# Mathematics

### Spearman's Rank

We are no strangers to the concept of correlation, as it is associated with the relationship between two quantitive variables. For example, the number of fans sold by a superstore with the rise of humidity during a summer period is a case in point.

Average Relative Humidity(%) Fans Sold
29 1200
32 1280
45 2200
65 3400
75 3800

In the above example, we can quantify the variables involved - they take numbers. So, we plot a graph of the two variables to see a relationship between the two and they go a step further to justify it in a mathematical manner to make it statistically appealing. In short, there is clearly a strong relationship between the two variables, the relative humidity and the sales of fans.

Some variables cannot be represented by numbers, though. yet, they can be arranged in a certain order so that the pattern makes sense to those who are interested in them.

This is what led Charles Edward Spearman, the British psychologist, to coming up with a method to rank the variables first and then find the correlation coefficient between the ranks, which came to be known as Spearman's Rank.
It's something best learned by following a real-life example, which is as follows:

E.g.

Suppose there is a beauty contest involving ten aspiring models and the enviable job of choosing the winner is in the hands of two judges.
Since beauty cannot be quantified, the judges have to rank them, say, 1 - 10, by taking into account a few factors, usually associated with beauty pageants.

Spearman's Rank(ρ), exactly like Pearson's Correlation Coefficient, can take any value between -1 and +1, indicating a strong negative correlation and a strong positive correlation respectively. If it's zero, there is no correlation between the ranks of the variables involved.

In the above example, ρ = 0.89. So, there is a strong correlation between the ranks given by Judge A and Judge B, in determining the beauty of the contestants of the pageant.

Here is an opportunity for you to practise Spearman's Rank interactively.

E.g.

Some bottles of wine can be arranged by the responses to their taste. The arrangement makes sense to people who are fond of wine, despite the absence of an index to measure it.

In these circumstances, Spearman's rank comes to our rescue. It can easily be used to determine the relationship between the two variables without numbers, but with ranks. Since it is universally accepted as a trusted method, we can easily cash in on this encouragingly simple method.

Since we do not have numbers for the variable, we assign numbers to them, in a sensible way. They are called ranks.

E.g.

Suppose there are five wine bottles A, B, C, D and E in the order of taste. We can assign ranks to them in the order of 5,4,3,2,1 or 10, 8.6, 4,2 . These are arbitrary values assigned to variables in a sensible way; there are not hard and fast rules about it. However, the simpler the better.

The ranking system must be extended to both sets of variables. Then a formula must be used to find Spearman's rank, the value of which determines the correlation.

Ranks of variable XRanks of variable YRanks of variable X - Ranks of variable Y
x1 y1 d1
x2 y2 d2
x3 y3 d3

Spearman's rank (rs) = 1 - 6∑d2 / [n(n2-1)]

The value of lies between o and 1 (inclusive)

Spearman's Rank Calculator

ItemABCDE
Rank By Judge-1
Rank By Judge-2

We can find the Spearman's Rank for quantitive data as well. The data, however, must be ranked exactly like we did it with qualitative data in the previous example first, though.

In order to rank the data, the smallest data value was given rank 1, the next value, rank 2 etc.

Exercises

Please work out the following questions to complement what you have just learnt.

1) The height of some baby girls are as follows:
87, 88.8, 90.9, 87.4, 88.7, 90.8, 91.5, 92.2, 88.4 and 94

The corresponding weight in (kg) are 12.7, 11.8, 12, 12.2, 12.4, 12.5, 12.6, 12.7, 12.8 and 13. Find Spearman's Rank.

2) The height of some baby boys are as follows:
92.1, 90.5, 89.8, 93.1, 89.2, 92.4, 95.1, 90.7, 93.9 and 93.8

The corresponding weight in (kg) are 14.1, 13.2, 13.3, 13.4, 12.5, 13.6, 13.7, 13.8, 14 and 14.4. Find Spearman's Rank.

Maths is challenging; so is finding the right book. K A Stroud, in this book, cleverly managed to make all the major topics crystal clear with plenty of examples; popularity of the book speak for itself - 7th edition in print.

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This is the best book available for the new GCSE(9-1) specification and iGCSE: there are plenty of worked examples; a really good collection of problems for practising; every single topic is adequately covered; the topics are organized in a logical order.

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This is the best book that can be recommended for the new A Level - Edexcel board: it covers every single topic in detail;lots of worked examples; ample problems for practising; beautifully and clearly presented.