Interpretation of the Correlation Coefficient

The closer r is to (plus or minus) one, the stronger the correlation. But there is another factor we must consider when we try to interpret the correlation coefficient – the number of points we have used.

• If we plotted only two points, then we would be bound to get a straight line between them

• With three points there is still a chance that the points will lie close to a straight line,
  purely by chance.

Clearly, a high correlation coefficient on only a few points is not very meaningful. A simpler alternative is to refer to statistical tables giving the significance of r.

These show that, to be, for example, 95 per cent sure of a significant value, when:

n = 5	we need r = 0.88 or larger
n = 10	we need r = 0.63 or larger
n = 20	we need r = 0.44 or larger

In the next section we will show how we can use a genuine relationship to predict the value of one variable from the value of the other.

Coefficient of Determination

This calculates the proportion of the variation in the actual values (y-values), which can be predicted by changes in the values of the independent variable (x-value).

Is denoted by r ², the square of the coefficient of correlation;

Example

Suppose that for turnover (y) measured against advertising expenditure (x) has its correlation coefficient calculated as r = 0.76

Then:  

Coefficient of Determination is r ²  = (0.76)² or = 0.58

This means that only 58% of the variation in turnover is due to advertising expenditure. Putting it in another way, it means 42% of variations in turnover is due to factors other than advertising expenditure; perhaps product quality, changing trends or productivity.

SPEARMAN’S RANK CORRELATION COEFFICIENT

This is an alternative method of measuring correlation, based on ranked data and is fairly easy to calculate. In many cases all the information you have to work with are rankings.

The procedure for obtaining Spearman’s rank correlation is as follows;

• Rank the x values (to give the r_x values)
• Rank the y values (to give the r_y values)

The Spearman’s Rank correlation is given by;

where d = r_x - r_y

Example 1

Calculate the Spearman’s rank correlation for the following data. (take 1 as the lowest value and 7 as the highest value)

X	Y
23	245
35	236
18	238
36	232
41	250
42	247
48	252

Solution

Rank x (rx)	Rank y (ry)	d  = (rx-ry)	d2 = (rx-ry)2
2	4	-2	4
3	2	1	1
1	3	-2	4
4	1	3	9
5	6	-1	1
6	5	1	1
7	7	0	0
-	-	-	∑d2 = 20

Tied Ranks

If one or more groups of data items have the same value (known as tied values), the ranks that would have been allocated separately must be averaged and this average rank given to each item with this equal value. For example, the following numbers 14, 26, 26 and 28 would be allocated ranks; 1, 2.5, 2.5 and 4 respectively. ^(i.e  )

EXERCISE 1

For the following two sets of data:

(a) Plot a scatter chart
(b) Explain in a sentence the relationship you observe
(c) Calculate the correlation coefficient and comment on your result

Table 5.4

Machine	Age of Machine (years)	Annual Maintenance Costs (MWK)
A	6	1220
B	2	500
C	7	1800
D	4	650
E	4	1100
F	1	200
G	3	600
H	5	1100
I	6	1450
J	5	950

Table 5.5

Panel Member	No. of pints of lager drunk in a week	No. of different brands of lager drunk in a week
1	24	3
2	42	2
3	38	1
4	57	3
5	10	6
6	48	4
7	41	1
8	55	6
9	31	4
10	56	2

EXERCISE 2

The following data relate to the number of vehicles owned and road deaths for the populations of 12 countries.

Vehicles per 100 Population	30	31	32	30	46	30	19	35	40	46	57	30
Road Deaths per 100,000 Population	30	14	30	23	32	26	20	21	23	30	35	26

Calculate Spearman’s rank correlation coefficient and comment on the result.