x: 0, 1, 2, 3, 4; P(x): 0.1, 0.2, 0.4, 0.2, 0.1, find the standard… Step by Step

Step by step

The question

Question: x can be 0, 1, 2, 3, 4, with probabilities 0.1, 0.2, 0.4, 0.2, 0.1: find the standard deviation.

x: 0, 1, 2, 3, 4; P(x): 0.1, 0.2, 0.4, 0.2, 0.1. 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 0.1 is 0.

1 times 0.2 is 0.2.

2 times 0.4 is 0.8.

3 times 0.2 is 0.6.

4 times 0.1 is 0.4.

0 plus 0.2 plus 0.8 plus 0.6 plus 0.4 is 2.

μ = 0 + 0.2 + 0.8 + 0.6 + 0.4 = 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 0.1: 0.4.

1 minus 2 is −1.

Squared: 1.

Times 0.2: 0.2.

2 minus 2 is 0.

Squared: 0.

Times 0.4: 0.

3 minus 2 is 1.

Squared: 1.

Times 0.2: 0.2.

4 minus 2 is 2.

Squared: 4.

Times 0.1: 0.4.

3 Add: the variance

0.4 plus 0.2 plus 0 plus 0.2 plus 0.4 is 1.2.

The variance, sigma squared, is 1.2.

σ² = 0.4 + 0.2 + 0 + 0.2 + 0.4 = 1.2

4 Square root

The SD is the square root of the variance: about 1.0954.

σ = √1.2 ≈ 1.0954

Careful!

1.2 is the variance.

The SD is its square root.

Careful: not σ = 1.2 ✗

5 Check

Let's check with the other formula.

Each x squared times P of x: 0 plus 0.2 plus 1.6 plus 1.8 plus 1.6 is 5.2.

x² × P(x): 0 + 0.2 + 1.6 + 1.8 + 1.6 = 5.2

Minus the mean squared: 5.2 minus 4 is 1.2, the variance again.

So it is right!

5.2 − 2² = 5.2 − 4 = 1.2

Let's look back.

First: the mean.

Then: distance from the mean.

Then: add: the variance.

Then: square root.

Last: check.

Answer