In: Statistics and Probability
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 |
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
= 6.75th Value of observation
= 6th observation + 0.75 ( 7th - 6th )
= 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
= 2.25th Value of observation
= 2nd observation + 0.25 ( 3rd - 2nd )
= 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