Question

Following data relates to density of population, number of deaths and population of various cities. Calculate death rate and Karl Pearson coefficient between density of population and death rate.

Cities	P	Q	R	S	T	U
Density of Population	200	500	700	500	600	900
Number of Deaths	840	300	312	560	1,140	1,224
Population	42,000	30,000	24,000	40,000	90,000	72,000

Answer 1

Cities	Density	Number of Deaths	Population	Death rate = $\frac{\mathrm{Number}\mathrm{of}\mathrm{Deaths}}{\mathrm{Population}}\times 100$
P Q R S T U	200 500 700 500 600 900	840 300 312 560 1440 1224	42000 30000 24000 40000 90000 72000	2 1 1.3 1.4 1.6 1.7

Density (X)	dx = X – A A = 500	dx2	Death Rate ​(Y)	dy = Y – B B = 1	dy2	dxdy
200 500 700 500 600 900	–300 0 200 0 100 400	90000 0 40000 0 10000 160000	2 1 1.3 1.4 1.6 1.7	1 0 0.3 0.4 0.6 0.7	1 0 0.09 0.16 0.36 0.49	–300 0 60 0 60 280
	∑dx = 400	∑dx2 = 300000		∑dy = 3	∑dy2 = 2.1	∑dxdy = 100

$$\begin{gathered} r=\frac{\sum dxdy-{\displaystyle \frac{\left(\sum dx\right)\left(\sum dy\right)}{n}}}{\sqrt{\sum d{x}^2-{\displaystyle \frac{{\left(\sum dx\right)}^2}{n}}}\sqrt{\sum d{y}^2-{\displaystyle \frac{{\left(\sum dy\right)}^2}{n}}}}=\frac{100-{\displaystyle \frac{400\times 3}{6}}}{\sqrt{300000-{\displaystyle \frac{{\left(400\right)}^2}{6}}}\sqrt{2.1-{\displaystyle \frac{{\left(3\right)}^2}{6}}}} \\ \mathrm{or},r=\frac{100-200}{\sqrt{273333.34}\times \sqrt{0.6}}=\frac{-100}{522.81\times 0.77}=\frac{-100}{402.56}=-0.248 \\ \mathrm{Hence},\mathrm{Karl}\mathrm{Pearson}\text{'}\mathrm{s}\mathrm{Coefficient}\mathrm{of}\mathrm{Correlation}\mathrm{between}\mathrm{density}\mathrm{of}\mathrm{population}\mathrm{and}\mathrm{death}\mathrm{rate}\mathrm{is}-0.248 \end{gathered}$$