Using Measures of Central Tendency (Edexcel GCSE Statistics)
Revision Note
Author
RogerExpertise
Maths
Choosing & Comparing Measures of Central Tendency
How do I decide which average to use?
When deciding how to present and interpret a data set it is important to select the right average to use
i.e. mode, median or (arithmetic) mean
Each average has advantages and disadvantages
Mode
Advantages
Can be used for all types of data
It is the only average that can be used for qualitative (non-numerical) data
But can also be used for quantitative (numerical) data
Usually easy to find
It is always a data value in the data set
Not affected by extreme values in the data set
Or by open-ended classes in grouped data
Disadvantages
There isn't always a mode
or there may be more than one mode
Cannot be used to calculate an associated measure of dispersion
Median
Advantages
Usually easy to calculate
Not affected by extreme values in the data set
Also a useful average when the data is skewed
Can help with calculating other things
Quartiles and interquartile range
Skew
Disadvantages
It may not be a data value in the data set
Mean
Advantages
It uses all the data in the data set
It can be used to calculate other things
Standard deviation
Skew
Disadvantages
It may not be a data value in the data set
It is affected by extreme values in the data set
Not always reliable when there are open-ended classes in a grouped data set
How do I use averages when comparing data sets?
When you compare two data sets it is important to compare their averages
Make sure you use the same average for both data sets
And choose an appropriate average for the context
Remark that 'on average' the values in one data set are greater or less than those in the other data set
You will also need to compare a measure of dispersion for the two data sets
The appropriate measure of dispersion to use will depend on the average you choose
See the 'Using Measures of Dispersion' revision note
Worked Example
(a) The weekly wages for a number of employees in a company are given below
£256 £256 £344 £344 £344 £458 £458 £458 £458 £3850
Karl works out that the median income of those employees is £401 and the mean income is £722.60.
Suggest with a reason which average would be most appropriate to use to describe the wages of those employees.
The mean here has been affected by the one extreme value (£3850)
Therefore the median would be a more appropriate average to use
(Unless you were an unscrupulous manager trying to lure employees to the company by claiming how high the 'average wage' is!)
The mean is quite high because of the one large value in the data set (£3850), but 9 of the 10 workers actually earn significantly less than the mean. Therefore the median would be the most appropriate average to use.
(b) An ice cream seller on a seaside promenade has collected the following data about ice creams sold during the previous summer:
Flavour | Vanilla | Chocolate | Tutti frutti | Pistachio | Blue cheese |
|---|---|---|---|---|---|
Number sold | 4920 | 5904 | 3936 | 3542 | 1378 |
State, with a reason, the best average to use for this data.
The 'data values' here (vanilla, chocolate, etc.) are qualitative
(The 'number sold' are the frequencies for each value - don't be fooled by this into thinking that the data is quantitative!)
So the mode is the only average that can be used
The data here (flavours of ice cream) is qualitative, so there is no way to calculate a mean or median. Therefore the mode is the best average to use.
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