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Well done, tip-top conference full of exciting idea, stimulating thoughts, and inspiring leaders of the arts community. I had really enjoyed the Scottish Arts Council's Web 2.0 conference last autumn and wasn't sure if this would match it, but it did.

Glasgow Grows Audiences Conference 2009 Delegate

Jargon Buster

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Average

What does this mean?

A measure of a data set’s central tendency which is the single value that best typifies the data set.

Three sorts of average are commonly used for differing purposes.

The mean is found by adding together the values for all the observations made, and dividing the resulting total by the number of observations. Best used when a commonly understood type of average is required. This is especially effective when all the available data needs to be included in the analysis, but can be distorted by any extreme values in the data.

The mode is the most frequently occurring answer – i.e. the one that crops up the most times. Use this if the answers given were categories rather than numbers.

Then the median is the mid-point of a range of values that have been put in order of size. Use this measure if there are a few very high or very low values that might distort the mean. See the relevant protocols and workings below.

How did we get this definition?

An ‘average’ is defined as ‘a measure of a data set’s central tendency which is the single value that best typifies the data set.’

In this sense then, averages are single, summary values that epitomise the essential flavour of a collection of data. But not all averages are equal. And not all averages are found in the same way. This is because there are a number of different ways of finding an average – basically because they each serve different purposes. So selecting the appropriate average to use is all about choosing the most appropriate tool for the job in hand.

The various averages are explained below, while the relevant formulae and worked examples are provided below.

Related and similar definitions

The three most common approaches to finding an average are:

  • the mean (also known as the ‘arithmetic’ mean)
  • the median
  • the mode

The mean

The mean uses a mathematical calculation to produce a value that represents the data’s ‘centre of gravity’. Neil Salkind [2000] says that:

‘The mean is like the fulcrum on a seesaw. It’s the centremost point where all the values on one side of the mean are equal in weight to all the values on the other side of the mean.’

That is, it summarises the essential flavour of the entire collection of data by providing a value that is based on all the data and, thus, to which each element of the data can be related.

The mean is easily worked out. This is done by first finding the total of all the values in the data set, and then dividing the result by the number of values being used.

The mean has a number of advantages. It is a commonly understood concept that is easily depicted graphically, and so can be useful in communicating findings. However, it does have a few pitfalls. For instance, it can be influenced by outlying or extreme values in the data. Equally, the fact that sometimes the actual mean may not occur as a value in the data set may seem contrary to common sense.

Nevertheless the mean is the appropriate average to use when all the data available needs to be taken into consideration. The formula for calculating the mean together with a worked example will be found below.

The mode

This is the most frequently occurring, and thus the most popular, value in the data set. That is, it is the value that crops up the most times.

When it comes to choosing between the different approaches to finding averages, the mode is especially appropriate when the data being used refers to categories rather than to numbers. See below for further details on finding the mode.

The median

This sort of average is the value that sits exactly in the middle of a data set. As such it is particularly useful when there are extreme values that otherwise could distort any calculated version of the average (such as the mean).

Once more the technicalities involved in finding the median together with a worked example are provided below.

This is an instance of choosing the most appropriate procedure for the job in hand – a case of ‘horses for courses’.

However, do not be perplexed if the mean, median or mode for the same data set occasionally produce different values. For instance, consider the following data on the total amount spent by customers on tickets in one year.

Example data set for total customer spend on tickets in one year

Annual spend No. of customers
Under £50 130
£50 but under £75 156
£75 but under £100 198
£100 but under £125 176
£125 but under £150 118
£150 but under £200 76
£200 but under £250 72
£250 but under £300 49
£300 but under £500 19
£500 and more 6

(After Curwin & Slater [1991])

Here, the three different averages for this data set are:

  • mean = £124.10
  • mode = £ 92.00
  • median = £102.27.

Thus visual depiction gives the following chart.

When to use

Averages should be used when a single figure summarising andepitomising a collection of data (and which therefore typifies the data)is required. However depending on the task in hand, it is always usefulto select the most appropriate approach to finding the data’s average.

Formulae & Worked Examples

The three different but commonly used forms of average (the mean, mode, and median) each require different approaches to working them out. These are set out below with worked examples.

