Population Standard Deviation

The population standard deviation σ of a discrete random variable
,
Measure how close a random variable tends to be the population mean μ, so you must understand μ before you understand σ

If you have a random variable like a bet at a casino or and investment then the standard deviation σ measure the risk, if there is a lot of risk then the standard deviation is high

The formulas for standard deviation are given below but you should look at the examples first

Population mean
Population variance method 1 ,
Population variance method 2 ,
(it does not matter what method you use both give the same answer, the only thing you need to know about variance is that it is population standard deviation squared so if variance is high then standard deviation is high

Population standard deviation σ =

Example 1 find the population mean μ and population standard deviation σ of the gambling game with distribution.
Profit x
-2
2
Probability p(x)
0.5
0.5

This is a gambling game were there is a 50% chance of losing $2 and a 50% chance of winning $2
Solution
mean μ= -2×0.5+2×0.5=0
There are two methods for finding the population standard deviation squared σ2, they give the same answer and you can use either method in the exam
Method 1 σ2= (-2-0)2×0.5+(2-0)2×0.5 = 4
Method 2 σ2= (-2)2×0.5+22×0.5-02 = 4
So population standard deviation σ==2

Example 2 Consider an ordinary dice the common singular of dice is “die”
If you throw a dice you can get the numbers 1,2,3,4,5 or 6.
The probability of each of these events is 1/6 so the value of a dice has distribution
Ff x
1
2
3
4
5
6
Prp(x)
1/6
1/6
1/6
1/6
1/6
1/6
find the population mean μ and population standard deviation σ
Solution
Mean= μ=1×1/6+2×1/6+3×1/6+4×1/6+5×1/6+6×1/6=3.5
There are two methods for finding the population standard deviation squared σ2, they give the same answer and you can use either method in the exam
Method 1 for finding population standard deviation squared.
σ2=