Question

Consider a sample with data values of 27, 24, 22, 15, 31, 33, 28, and 24. Compute the range, interquartile range, variance, and standard deviation (Round to 2 decimals, if necessary).

Answer 1

Solution:
Lets sort the data from smallest to largest data
15, 22, 24, 24, 27, 28, 31, 33
Range of data can be calculated as
Range = Largest Data - Smallest data = (33-15) = 18
For Interquartile range we need to Calculate Quartile 1 and Quartile 3 which can be calculated as
Quartile 1 = (n+1)/4 = (8+1)/4 = 2.25th value i.e. 22.5
Quartile 3 = 3*(n+1)/4 = 3*(8+1)/4 = 6.75th i.e. 30.25
Interquartile range can be calculated as
Interquartile range = (Quartile 3 - Quartile 1) = 30.25-22.5 = 7.75

Mean = (15+22+24+24+27+28+31+33)/8 = 204/8 = 25.5
Variance can be calculated as
Variance = Summation(Xi-mean)^2/(n-1)

X	(Xi-mean)	(Xi-mean)^2
27	1.5	2.25
24	-1.5	2.25
22	-3.5	12.25
15	-10.5	110.25
31	5.5	30.25
33	7.5	56.25
28	2.5	6.25
24	-1.5	2.25

Variance = (2.25+2.25+12.25+110.25+30.25+56.25+6.25+2.25)/7 = 222/7 = 31.71
Standard deviation = sqrt(Variance) = sqrt(31.71) = 5.63