Measures of Central Tendency
Mean, Median, Mode and Range

In the units on Data Collection and Data Presentation, you were looking at ways of collecting and representing data. In this unit, you will go one step further and find out how to calculate statistical quantities which summarise the important characteristics of the data.

The mean, median and mode are three different ways of describing the average.

Worked Examples

1

Find

(a)   the mean     (b)   the median     (c)   the mode     (d)   the range

of this set of data.

5,   6,   2,   4,   7,   8,   3,   5,   6,   6

(a)
Show me

The mean is

=
= 5.2
(b)
Show me

To find the median, place all the numbers in order.

2,   3,   4,   5,   5,   6,   6,   6,   7,   8

As there are two middle numbers in this example, 5 and 6,

median =
=
= 5.5
(c)
Show me

From the list above it is easy to see that 6 appears more than any other number, so

mode = 6
(d)
Show me

The range is the difference between the smallest and largest numbers, in this case 2 and 8. So the range is 8 − 2 = 6.
2

Five people play golf and at one hole their scores are

3,   4,   4,   5,   7

For these scores, find

(a)   the mean     (b)   the median     (c)   the mode     (d)   the range.

(a)
Show me

The mean is

=
= 4.6
(b)
Show me

The numbers are already in order and the middle number is 4. So

median = 4
(c)
Show me

The score 4 occurs most often, so,

mode = 4
(d)
Show me

The range is the difference between the smallest and largest numbers, in this case 3 and 7, so

range = 7 − 3
= 4
3

In a survey of 10 households, the number of children was found to be

4,   1,   5,   4,   3,   7,   2,   3,   4,   1
(a)

State the mode.

Show me

Mode = 4 (as its frequency is highest)
(b)

Calculate

(i)

the mean number of children per household

Show me
Mean = (4 + 1 + 5 + 4 + 3 + 7 + 2 + 3 + 4 + 1) ÷ 10
=
= 3.4
(ii)

the median number of children per household.

Show me

Median: first put the data in numerical order.

1,   1,   2,   3,   3,   4,   4,   4,   5,   7

median = = 3.5
(c)

A researcher says: "The mode seems to be the best average to represent the data in this survey." Give ONE reason to support this statement.

Show me

The mode gives the value that occurs most frequently.
(d)

Calculate the probability that a household chosen at random from those in the survey would have

(i)

exactly 4 children

Show me

p(4 children) = = 0.3 (4 occurs 3 times)
(ii)

more than 4 children.

Show me

p(more than 4 children) = = 0.2 (5 and 7)

Exercises

Find the mean, median, mode and range of each set of numbers below.

(a)

3,   4,   7,   3,   5,   2,   6,   10

mean =

median =

mode =

range =

(b)

8,   10,   12,   14,   7,   16,   5,   7,   9,   11

mean =

median =

mode =

range =

(c)

17,   18,   16,   17,   17,   14,   22,   15,   16,   17,   14,   12

mean =

median =

mode =

range =

(d)

108,   99,   112,   111,   108

mean =

median =

mode =

range =

(e)

64,   66,   65,   61,   67,   61,   57

mean =

median =

mode =

range =

(f)

21,   30,   22,   16,   24,   28,   16,   17

mean =

median =

mode =

range =

Twenty students were asked their shoe sizes. The results are given below.

8,   6,   7,   6,   5,   4,   7,   6,   8,   10
7,   5,   5,   8,   9,   7,   5,   6,   8,   6

For this data, find

(a)
the mean
(b)
the median
(c)
the mode
(d)
the range.

Eight people work in an office. They are paid hourly rates of

£12,   £15,   £15,   £14,   £13,   £14,   £13,   £13

(a)

Find

(i)
the mean £
(ii)
the median £
(iii)
the mode. £
(b)

Which average would you use if you wanted to claim that the staff were:

(i)
well paid
(ii)
badly paid?
(c)

What is the range?

£

A gardener buys 10 packets of seeds from two different companies. Each pack contains 20 seeds and he records the number of plants which grow from each pack.

Company A20520202062020208
Company B17181516181817151718
(a)

Find the mean, median and mode for each company's seeds.

Company A

mean =

median =

mode =

Company B

mean =

median =

mode =

(b)

Which company does the mode suggest is best?

(c)

Which company does the mean suggest is best?

(d)

Find the range for each company's seeds.

Company A:

Company B:

A petrol station owner records the number of cars which visit his premises on 10 days. The numbers are:

204,   310,   279,   314,   257,   302,   232,   261,   308,   217

(a)

Find the mean number of cars per day.

(b)

The owner hopes that the mean will increase if he includes the number of cars on the next day. If 252 cars use the petrol station on the next day, will the mean increase or decrease?

The mean .

In a singing contest, the scores awarded by eight judges were:

5.9   6.7   6.8   6.5   6.7   8.2   6.1   6.3

(a)

Using the eight scores, determine:

(i)
the mean
(ii)
the median
(iii)
the mode
(b)

Only six scores are to be used. Which two scores may be omitted to leave the value of the median the same?

and

The weights, in grams, of seven sweet potatoes are

260,   225,   205,   240,   232,   205,   214

What is the median weight? (to the nearest whole gram)

grams

Here are the number of goals scored by a school football team in their matches this term.

3,   2,   0,   1,   2,   0,   3,   4,   3,   2

(a)

Work out the mean number of goals.

goals
(b)

Work out the range of the number of goals scored.

goals
(a)

The weights, in kilograms, of the 8 members of Hereward House tug-of-war team at a school sports event are

75,   73,   77,   76,   84,   76,   77,   78.

Calculate the mean weight of the team.

kg
(b)

The 8 members of Nelson House tug of war team have a mean weight of 64 kilograms.

Which team do you think will win a tug-of-war between Hereward House and Nelson House?