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Unit E4 Section 1
Cumulative Frequency

Cumulative frequencies are useful if more detailed information is required about a set of data. In particular, they can be used to find the median and inter-quartile range.

The inter-quartile range contains the middle 50% of the sample and describes how spread out the data are. This is illustrated in Worked Example 2.

Worked Examples

1

For the data given in the table, draw up a cumulative frequency table and then draw a cumulative frequency graph.

The table opposite shows how to calculate the cumulative frequencies.

A graph can then be plotted using points as shown opposite.

Note

A more accurate graph is found by drawing a smooth curve through the points, rather than using straight line segments.
2

The cumulative frequency graph below gives the results of 120 students on a test.

Use the graph to find:

(a)

the median score,

Since of 120 is 60, the median can be found by starting at 60 on the vertical scale, moving horizontally to the graph line and then moving vertically down to meet the horizontal scale.

In this case the median is 53.
(b)

the inter-quartile range,

To find out the inter-quartile range, we must consider the middle 50% of the students.

To find the lower quartile, start at of 120, which is 30.

This gives

Lower quartile = 45

To find the upper quartile, start at of 120, which is 90.

This gives

Upper Quartile = 67

The inter-quartile range is then

Inter - quartile Range = Upper Quartile − Lower Quartile
= 67 − 45
= 22
(c)

the mark which was attained by the top 10% of the students,

Here the mark which was attained by the top 10% is required.

10% of 120 = 12

so start at 108 on the cumulative frequency scale.

This gives a mark of 78.
(d)

the number of students who scored more than 75 on the test.

To find the number of students who scored more than 75, start at 75 on the horizontal axis.

This gives a cumulative frequency of 105.

So the number of students with a score greater than 75 is

120 − 105 = 15

As in Worked Example 1, a more accurate estimate for the median and inter-quartile range is obtained if you draw a smooth curve through the data points.
3

The table below shows the distribution of marks on a test for a group of 70 students.

(a)

Copy and complete the table to show the cumulative frequency for the distribution.
(b)
(i)

Using a scale of 1 cm to represent 5 marks on the horizontal axis and 1 cm to represent 5 students on the vertical axis, draw the cumulative frequency curve for the scores.
(ii)

What assumption have you made in drawing your curve through the point (0, 0)?

Assumption that no student scored zero.
(c)

The pass mark for the test is 47. Use your graph to determine the number of students who passed the test.

70 − 42 = 28 students passed the test.
(d)

What is the probability that a student chose at random had a mark less than or equal to 30 ?

Probability = = .
4

The heights, in metres, of a random sample of 40 soldiers from a regiment were measured. The heights are summarised in the following table.

(a)

Copy and complete the cumulative frequency column in the table.

The completed cumulative frequency table is given below.
(b)

Construct a cumulative frequency curve for the data.
(c)

Estimate from this cumulative frequency curve:

(i)
the median (Q2) 1.955
(ii)
the upper quartile (Q3) 1.99
(iii)
the lower quartile (Q1). 1.915
(d)
(i)

Compare your values of (Q3 − Q2) and (Q2 − Q1).

Q3 − Q2 = 0.035 , Q2 − Q1 = 0.04
(ii)

What does this indicate about the shape of the distribution?

Q2 − Q1 > Q3 − Q2 , hence (positive) skew to the data; that is, it is not symmetric.

Exercises

Make a cumulative frequency table for each of the three sets of data given below.

Then, for each set of data, draw a cumulative frequency graph.
Use it to find the median and inter-quartile range.

(a)

John weighed each apple in a large box. His results are given in this table.

Weight of
apple (g)		60 < w ≤ 8080 < w ≤ 100100 < w ≤ 120120 < w ≤ 140140 < w ≤ 160
Frequency42833278
(i)
Weight
in grams (w)		Cumulative
Frequency
60 < w ≤ 80
80 < w ≤ 100
100 < w ≤ 120
120 < w ≤ 140
140 < w ≤ 160
(ii)

Median ≈

Inter-quartile range ≈

(b)

Paul asked the students in his class how far they travelled to school each day. His results are given below.

