x: 0, 1, 2; P(x): 0.25, 0.5, 0.25, find the standard deviation Step by Step

Step by step

The question

Question: x can be 0, 1, 2, with probabilities 0.25, 0.5, 0.25: find the standard deviation.

x: 0, 1, 2; P(x): 0.25, 0.5, 0.25. SD?

1 The mean

The table lists each value of x and its probability, P of x.

The mean: each x times its probability.

0 times 0.25 is 0.

1 times 0.5 is 0.5.

2 times 0.25 is 0.5.

0 plus 0.5 plus 0.5 is 1.

μ = 0 + 0.5 + 0.5 = 1

2 Distance from the mean

The variance: how far each x is from the mean, squared, times its probability, all added up.

variance σ² = add up all (x − μ)² × P(x)

0 minus 1 is −1.

Squared: 1.

Times 0.25: 0.25.

1 minus 1 is 0.

Squared: 0.

Times 0.5: 0.

2 minus 1 is 1.

Squared: 1.

Times 0.25: 0.25.

3 Add: the variance

0.25 plus 0 plus 0.25 is 0.5.

The variance, sigma squared, is 0.5.

σ² = 0.25 + 0 + 0.25 = 0.5

4 Square root

The SD is the square root of the variance: about 0.7071.

σ = √0.5 ≈ 0.7071

Careful!

0.5 is the variance.

The SD is its square root.

Careful: not σ = 0.5 ✗

5 Check

Let's check with the other formula.

Each x squared times P of x: 0 plus 0.5 plus 1 is 1.5.

x² × P(x): 0 + 0.5 + 1 = 1.5

Minus the mean squared: 1.5 minus 1 is 0.5, the variance again.

So it is right!

1.5 − 1² = 1.5 − 1 = 0.5

Let's look back.

First: the mean.

Then: distance from the mean.

Then: add: the variance.

Then: square root.

Last: check.

Answer