Solve the following : Calculate of Q3, D5, and P35 for the given data Income (in lakhs ) 1 2 3 4 5 6 No. of family 2 5 20 25 15 12 - Economics

QUESTION

Solve the following :

Calculate of Q₃, D₅, and P₃₅ for the given data

Income (in lakhs ₹)		1	2	3	4	5	6
No. of family		2	5	20	25	15	12

SOLUTION

Arrange the data of the income in ascending order. 
(i.e. from lowest number to highest as shown below) and the out the cumulative frequency.

Income (in lakhs RS)	no. of family	cumulative frequency(cf)
1	2	2
2	5	7
3	20	27
4	25	52
5	15	67
6	12	79
	N = 79

Q₃ = Size of 3${\displaystyle {\left(\frac{n+1}{4}\right)}^{\text{th observation}}}$

= size of 3${\displaystyle {\left(\frac{79+1}{4}\right)}^{\text{th observation}}}$

= size of 3${\displaystyle {\left(\frac{80}{4}\right)}^{\text{th observation}}}$

= size of 3 ${\displaystyle {\left(20\right)}^{\text{th observation}}}$

= size of 60^{th observation}  lies in cf 67 

Hence, quartile value = Rs .5 lakhs Q₃ = Rs. 5 Lakhs

D₅ = size of 5${\displaystyle {\left(\frac{n+1}{10}\right)}^{\text{th observation}}}$

=size of 5${\displaystyle {\left(\frac{79+1}{10}\right)}^{\text{th observation}}}$

 = size of 5${\displaystyle {\left(\frac{80}{10}\right)}^{\text{th observation}}}$

 = size of 5${\displaystyle {\left(8\right)}^{\text{th observation}}}$

= size of ${\displaystyle {\left(40\right)}^{\text{th observation}}}$

Size of 40^{th observation} lies in cf 52 , hence decile value =Rs. 4 Lakhs

D₅ =Rs. 4 Lakhs

P₃₅ =Size of 35${\displaystyle {\left(\frac{n+1}{100}\right)}^{\text{th observation}}}$

P₃₅ =Size of 35${\displaystyle {\left(\frac{79+1}{100}\right)}^{\text{th observation}}}$

P₃₅ =Size of 35${\displaystyle {\left(\frac{80}{100}\right)}^{\text{th observation}}}$

=Size of 35 ${\displaystyle {\left(0.80\right)}^{\text{th observation}}}$

=Size of 28^{th observation} lies in cf 52

Hence, percentile value = Rs.4 lakhs P₃₅ = Rs. 4 lakhs.