# Unit 18 Section 2 : Measures of Central Tendency

In this section we will look at three different types of 'average'. These are the mean, the median and the mode.
They are used by statisticians as a way of summarising where the 'centre' of the data is. ## Finding the Mean, Median and Mode

We want to work out the mean, median and mode for the data below.
5, 9, 12, 4, 5, 14, 19, 16, 3, 5, 7

The Mean
To calculate the mean, we need to add all the values up and divide by the number of values.
 5 + 9 + 12 + 4 + 5 + 14 + 19 + 16 + 3 + 5 + 7 = 99 = 9  11 11
In this case the mean is 9 which is one of the values in the list.
Sometimes the mean will not appear in the original list. It might even be a decimal value.

The Median
To calculate the median, we need to put the numbers in order and find the middle value.
34555
 7
912141619
Here the median is 7 because this is the middle value.
Half of the other values in the list are below 7 and half are above 7.

The Mode
To calculate the mode, we need to look at which value appears the most often. It can help if the numbers are in order.
34
 5 5 5
7912141619
In this list the mode is 5, because it appears most often.
Sometimes there will be more than one mode, because two or more values appear the same number of times.

For the list: "5, 9, 12, 4, 5, 14, 19, 16, 3, 5, 7", the mean is 9, the median is 7 and the mode is 5.

## Finding the median where there are an even number of values

When there are an even number of values, there is no clear middle value.

For example, what is the median of: 3, 6, 7, 8, 11, 15?

In this case, there are two middle values.
36
 7 8
1115
The median is the mean of these two middle numbers.
 7 + 8 =7.5 2
So the median for this set of values is 7.5.
Like the mean, the median value does not always appear in the original list of values.

## Example Question

Look at this list of values:
 9 8 5 9 12 8 7 6 5 9
Work out the answer to each of the questions below then click to see whether you are correct.
You may find it helpful to do some working out on paper separately.
 (a) What is the mean of the values above? (b) What is the median of the values above? (c) What is the mode of the values above? ## Exercises

Work out the answers to the questions below and fill in the boxes. Click on the button to find out whether you have answered correctly. If you are right then will appear and you should move on to the next question. If appears then your answer is wrong. Click on to clear your original answer and have another go. If you can't work out the right answer then click on to see the answer.

Question 1
Work out the mean, median and mode of each set of numbers below:
 (a) 4 4 6 8 5 The mean is  The median is  The mode is  (b) 6 7 7 7 7 5 6 2 9 8 The mean is  The median is  The mode is  (c) 8 4 3 3 5 7 The mean is  The median is  The mode is  (d) 6 6 7 7 4 9 1 7 10 1 The mean is  The median is  The mode is  Question 2
The owner of a shoe shop recorded the sizes of the feet of all the customers who bought shoes in his shop in one morning.
These sizes are listed below:
 8 7 4 5 9 13 10 8 8 7 6 5 3 11 10 8 5 4 8 6
(a) What is the mean of these values?  (b) What is the median of these values?  (c) What is the mode of these values?  Question 3
Eight people work in a shop. Their hourly rates of pay are:
 £4 £14 £6 £5 £4 £5 £4 £4
(a) Work out the mean, median and mode for the values above.
Mean = £  Median = £  Mode = £  (b) The owner of the shop wants to argue that the staff are paid well. Which measure would they use?  (c) The staff in the shop want to argue that they are badly paid. Which measure would they use?  Question 4
The table below gives the number of accidents each year at a particular road junction:
 1991 1992 1993 1994 1995 1996 1997 1998 4 5 4 2 10 5 3 5

(a) Work out the mean, median and mode for the values above.
Mean = accidents  Median = accidents  Mode = accidents  (b) A road safety group want to get the council to make this junction safer.
Which measure will they use to argue for this?  (e) The council don't want to spend money on the road junction.
Which measure will they use to argue that safety work is not necessary?  Question 5
One day the number of minutes that trains were late to arrive at a station was recorded.
The times are listed below:
 0 7 0 0 1 2 5 0 0 0 6 0 1 52 0 10 1 1 8 22
(a) Work out the mean, median and mode for the values above.
Mean = late trains  Median = late trains  Mode = late trains  (b) Which measure would you use to argue that too many trains are late each day?  Question 6
Mr Hall grows two different types of tomato plant in his greenhouse.
One week he keeps a record of the number of tomatoes he picks from each type of plant.
 Day Mon Tue Wed Thu Fri Sat Sun Type A 5 5 4 1 0 1 5 Type B 3 4 3 3 7 9 6

(a) Calculate the mean, median and mode for the Type A plants.
Mean = plants  Median = plants  Mode = plants  (b) Calculate the mean, median and mode for the Type B plants.
Mean = plants  Median = plants  Mode = plants  (c) Which measure would you use to argue that there is no difference between the types?  (d) Which measure would you use to argue that Type A is the best plant?  (e) Which measure would you use to argue that Type B is the best plant?  You have now completed Unit 18 Section 2
 Your overall score for this section is Correct Answers You answered questions correctly out of the questions in this section. Incorrect Answers There were questions where you used the Tell Me button. There were questions with wrong answers. There were questions you didn't attempt.
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