Unit 16 Section 1 : Averages

In this section we revisit the three types of average, the mean, median and mode. We also use the range of a set of data.

Mean =
Median = middle value (when the data is arranged in order); where there are two central values, the median is their mean
Mode = most common value
Range = difference between largest and smallest values

Example 1

1782365103

For this sample,

(a)

calculate the mean,

Mean = = = 5
(b)

determine the median,

To find the median, first write the numbers in order.

1233567810
Median

As the number of data items is odd, the median will be the middle number, which is 5 in this case, so

Median = 5

(c)

state the mode,

The mode is the most common value, which is 3 for this set of values.
(d)

calculate the range.

Range = 10 – 1 = 9

Example 2

Determine the median of the following set of values:

44328819337462313356

First write the numbers in order:

19313233334456627488

In this case, there are 2 middle numbers, 33 and 44. The median will be the mean of these.

Median = = = 38.5

Example 3

A class collected data on the number of people living in their home, which is shown in the following table:

Number of People
Living in Home
Frequency
23
39
410
52
63
71
81
90
101
(a)

Calculate the mean number of people living in each home.

The first step is to complete the table below:

Number of People
Living in Home
FrequencyNumber of People × Frequency
236
3927
41040
5210
6318
717
818
900
10110
TOTALS30126

Mean = = 4.2 people per home

(b)

Determine the median of the data.

As there are 30 values, the median is the mean of the 15th and 16th values.
From the first table we can see that both the 15th and 16th values are 4, so the median is 4 people per home.

(c)

State the mode of the data.

The most common value is 4 so the mode is 4 people per home.

Exercises

Question 1

Calculate the mean and the range of each of the following sets of data:

(a)
3175612

Mean =

Range =

(b)
304219213362

Mean =

Range =

(c)
78314313891322

Mean =

Range =

(d)
114115110119114118123133

Mean =

Range =

Question 2

Determine the median and the mode of each of the following sets of data:

(a)
85193219

Median =

Mode =

(b)
331416192233163322

Median =

Mode =

(c)
59193142157

Median =

Mode =

(d)
11211911131611192220

Median =

Mode =

Question 3

In which of the following data sets is the mean the same as the median:

A 34 6 19 17 9
B 29 12 17 18 44 13 17 40
C 101 107 183 51 57 77 100 92
D 27 92 56 83 45
Question 4

Which of the following data sets has the largest range:

A 14 27 88 73 56 61
B 374 521 628 314 729
C 888 912 897 907 887 893
Question 5

The following table gives the results of a survey question asking people how many television sets they had in their home.

Number of TelevisionsFrequency
03
118
264
373
422
514
66

For this data,

(a)

calculate the mean,

(b)

determine the median,

(c)

state the mode.

Question 6

A car park manager recorded the number of cars entering her car park each hour. The data she collected is listed below.

16 22 17 6 5 8 32 15 9 7 14 33
21 11 6 5 11 14 12 22 19 11 3 14
14 7 23 41 32 16 5 19 14 33 7 12

For this data:

(a)

calculate the mean,

(to 1 d.p.)
(b)

determine the median,

(c)

determine the mode,

(d)

calculate the range.

The manager should use the to convince her employees that the car park is heavily used and will therefore make large profit.
Question 7

John looks at the price of a computer game in 8 different shops. The prices he sees are:

£29.99 £25.00 £34.99 £29.00
£24.99 £29.99 £31.00 £29.95
(a)

Calculate the mean of this data.

(to the nearest penny)
(b)

State the mode of this data.

(c)

Determine the median.

Which of these averages should he use to argue that the computer game is too expensive?

Question 8

For the set of data given below, calculate the mean and determine the median.

4 7 3 9 5 6 142 3 7 11

Mean =

Median =

Is there any advantages of using the median, rather than the mean in this case?

The advantage of using the median is that the mean is heavily influenced by the one large value, 142, making it unrepresentative of the data set as a whole.

Question 9

A student collected data on the number of visits to the dentist made by members of his class in one school year. His results are shown in the following bar chart:

His results are shown in the following bar chart:

For the data:

(a)

state the mode,

(b)

calculate the mean,

Number of VisitsFrequencyNo. of Visits × Frequency
0
1
2
3
4
5
6
7
8
TOTALS
The mean =
(c)

determine the median.

Question 10

A set of three numbers has mean 11, median 12 and range 13. What are the 3 numbers?

, and