Question

Give three examples of perfect correlation. Find out rank difference coefficient of correlation with the help of the following data:

X	48	33	40	9	16	65	26	15	57
Y	13	13	22	6	14	20	9	6	15

Answer 1

Three examples of perfect correlation are:
1. T.V. viewing and Study hours (-ve correlation). That is, as the hours spent in T.V. viewing increases, the numbers of hours that can be devoted to study decreases and vice-versa.
2. Income used for consumption and amount of saving (-ve correlation). That is, greater the portion of income used for consumption purposes, smaller is the portion of income left for saving purposes and vice-versa.
3. Amount deposited in bank and interest earned (+ve correlation). That is, as the amount deposited in the bank increases, the amount of interest that is earned increases and vice-versa.

X	R1	Y	R2	D = R1 – R2	D2
48 33 40 9 16 65 26 15 57	3 5 4 9 7 1 6 8 2	13 13 22 6 14 20 9 6 15	5.5 5.5 1 8.5 4 2 7 8.5 3	–2.5 –.5 3 .5 3 –1 –1 –.5 .1	6.25 .25 9 .25 9 1 1 .25 1
N = 9					${\sum }_{}{D}^2=28$

$$\begin{gathered} r_k=1-\frac{6\left[\sum D^2+{\displaystyle \frac{1}{12}}\left(M_1^3-M_1\right)+{\displaystyle \frac{1}{12}}\left(M_2^3-M_2\right)\right]}{N^3-N} \\ \mathrm{Here,} \\ {M}_1=\mathrm{2,}\mathrm{as}\mathrm{the}\mathrm{item}13\mathrm{is}\mathrm{appearing}\mathrm{twice}\mathrm{in}Y-\mathrm{series}. \\ {M}_2=2,\mathrm{as}\mathrm{the}\mathrm{item}6\mathrm{is}\mathrm{appearing}\mathrm{twice}\mathrm{in}Y-\mathrm{series}. \\ \mathrm{or},r_k=1-\frac{6\left[28+{\displaystyle \frac{1}{12}}(2^3-2)+{\displaystyle \frac{1}{12}}(2^3-2)\right]}{729-9} \\ \therefore ,r_k=\frac{720-147}{720}=0.758 \end{gathered}$$