The mean

As the arithmetic centre of a set of data, the mean is found by finding the total of all the values in the data and then dividing this by thenumber of values. Hence the formula for this is:

M = ∑x ÷ N

Where:

  • M is the mean
  • ∑ indicates finding the sum or total
  • X is each of the values in the collection of data
  • N is the number of values in the data.

To work this out entails the following steps:

  1. List the entire set of values.
  2. Work out the total of all the values (i.e. ∑X).
  3. Then divide this total by the number of values (i.e. N).
Worked example

Say that the data for attendances at three different exhibitions shows the following:

Exhibition Total attendances
Total 7,523
Renaissance masterpieces 2,563
Monet, the impressionists and light 4,061
Local watercolours 899

As can be seen, this is based on 3 values. So the average (mean) attendance per exhibition is:

M = ∑x ÷ N
= (2,563 + 4,061 + 899) ÷ 3
= 7,523 ÷ 3
= 2,507.7

A variation on the mean to take account of frequency

Means can also be used when issues of frequency are involved (such as when an average figure for attendance is required).

Here the formula to use is:

FM = ∑FX ÷ ∑F

Where:

  • FM is the frequency mean
  • ∑ indicates the sum or total
  • F is the frequency or number of times a value appears
  • X is the various values.

So to use this as a calculation requires the following steps:

  1. Take the data to be analysed and identify the categories it applies to.
  2. Work out the number of times each value appears (this is the frequency F).
  3. Find the total of all the frequencies (i.e. ∑F).
  4. Now multiply the value for each category by the frequency withwhich it occurs (this is FX).
  5. Find the total for all the FX numbers (i.e. ∑FX).
  6. Lastly, divide the total for the FX numbers (i.e. ∑FX) by the totalfrequencies (i.e. ∑F).
Worked example

Say that the attendance pattern of people coming from a particular postal sector is examined. There are 45 such individuals and the data for the number of times they have been to the venue look like this:

{3, 1, 4, 2, 3, 1, 2, 1, 4, 2, 5, 1, 2, 3, 1, 5, 1, 4, 1, 1, 3, 1, 4, 1, 2, 1, 4, 5, 1, 2, 3, 1, 2, 3, 1, 2, 1, 5, 1, 2, 3, 1, 1, 2, 2}

Identifying the categories for ‘numbers of time attended a year’ and finding the number of times these frequencies occur gives the following:

Number of attendances made in yeah (X) Frequency of occurence (F)
1 18
2 11
3 7
4 5
5 4

Now the total for the frequencies can be worked out (∑F), as can the frequencies multiplied by the number of attendances made (FX), and the total for this (∑FX):

Number of attendances made in yeah (X) Frequency of occurence (F) Attendances × Frequency (FX)
Totals 45 101
1 18 18
2 11 22
3 7 21
4 5 20
5 4 20

Finally the average (mean) frequency of attendance for these people is found by dividing the total for FX by the total for F. Thus this equals:

101 ÷ 45 = 2.2

So the average (mean) frequency of attendance for people from this postal sector is a 2.2 times a year.

The mode

The mode is the most frequently occurring value (i.e. the most popular one). This can be found as follows:

  1. Listing all the values in the data, but list each one only once.
  2. Tally or count the number of times each value occurs.
  3. Find the value that appears most often – this is the mode.

So in the attendance data used above, the mode is once a year (since 1 time a year appears the most times ie on 18 occasions).

The median

The median is the mid-point of a set of data. So to find this entails working as follows:

  1. Sort or arrange the data in order – either from lowest to highest, or from highest to lowest.
  2. Now find the value that’s in the middle – this is the median.

Hence in the attendance data example used above, the sorted list of values (together with the count order in which they appear) is shown on the below. Since there are 45 observations or values here, the median will be the value that is midway between the 22nd and the 23rd observation. So the median here is 2 times a year.

Count Values
1 1
2 1
3 1
4 1
5 1
6 1
7 1
8 1
9 1
10 1
11 1
12 1
13 1
14 1
15 1
16 1
17 1
18 1
19 2
20 2
21 2
22 2
The median
23 2
24 2
25 2
26 2
27 2
28 2
29 2
30 3
31 3
32 3
33 3
34 3
35 3
36 3
37 4
38 4
39 4
40 4
41 4
42 5
43 5
44 5
45 5