x: 0, 5, 10; P(x): 0.5, 0.3, 0.2, find the standard deviation Step by Step

Step by step

The question

Question: x can be 0, 5, 10, with probabilities 0.5, 0.3, 0.2: find the standard deviation.

x: 0, 5, 10; P(x): 0.5, 0.3, 0.2. 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 0.5 is 0.

5 times 0.3 is 1.5.

10 times 0.2 is 2.

0 plus 1.5 plus 2 is 3.5.

μ = 0 + 1.5 + 2 = 3.5

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 3.5 is −3.5.

Squared: 12.25.

Times 0.5: 6.125.

5 minus 3.5 is 1.5.

Squared: 2.25.

Times 0.3: 0.675.

10 minus 3.5 is 6.5.

Squared: 42.25.

Times 0.2: 8.45.

3 Add: the variance

6.125 plus 0.675 plus 8.45 is 15.25.

The variance, sigma squared, is 15.25.

σ² = 6.125 + 0.675 + 8.45 = 15.25

4 Square root

The SD is the square root of the variance: about 3.9051.

σ = √15.25 ≈ 3.9051

Careful!

15.25 is the variance.

The SD is its square root.

Careful: not σ = 15.25 ✗

5 Check

Let's check with the other formula.

Each x squared times P of x: 0 plus 7.5 plus 20 is 27.5.

x² × P(x): 0 + 7.5 + 20 = 27.5

Minus the mean squared: 27.5 minus 12.25 is 15.25, the variance again.

So it is right!

27.5 − 3.5² = 27.5 − 12.25 = 15.25

Let's look back.

First: the mean.

Then: distance from the mean.

Then: add: the variance.

Then: square root.

Last: check.

Answer