18.2 CHAPTER 18. STATISTICS
Example 1: Variance and StandardDeviation
QUESTION
What is the variance and standard deviation ofthe population of possibilities associated with
rolling a fair die?
SOLUTION
Step 1 : Determine how many outcomes make up the population
When rolling a fair die,the population consists of 6 possible outcomes. Thedata
set is therefore x ={1;2;3;4;5;6}. and n = 6.Step 2 : Calculate the population mean
The population mean iscalculated by:̄x =1
6
(1 + 2 + 3 + 4 + 5 + 6)
= 3, 5
Step 3 : Calculate the population variance
The population varianceis calculated by:σ^2 =�
(x− ̄x)^2
n
=1
6
(6,25 + 2,25 + 0,25 + 0,25 + 2,25 + 6,25)
= 2, 917
Step 4 : Alternately the population variance is calculated by:X (X−X ̄) (X−X ̄)^2
1 − 2. 5 6. 25
2 − 1. 5 2. 25
3 − 0. 5 0. 25
4 0. 5 0. 25
5 1. 5 2. 25
6 2. 5 6. 25
�
X = 21
�
x = 0�
(X−X ̄)^2 = 17. 5
Step 5 : Calculate the standarddeviation
The (population) standard deviation is calculated by:σ =�
2 , 917
= 1, 708.
Notice how this standard deviation is somewhere in between the possible devia-
tions.