Question

Consider the following sample. 18, 20, 11, 22, 20, 21

(a) Find the range. 11 (b) Find the variance. (Give your answer correct to two decimal places.) (c) Find the standard deviation. (Give your answer correct to two decimal places.)

Answer 1

Solution:
Let's sort the data from Smallest to largest data set as follows:
11, 18, 20, 20, 21, 22
Solution(a)
Range of sample can be calculated as
Range = Largest value - Smallest value = 22 - 11 = 11
Solution(b)
The variance of the sample can be calculated as
Variance = (X-mean)^2 /(n-1)
Mean can be calculated as
Mean = Xi/n = (11+18+20+20+21+22)/6 = 112/6 = 18.67
Variance = ((11-18.67)^2 + (18-18.67)^2 + (20-18.67)^2 +(20-18.67)^2 + (21-18.67)^2 +(22-18.67)^2)/5) = (0.4489+1.7689+58.8289+11.0889+1.7689+5.4289)/5 = 15.87

Data	(Xi-mean)	(Xi-mean)^2
18	-0.67	0.4489
20	1.33	1.7689
11	-7.67	58.8289
22	3.33	11.0889
20	1.33	1.7689
21	2.33	5.4289

Solution(c)
Standard deviaton can be calculated as
Standard deviation = sqrt(Variance) = sqrt(15.87) = 3.98