### 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 X | Ranks of variable Y | Ranks of variable X - Ranks of variable Y |

x_{1} |
y_{1} |
d_{1} |

x_{2} |
y_{2} |
d_{2} |

x_{3} |
y_{3} |
d_{3} |

**Spearman's rank (r**_{s}) = 1 -
6∑d^{2} / [n(n^{2}-1)]

The value of lies between o and 1 (inclusive)

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.