graph that shows how frequency changes across class intervals. Points are plotted at the midpoint of each class, and lines join them. A cumulative frequency diagram (ogive) plots running totals: add up all frequencies up to and including each class boundary. The shape of a cumulative frequency curve is typically S-shaped. From a cumulative frequency diagram, you can read the median, lower quartile (Q1), upper quartile (Q3), and identify outliers.

Constructing a frequency polygon

Plot the frequency polygon for test scores grouped into intervals: 0-20 (3 students), 20-40 (7 students), 40-60 (12 students), 60-80 (6 students), 80-100 (2 students).

Interval	Midpoint	Frequency
0-20	10	3
20-40	30	7
40-60	50	12
60-80	70	6
80-100	90	2

MethodPlot points at the midpoint of each interval, then join with straight lines

The midpoint of 0-20 is 10, the midpoint of 20-40 is 30, etc. Always use midpoints, not endpoints.

Constructing a cumulative frequency diagram

Build a cumulative frequency table and diagram for: 0-20 (3), 20-40 (7), 40-60 (12), 60-80 (6), 80-100 (2).

Interval	Frequency	Cumulative Frequency
0-20	3	3
0-40	7	10
0-60	12	22
0-80	6	28
0-100	2	30

MethodAdd up all frequencies up to each class boundary and plot points at the upper boundary

Cumulative means 'running total'. Plot at the upper boundary of each class, not the midpoint.

Reading the median from a cumulative frequency diagram

100 students sit a test, grouped as: 0-20 (10), 20-40 (15), 40-60 (30), 60-80 (25), 80-100 (20). Where is the median?

MethodFind $\frac{n}{2}$ on the cumulative frequency axis, read across to the curve, then down to the value axis

Q1 position $\frac{n}{4}$ (25 in this case)

Q3 position $\frac{3n}{4}$ (75 in this case)

The median is always at cumulative frequency = n/2. Lower quartile at n/4, upper quartile at 3n/4.

Worked example — finding the median and quartiles

A cumulative frequency diagram shows heights of 60 students. Draw the diagram and find the median, Q1 and Q3.

Height (cm)	Frequency	Cumulative Frequency
140-150	8	8
150-160	15	23
160-170	22	45
170-180	10	55
180-190	5	60

From diagramMedian ≈ 163 cm, Q1 ≈ 155 cm, Q3 ≈ 170 cm

List the data in a frequency table with cumulative frequencies.
Plot points at the upper class boundary and cumulative frequency.
Join the points with a smooth, increasing curve (S-shaped).
Find the median position: n/2 = $\frac{60}{2}$ = 30.
On the diagram, find cumulative frequency 30, read across to the curve, then down to find the median height.
Similarly find Q1 (position 15) and Q3 (position 45).
Interquartile range IQR = Q3 - Q1 ≈ 170 - 155 = 15 cm.

Try it

Construct frequency polygons or cumulative frequency diagrams and use them to find medians, quartiles and compare distributions.

Question 1

In a frequency polygon, where should points be plotted?

💡 Frequency polygons plot at the midpoint to represent the centre of each class.

Question 2

In a cumulative frequency diagram, where are points plotted?

💡 Cumulative frequency graphs plot at the upper class boundary because cumulative means 'up to and including'.

Question 3

If a cumulative frequency diagram has a total of 80 students, at what cumulative frequency position is the median?

💡 Median position = n/2 = $\frac{80}{2}$ = 40.

Question 4

Which statistic is found at the cumulative frequency position of n/4?

💡 Q1 (lower quartile) is at position n/4, median at n/2, Q3 at 3n/4.

Question 5

A cumulative frequency diagram has 120 data values. Find the position of the upper quartile (Q3).

💡 Q3 position = 3n/4 = (3 × 120) / 4 = 360 / 4 = 90.

Common mistakes

Watch for these when working through the lesson.

Using class boundaries instead of midpoints in frequency polygons: frequency polygons must plot at the midpoint of each interval.
Plotting at midpoints instead of upper boundaries in cumulative frequency diagrams: these diagrams always plot at the upper class boundary.
Not recognizing that cumulative frequency curves are typically S-shaped: they should be smooth and increasing (never decreasing).

Frequency polygons and cumulative frequency diagrams

Lesson overview

Worked example — finding the median and quartiles

Try it

Common mistakes

Related topics

Independent practice will plug in here