SOLUTION: Verify that the variance of the sample 4, 9, 3, 6, 4 and 7 is 5.1 and using this fact find the variance of the sample 12, 27, 9, 18, 12 and 21.

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Question 885158: Verify that the variance of the sample 4, 9, 3, 6, 4 and 7 is 5.1 and using this fact find the variance of the sample 12, 27, 9, 18, 12 and 21.

Answer by Theo(13342) (Show Source):
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the second data set is equal to to each element of the first data set multiplied by 3.
the statistics for both data sets are shown in the following picture.
all the statistics in data set 2 are 3 times the statistics in data set 1 except for sum of squares and variance.
those statistics in data set 2 are 9 times the statistics in data set which is the same as 3^2 times the statistics in data set 1.
this is because the sum of squares calculation takes the difference between each data set value and the mean and then squares it.
this makes the sum of squares of data set 2 equal to 3^2 times the sum of squares of data set 1.
a sample calculation shows you what i mean.

take:
x = 5
m = 3
(x-m) = (5-3) = 2
(x-m)^2 = 2^2 = 4

now take:
x = 15
m = 9
(x-m) = 6
(x-m)^2 = 6^2 = 36

x from data set 2 is 3 times x from data set 1.
(x-m) from data set 2 is 3 times (x-m) from data set 1.
(x-m)^2 from data set 2 is 9 times x from data set 1 = 3^2 times x from data set 1.

this is because sum of squares is the difference squared.
variance follows suit because variance is simply sum of squares divided by (n-1) for sample, or sum of squares divided by n for population.