Quartile

Q1 = (n+1/4)th term

Q2 = (n+1/2)th term

Q3 = 3(n+1/4)th term

Quartile deviation (Q.D) = Q3-Q1/2

Interquartile range = Q3-Q1

For continuous series,

Q1 = l + (N/4-cf )/ f x h

Q3 = l+ ( 3N/4 – cf) /f x h

Where,

l= lower limit

h= width of the quartile

c.f. = Cumulative frequency of class interval

f = Frequency

N = no. of observation

Q. 28,18,20,24,30,15,47,27. Find Quartiles and Interquartile range.

Solution,

Here, Arranging the values in ascending order, we get

15,18,20,24,27,28,30,47

n = 8

Now, Q 1 = (n+1/4)th term = 2.25^th term = 2^nd term + 0.25( 3^rd term – 2^nd term)

                                                                  = 18+0.25(20-18) = 18.5

Q3 = 3(n+1/4)th term = 6.75 term = 6^th term + 0.75( 7^th term – 6^th term)

                                                        = 28 + 0.75 ( 30-28) = 29.5

Inter Quartile range = Q3-Q1 = 29.5 – 18.5 = 11

Q. 7 farmers of a village had 22,20,29,26,27,21,25 cattle. Calculate the following. Mean, median, Q3 and Variance.

Solution,

n=7, ΣX = 170 , Hence, mean = ΣX/n = 170/7 = 24.28

Also, arranging the data in ascending order, we get

20,21,22,25,26,27,29

Now,
Median – ( n+1/2)th term = ( 7+1/2)th term = 4^th term = 25.

Q3 = 3(n+1/4)th term = 3 x 8/4^th term = 6^th term = 27

Variance =  Σ (X- X̄)²/N-1 = 67.37/7-1 = 11.22