Spearman’s Rank correlation coefficient used with ordinal data . . .

Inferential statistics We move from descriptive to inferential statistics. Nor longer are we simply comparing data sets ; we are now seeking ‘cause and effect’ relationships. Note that a ‘statistical relationship’ can occur where no ‘meaningful relationship’ is possible. Such a relationship is spurious. So any positive statistical result must always be backed up by sound reasoning.

Scatter graphs It is usually wise to draw a scatter graph, if undertaking any correlation. It is an easy way to highlight any relationship that may exist and its type, whether direct or inverse.

GNP and adult literacy Let us test whether there is a relationship between GNP per capita and educational provision.   GNP per capita % adult literacy Nepal 210 39.7 Sudan 290 55.5 Gambia 340 34.5 Peru 2460 89 Turkey 3160 81.4 Brazil 4570 84 Argentina 8970 97 Israel 15940 96 U.A.E. 18220 74.3 Netherlands 24760 100

GNP and adult literacy First, construct a null hypothesis (Ho) – that there is no relationship between GNP per capita and % adult literacy.   GNP per capita % adult literacy Nepal 210 39.7 Sudan 290 55.5 Gambia 340 34.5 Peru 2460 89 Turkey 3160 81.4 Brazil 4570 84 Argentina 8970 97 Israel 15940 96 U.A.E. 18220 74.3 Netherlands 24760 100

GNP and adult literacy Spearman’s Rank correlation coefficient (Rs) is the best method to use, as the GNP data is skewed.   GNP per capita % adult literacy Nepal 210 39.7 Sudan 290 55.5 Gambia 340 34.5 Peru 2460 89 Turkey 3160 81.4 Brazil 4570 84 Argentina 8970 97 Israel 15940 96 U.A.E. 18220 74.3 Netherlands 24760 100 Remember, Spearman’s Rank can only be used with ordinal data. It is necessary, therefore, to rank-order the data first.

Setting out Spearman’s Rank . . . Set the data out as shown with columns for ‘ranking’ the variables and ‘n =’ and ‘Σd2’ at base :

Ranking the X data . . . The GNP (X) is already rank-ordered ; all it needs is to enter the ranking, in this case from lowest to highest.

Ranking the Y data . . . Ranking is now complete for GNP. Do the same for % adult literacy (again start with the lowest value. . .)

Rank ordering completed . . . Both variables X and Y are now ranked. It is time to find the difference of (Rank X - Rank Y) or ‘d’.

Squaring ‘d’ . . . To get rid of the minuses, square (d) . . .

Summing ‘d^2’ . . . Now sum ‘d^2’ . . . . . . which gives the answer 44.

Getting ‘n’ . . . There are ten countries – so ‘n’ = 10

Calculating Spearman’s Rank . . . Insert the figures into the equation . . . ( 6 x 44 ) (1000 – 100) Rs = 1 -

The answer to Spearman’s . . . . . . and, he presto, the answer to Rs is 0.733. 264 990 Rs = 1 - Rs = 1 – 0.267 Rs = 0.733

Your (Rs) findings . . . Null hypothesis (Ho) was that there was no relationship between GNP per capita and % adult literacy. The degrees of freedom are (n – 1). So ‘df’ = 9. Spearman’s Rank correlation coefficient (Rs) result of 0.733 exceeds the 95% probability value of 0.60 at 9 degrees of freedom. Therefore the Ho must be rejected and replaced by the alternative hypothesis (H1) – that there is a relationship between GNP per capita and % adult literacy.