Distance (km)0 < d ≤ 11 < d ≤ 22 < d ≤ 33 < d ≤ 44 < d ≤ 55 < d ≤ 6
Frequency5125653
(i)
Distance
in km (d)		Cumulative
Frequency
0 < d ≤ 1
1 < d ≤ 2
2 < d ≤ 3
3 < d ≤ 4
4 < d ≤ 5
5 < d ≤ 6
(ii)

Median ≈

Inter-quartile range – about (to 1 d.p.)

(c)

An athletics coach recorded the distances students could reach in the long jump event. His records are summarised in the table below.

Length of jump (m)1 < d ≤ 22 < d ≤ 33 < d ≤ 44 < d ≤ 55 < d ≤ 6
Frequency512565
(i)
Length
in metres (d)		Cumulative
Frequency
1 < d ≤ 2
2 < d ≤ 3
3 < d ≤ 4
4 < d ≤ 5
5 < d ≤ 6
(ii)

Median ≈

Inter-quartile range – about (to 1 d.p.)

A consumer group tests two types of batteries using a DVD player.

Lifetime (hours)2 < l ≤ 33 < l ≤ 44 < l ≤ 55 < l ≤ 66 < l ≤ 77 < l ≤ 8
Frequency Type A13102284
Frequency Type B0223860
(a)

Use cumulative frequency graphs to find the median and inter-quartile range for each type of battery.

Type A:

Median ≈

Inter-quartile range ≈

Type B:

Median ≈

Inter-quartile range ≈

(b)

Which type of battery would you recommend and why?

The table below shows how the height of children of a certain age vary. The data was gathered using a large-scale survey.

Height (cm)50 < h ≤ 5555 < h ≤ 6060 < h ≤ 6565 < h ≤ 7070 < h ≤ 7575 < h ≤ 8080 < h ≤ 85
Frequency1003002400130070015050

A doctor wishes to be able to classify children as:

CategoryPercentage of Population
Very Tall5%
Tall15%
Normal60%
Short15%
Very short5%

Use a cumulative frequency graph to find the heights of children in each category.

The heights of children in each category (to the nearest cm) are:

Very Tall < h ≤
Tall < h ≤
Normal < h ≤
Short < h ≤
Very short < h ≤

The histogram shows the cost of buying a certain toy in a number of different shops.

(a)

Use cumulative frequency graph to answer the following questions.

(i)

How many shops charged more than £2.65?

shops
(ii)

What is the median price?

About £
(iii)

How many shops charged less than £2.30?

shops
(iv)

How many shops charged between £2.20 and £2.60?

shops
(v)

How many shops charged between £2.00 and £2.50?

or shops
(b)

Comment on which of your answers are exact and which are estimates.

(i)

(ii)

(iii)

(iv)

(v)

Darrita and Jenine played 40 games of golf together. The table below shows Darrita's scores.

Scores (x)70 < x ≤ 8080 < x ≤ 9090 < x ≤ 100100 < x ≤ 110110 < x ≤ 120
Frequency1415173

Draw a cumulative frequency diagram to show Darrita's scores.

(a)

Use your graph to find

(i)

Darrita's median score,

(ii)

the inter-quartile range of her scores.

About
(b)

Jenine's median score was 103. The inter-quartile range of her scores was 6.

(i)

Who was the more consistent player?

(ii)

The winner of a game of golf is the one with the lowest score. Who won most of these 40 games?

Mr Isaacs is checking the time taken for documents to be word-processed. The first 12 documents which he samples take the following times, in minutes, to process.

51213141618192334354043
(a)

Calculate

(i)
the median time,
(ii)
the interquartile range.

His completed survey gives him the following information.

Time
(minutes)		0 - 10 - 20 - 30 - 40 - 50 - 60 - 70 - 80
Number of
documents		35121416721
(b)

Complete the cumulative frequency table for this information.

Time less than
(minutes)
Number of
documents		3820
(c)

Use your cumulative frequency table to draw a cumulative frequency graph.

Use your graph to find

(i)
the median time,
(ii)
the interquartile range.

Information

A quartile is one of 3 values (lower quartile, median and upper quartile) which divides data into 4 equal groups.

A percentile is one of 99 values which divides data into 100 equal groups.

The lower quartile corresponds to the 25th percentile. The median corresponds to the 50th percentile. The upper quartile corresponds to the 75th percentile.