Step by step
The question
Question: x can be 0, 1, 2, 3, with probabilities 0.2, 0.4, 0.3, 0.1: find the standard deviation.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 The mean
x is a random variable: a number that comes out by chance.
The table lists each value of x and its probability, P of x.
The mean: each x times its probability.
0 times 0.2 is 0.
1 times 0.4 is 0.4.
2 times 0.3 is 0.6.
3 times 0.1 is 0.3.
0 plus 0.4 plus 0.6 plus 0.3 is 1.3.
2 Distance from the mean
The standard deviation, SD, tells how spread out x is.
First the variance: how far each x is from the mean, squared, times its probability, all added up.
Squaring stops the minus and plus distances from cancelling.
0 minus 1.3 is −1.3.
Squared: 1.69.
Times 0.2: 0.338.
1 minus 1.3 is −0.3.
Squared: 0.09.
Times 0.4: 0.036.
2 minus 1.3 is 0.7.
Squared: 0.49.
Times 0.3: 0.147.
3 minus 1.3 is 1.7.
Squared: 2.89.
Times 0.1: 0.289.
3 Add: the variance
0.338 plus 0.036 plus 0.147 plus 0.289 is 0.81.
The variance, sigma squared, is 0.81.
4 Square root
The SD is the square root of the variance.
The root undoes the squaring, so the SD is in the same units as x: 0.9.
Careful!
0.81 is the variance.
The SD is its square root.
The right way: σ = √0.81 = 0.9.
5 Check
Let's check with a second formula for the variance.
It comes from opening the bracket of (x − μ)², and must give the same number.
Each x squared times P of x: 0 plus 0.4 plus 1.2 plus 0.9 is 2.5.
Minus the mean squared: 2.5 minus 1.69 is 0.81, the variance again.
So it is right!
6 Pick the choice
Our answer is σ = 0.9.
That is choice B.
7 Why not the others
A is 0.81.
Wrong: that is the variance: take its square root.
That is the Careful mistake from before.
C is 1.3.
Wrong: that is the mean.
D is 2.5.
Wrong: used E(X²) only: take away the mean squared.
Let's look back.
First: the mean.
Then: distance from the mean.
Then: add: the variance.
Then: square root.
Then: check.
Then: pick the choice.
Last: why not the others.