Economics · Ch 6 — Correlation

Step Deviation Method To Calculate Correlation Coefficient

6.3.4

Step Deviation Method To Calculate Correlation Coefficient

When the values of the variables are large, computing $r$ directly is tedious. The step deviation method cuts the labour by exploiting the property that $r$ is independent of change of origin and scale.

The variables are transformed as

U = \frac{X - A}{h}, \qquad V = \frac{Y - B}{k}

where $A$ and $B$ are assumed means and $h$ , $k$ are common factors of the same sign. Then $r_{UV} = r_{XY}$ , so we may compute $r$ from the simpler $U$ , $V$ values.

Worked example (Example 2): price index (X) vs money supply in Rs crore (Y).

Original data: X = 120, 150, 190, 220, 230; Y = 1800, 2000, 2500, 2700, 3000. Choosing $A = 100$ , $h = 10$ , $B = 1700$ , $k = 100$ gives the transformed table:

U	V	$U^2$	$V^2$	$UV$
2	1	4	1	2
5	3	25	9	15
9	8	81	64	72
12	10	144	100	120
13	13	169	169	169
ΣU = 41	ΣV = 35	ΣU² = 423	ΣV² = 343	ΣUV = 378

With $N = 5$ , applying formula (3) in terms of $U$ and $V$ :

r = \frac{\sum UV - \dfrac{(\sum U)(\sum V)}{N}}{\sqrt{\sum U^2 - \dfrac{(\sum U)^2}{N}} \; \sqrt{\sum V^2 - \dfrac{(\sum V)^2}{N}}} = \frac{378 - \dfrac{41 \times 35}{5}}{\sqrt{423 - \dfrac{41^2}{5}} \; \sqrt{343 - \dfrac{35^2}{5}}}

$= \frac{378 - 287}{\sqrt{423 - 336.2}\;\sqrt{343 - 245}} = \frac{91}{\sqrt{86.8}\;\sqrt{98}} \approx 0.98$ …

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