Question

Consider a sample with data values of 26, 25, 22, 17, 32, 33, 29, and 25. Compute the range, interquartile range, variance, and standard deviation (Round to 2 decimals, if necessary).

Range
Interquartile range
Variance
Standard deviation

Answer 1

Solution:

Given that

a ) Range = Maximum value - Minimum value

= 33 - 17

= 16

Range = 16

b ) Interquartile range = Q₃ - Q₁

Lower quartile

Arranging Observations in the ascending order, We get :
17,22,25,25,26,29,32,33

Here, n = 8

Q₃ = (3(n + 1) / 4 )^th Value of observation

   = (3*9/ 4 )^th Value of observation

    = 6.75^th Value of observation

= 6^th observation + 0.75 ( 7^th - 6^th )

= 29 observation +0.75 (32 - 29 )

= 29 + 0.75 ( 3 )

= 29 + 2.25

= 31.25

Q₃ = 31.25

Q₁ = ((n + 1) / 4 )^th Value of observation

   = (9/ 4 )^th Value of observation

    = 2.25^th Value of observation

= 2^nd observation + 0.25 ( 3^rd - 2^nd )

= 22 observation +0.25 (25 - 22 )

= 22 + 0.25 ( 3 )

= 22 + 0.75

= 22.75

Q₁ = 22.75

Interquartile range = Q₃ - Q₁

= 31.25 - 22.75

= 8.50

Interquartile range = 8.50

x	x2
26	676
25	625
22	484
17	289
32	1024
33	1089
29	841
25	625
x = 209	x2 = 5653

c ) The  sample variance S²

  S² = ( x² ) - (( x)² / n ) / 1 -n )

= ( 5653 ( (209 )² / 8 ) / 7

   = ( 5653 - 5460.125 / 7 )

= 192.875 / 7

= 27.55

The  sample variance S² is 27.55

d ) The sample standard deviation is s

s = sample variance

= 27.55

= 5.25

The sample standard deviation is 5.25