x: 0, 2, 4; P(x): 0.3, 0.4, 0.3, find the standard deviation Step by Step

Step by step

The question

Question: x can be 0, 2, 4, with probabilities 0.3, 0.4, 0.3: find the standard deviation.

x: 0, 2, 4; P(x): 0.3, 0.4, 0.3. 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 0.3 is 0.

2 times 0.4 is 0.8.

4 times 0.3 is 1.2.

0 plus 0.8 plus 1.2 is 2.

μ = 0 + 0.8 + 1.2 = 2

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 2 is −2.

Squared: 4.

Times 0.3: 1.2.

2 minus 2 is 0.

Squared: 0.

Times 0.4: 0.

4 minus 2 is 2.

Squared: 4.

Times 0.3: 1.2.

3 Add: the variance

1.2 plus 0 plus 1.2 is 2.4.

The variance, sigma squared, is 2.4.

σ² = 1.2 + 0 + 1.2 = 2.4

4 Square root

The SD is the square root of the variance: about 1.5492.

σ = √2.4 ≈ 1.5492

Careful!

2.4 is the variance.

The SD is its square root.

Careful: not σ = 2.4 ✗

5 Check

Let's check with the other formula.

Each x squared times P of x: 0 plus 1.6 plus 4.8 is 6.4.

x² × P(x): 0 + 1.6 + 4.8 = 6.4

Minus the mean squared: 6.4 minus 4 is 2.4, the variance again.

So it is right!

6.4 − 2² = 6.4 − 4 = 2.4

Let's look back.

First: the mean.

Then: distance from the mean.

Then: add: the variance.

Then: square root.

Last: check.

Answer