Measures of Average and of Dispersion

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1 Introduction

It is often of interest to be know the average of a set of data. For example we may have asked a sample of people what their wages are, and want to know what the average wage is. In this chapter we will look at different ways we can calculate an average. Additionally, even if we have calculated the average wage, it might be on interest to know whether all of the sample had a wage close to the average or whether some earned a lot more and some a lot less than the average. This is known as the dispersion and we will look at different ways of measuring this.

2 Frequency distributions

A frequency distribution is a table showing the number of observations of each variable. They may be discrete variables which can only consist of certain values, or continuous variables where we group the variables.

Discrete variables:

A company has recorded the number of complaints received per week over the last year, and has produced the following table:

Number of complaints	Frequency
	12
1	16
2	20
3	4
	52

Continuous variables:

A company has recorded the total amount paid to employees each week over the last year, and has produced the following table:

Total paid ($)	Frequency
0 - under $500	1
500 - under 1,000	4
1,000 - under 1,500	8
1,500 - under 2,000	19
2,000 - under 2,500	14
2,500 - under 3,000	6
	52

3 The histogram and the ogive

We looked in an earlier chapter at various ways of presenting data. We are now going to look at two more ways of presenting grouped data that can be particularly helpful for the calculations that follow in both this and the next chapter.

A company has recorded the total amount paid to employees each week over the last year, and has produced the following table:

Total paid ($)	Frequency
0 - under $500	1
500 - under 1,000	4
1,000 - under 1,500	8
1,500 - under 2,000	19
2,000 - under 2,500	14
2,500 - under 3,000	6
	52

Present the above table in the form of

a histogram
an ogive

(a)

(b)

4 Measures of average

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You need to be aware of the following different measures of determining the average of a set of observations:

Arithmetic mean

This is calculated by adding up all of the observations and dividing by the number of observations

Median

This is the centrally occurring observation when all of the observations are arranged in order of magnitude

Mode

This is the most frequently occurring observation

A company has recorded the number of complaints received per week over the past thirteen weeks, and has produced the following table:

Calculate:

Number of complaints	Frequency
	1
1	6
2	4
3	2
	13

the arithmetic mean
the median
the mode

Number of complaints x	Frequency f	fx
	1
1	6	6
2	4	8
3	2	6
	13	20

(a) arithmetic mean = 20/13 = 1.54

(b) median = value of 7th observation = 1

A company has recorded the total amount paid to employees each week over the last year, and has produced the following table:

Calculate:

Total paid ($)	Frequency
0 - under $500	1
500 - under 1,000	4
1,000 - under 1,500	8
1,500 - under 2,000	19
2,000 - under 2,500	14
2,500 - under 3,000	6
	52

the arithmetic mean
the median
the mode

Total paid ($)	Mid-point x	Frequency f	fx
0 - under $500	250	1	250
500 - under 1,000	750	4	3,000
1,000 - under 1,500	1,250	8	10,000
1,500 - under 2,000	1,750	19	33,250
2,000 - under 2,500	2,250	14	31,500
2,500 - under 3,000	2,750	6	16,500
		52	94,500

(a) arithmetic mean = 94,500 / 52 = $1,817

(b) median = value of the 25.5th item, which is in the range $1,500 to $2,000

(watch lecture for more)

5 Measures of dispersion

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Dispersion is looking at the spread of the observations.

You need to be aware of the following measures of dispersion:

Range

This is simply the difference between the highest and the lowest of the observations

Variance

Here we measure the differences between the observations and the arithmetic mean, square the differences, and then take the average of these squared differences.

Standard deviation

This is the square root of the variance

Coefficient of variation

The standard deviation divided by the arithmetic mean

For the information in example 2, calculate:

the range
the variance
the standard deviation
the coefficient of variation

Number of complaints X	Frequency f	fX	X-x̄	(X-x̄)2	f(X-x̄)2
	1		(1.54)	2.37	2.37
1	6	6	(0.54)	0.29	1.74
2	4	8	+0.46	0.21	0.84
3	2	6	+1.46	2.13	4.26
	13	20			9.21

(a) Range = 3 - 0 = 3

(b) Variance = 9.21 / 13 = 0.71

(d) Coefficient of variation = 0.84 / 1.54 = 0.55

For the information in example 3, calculate:

the range
the variance
the standard deviation
the coefficient of variation

Total paid ($)	Mid-point x	Frequency f	fX	X-x̄	(X-x̄)2	f(X-x̄)2
0 - under $500	250	1	250	(1,567)	2,455,489	2,455,489
500 - under 1,000	750	4	3,000	(1,067)	1,138,489	4,553,956
1,000 - under 1,500	1,250	8	10,000	(567)	321,489	2,571,912
1,500 - under 2,000	1,750	19	33,250	(67)	4,489	85,291
2,000 - under 2,500	2,250	14	31,500	+433	187,489	2,624,846
2,500 - under 3,000	2,750	6	16,500	+933	870,489	5,222,934
		52	94,500			17,514,428

(a) Range = 3,000 = 0 = 3,000

(b) Variance = 17514428/52 = 336816

(d) Coefficient of variation = 580/1817 = 0.32

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