Averages


Introduction


You might be quite familiar with the idea of an "average" from everyday life.

In Mathematics the situation is slightly more complicated, we have three separate ways of expressing an average -

  • the mean
  • the median
  • the mode

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Range


The range is the difference between the highest and lowest figure in a set of data

For example, for the data

4, 7, 12, 3, 6, 3, 7, 11, 8, 6

The highest-valued data is 12 and the lowest-valued data is 3, so therefore the range is

12 - 3 = 9

Quick Quiz     What is the range of the following groups of data

1.     1, 6, 7, 12

2.   3, 8, 12, 11

3.   5, 23, 2, 45

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Mean


To compute the mean (which is normally just called the "average" in everyday life - the average that you most commonly come across), sum all the data items and then divide by the number of data items.

Mean = sum of all the data items / number of data items

Warning : The mean can be a bit unsuitable when large ranges are involved, e.g. average salary calculations could be distorted by a small number of extremely high salaries. We could ameliorate this situation, maybe, by 'throwing away' one-sixth of the readings from the top and the bottom of the range, and finding the mean of the remaining figures. Alternatively, the median might be more appropriate.

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Median


The median is the central value of an ordered distribution.

To obtain the median, order the values from the smallest to the largest. Then pick the middle score. If the data set has an even number of entries, add the two middle values and divide by 2.

or alternatively

The median is that value that has the same number of results above it as below it.

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Mode


The mode is the value that occurs most frequently in the data.

This average can only be computed if the data has some repeated values (and note that there could be more than one mode).

Although traditionally the mode is presented as the 'least useful' average, it does actually have great significance for various sectors, for example the clothing industry.

For example

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Grouped Data


Finding the averages of grouped data can cause difficulties to some students.

Consider the grouped data below


The important thing to realize, right from the beginning, is that there are 27 pieces of data here.

So the mean would involve adding up the 27 pieces of data, and then dividing by 27.

The adding of the data would proceed as follows

(1 × 40) + (1 × 38) + (6 × 37) + (5 × 36) + (1 × 35) + (5 × 34) + (3 × 33) + (2 × 32) + (3 × 31)

The median is the middle piece of data. The data is already effectively ordered, so you have to consider where the middle piece of data might lie - there are 27 pieces of data, so we are looking for the 14th. item of data. Here this will be 35 (the fact that there is only one piece of data corresponding to 35 is purely coincidental).

The mode can be read off straight away. The most commonly occuring value is 37.


When dealing with frequency distributions, things get a bit harder, but for the sake of example, let us consider where the median of the following data might lie.

AgeTallyTotal Frequency
1-10 8
11-20 13
21-30 18
31-40   5
41-50 5
51-60 18
61-70   3

As always, the first thing to notice is that we have 70 pieces of data.

To find the median we require the middle value, and since we have an even number of pieces of data, we would actually require half the sum of the 35th. and 36th values. Both of these values will be in the 21-30 range, so that's where the median will lie. We do not know its exact value but we know the range within which it lies.

Quick Quiz     What is the median and mean of the data presented in this frequency table ?

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Links to Other Sites


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Past Exam Questions


Four typists who are applying for a job are being assessed. Each has to type 10 items and the numbers of mistakes are counted


1.   The number of mistakes for James' tenth item was incorrectly recorded as zero. His mean score is 4 - what was the correct result for his tenth assessment?

  • A     2
  • B     3
  • C     4
  • D     5

2. The assessor wants to know which applicant is the most inconsistent. Which measure would be the most appropriate to use?

  • A     mode
  • B     median
  • C     range
  • D     maximum value


The bar charts show the results of two different groups of students, group M and group P, doing the same five test questions.

1.   Which of the following statements is true ?

  • A     The results of both groups are equally good
  • B     Group M's results are better than those of group P
  • C     Group P's results are better than those of group M
  • D     You cannot say which group is better

2. Which of these statements is correct ?

  • A     The median result for group M is 2.5
  • B     The median result for group P is 3
  • C     The median result for group M is higher than the median for group P
  • D     The median result for group P is higher than the median for group M


A firm has 20 employees. The annual salaries for the employees are shown in the table below.

1.   The percentage of the firm's employees earning less than 15 000 is ?

  • A     6%
  • B     15%
  • C     30%
  • D     70%

2. The mode of the employees' salaries is

  • A     17 500
  • B     19 000
  • C     21 000
  • D     25 000

3. The range of the salaries is

  • A     7 000
  • B     17 500
  • C     36 000
  • D     43 000


Stage 3

The bar chart below shows the number of children per family in 50 families

1.   What is the mean number of children per family ?

2. Another sample of 50 families has a mean number of children of 2.36 children per family and a mode of one child per family

What does this tell you about the number of families with 3 or more chilfren in this second sample, compared with the number in the first sample ?

Give a reason to support your answer.

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