Using Measures of Central Tendency (Edexcel GCSE Statistics)

Revision Note

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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Roger B

Author: Roger B

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Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

Dan Finlay

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Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.