Tutor’s Toolkit: Application of Number

Carrying Out Calculations

C8	Compare sets of data with a minimum of 20 items

Useful Web Links:

BBC Bitesize Revision: Maths: Measures of central tendency and measures of spread
http://www.bbc.co.uk/schools/gcsebitesize/maths/datahandlingfi/centraltendencyrev1.shtml

BBC Skillswise: Numbers: Understanding averages
http://www.bbc.co.uk/skillswise/numbers/handlingdata/numericalanalysis/mean/

C8.1	Using percentages

Scope: 

Students need to be able to:
•	Make comparisons using percentages. [NOE]

What students need to know:

Make comparisons using percentages.

The most common way of comparing two sets of data is to use percentages. For example, we can compare two TV programmes by saying that 30% of the audience for Programme A is under 20, while for Programme B, 50% are under 20. This tells us that Programme B is more popular with the under 20s than Programme A.

Example Question:
25 people were asked about the TV programmes they watched.
15 said they watched “Favourite TV Moments”.
8 said they watched “TV Greats”.
Make a percentage comparison.

Solution:

15 out of 25 is: 

8 out of 25 is: 

So, 32% watched “TV Greats”, but 60% watched “Favourite TV Moments”.

C8.2	Mean, median and mode

Scope: 

Students need to be able to:
•	Calculate the mean of a set of data given as a list.
•	Calculate the mean of a set of data taken from a table.
•	Interpret a given mean.
•	Calculate the median.
•	Calculate the mode.
•	Be able to select most suitable measure of average.

What students need to know:

Mean, median and mode are all different types of average. They are indications of the centre of a collection of data. A sensible average can be used to represent the data. We can make statements like; “On average it took 20 minutes to walk to town” or “The average crowd at matches was roughly 42 000”. It is important to be able to work out averages from a list of data or taken from a table of frequencies. Also, it is important to choose which average is best in different circumstances.

Calculate the mean of a set of data given as a list

The mean average is the one where you add up all of the values and divide by how many there are.

The average of 12, 10, 7 and 3 is (12 + 10 + 7 + 3)  4 = 32  4 = 8

Calculate the mean of a set of data taken from a table

If the table shows how often each number crops up, then it is called a frequency table.
This table shows the sizes of the shoes worn by a group of teachers.

Shoe Size	Frequency	
5	3	
6	6	
7	9	
8	2	

The table says that 3 teachers wear size 5 shoes, 6 teachers wear size 6 and so on. 

Notice that there are 3 + 6 + 9 + 2 = 20 teachers altogether.
So, the show size list is: 5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8.

Much easier to work out: 
5  3 + 6  6 + 7  9 + 8  2 = 15 + 36 + 63 + 16 = 130
So, the mean average is 130  20 = 6.5.

Interpret a given mean

The mean average tells you an amount (say) per person. If we know the mean and how many people there were, we can work out the total amount.

Example Question:
In a sponsored run, 500 runners run for an average of 4 hours. What was the total running time for the group. 

Solution:

To work out the average, we must have calculated the total time  500 and got the answer 4. So, we can work backwards:

Total time = 500  4 = 2000 hours.

Calculate the median
The median is the middle one in an ordered list. There are two issues: 
•	The list must be put in order.
•	If there are an even number of items – that is, there are two numbers in the middle – we have to take the middle of these.

Example Question:
Find the median of these hand spans (given in cm):
12, 18, 21, 17, 15, 20, 19, 17, 14, 18.

Solution:

First, we put them in order. (It is a good idea to cross the numbers off the list as you go).
12, 14, 15, 17, 17, 18, 18, 19, 20, 21.

The two values in the middle are 17 and 18.

So, the median is 17.5cm.

Calculate the mode

The mode is the most common item in a list.

Take these lengths: 2m, 5m, 3m, 4m, 5m, 2m, 5m.
The mode is 5m, which occurs 3 times.

The mode is easy to spot in a frequency table, because frequency tells you immediately how many there are. 

Look at this table of teachers shoe sizes:

Shoe Size	Frequency	
5	3	
6	6	
7	9	
8	2	

The mode is size 7, because the frequency for size 7 is the largest.

Be able to select most suitable measure of average.

•	The mode is the easiest to work out. However, the mode is often a bad indicator. For example, there may be so many different values that none of them are significantly more common than the others.

•	The median is fairly easy to work out and is a good choice unless the data is skewed. That means that for some reason most of the data occurs at one end of the possible range. For example, the salaries of employees in a cleaning firm: most employees will get low wages, but the management and the boss can get very high wages indeed. The median will make it look as if the average is as low as the lowest paid staff.

•	The mean is the hardest to work out, but overcomes the problems of the median and the mode.