Q.The following data is obtained to know the relation between maximum day temperature and the sale of ice-cream in Ahmedabad city: | Maximum Temperature (Celsius) | 35 | 42 | 40 | 39 | 44 | 40 | 45 | 40 |
| Sales of ice-cream (kg) | 600 | 680 | 750 | 630 | 920 | 750 | 900 | 720 | Calculate the rank correlation coefficient.

When the ranks match exactly for every observation, $\Sigma d^2 = 0$ and Spearman's rank correlation coefficient becomes $r = 1$ — option (c).

Spearman's rank correlation coefficient (GSEB Class-12 Statistics, Linear Correlation) is:

$r = 1 - \frac{6\,\Sigma d^2}{n(n^2 - 1)}, \qquad d = R_x - R_y$

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They are equal when the data are ranks $1,2,\dots,n$ with no ties, since Pearson's $r$ on such ranks reduces exactly to Spearman's formula.

Spearman's rank correlation coefficient

$r_R = 1 - \frac{6\sum d^2}{n(n^2 - 1)}$

is actually Karl Pearson's coefficient of correlation computed on the ranks of the two variables, under the assumption that the ranks are the integers $1, 2, \dots, n$ each occurring once (no ties).

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