x: 1, 2, 3; P(x): 0.2, 0.6, 0.2, find the standard deviation Step by Step

Step by step

The question

Question: x can be 1, 2, 3, with probabilities 0.2, 0.6, 0.2: find the standard deviation.

x: 1, 2, 3; P(x): 0.2, 0.6, 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.

1 times 0.2 is 0.2.

2 times 0.6 is 1.2.

3 times 0.2 is 0.6.

0.2 plus 1.2 plus 0.6 is 2.

μ = 0.2 + 1.2 + 0.6 = 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)

1 minus 2 is −1.

Squared: 1.

Times 0.2: 0.2.

2 minus 2 is 0.

Squared: 0.

Times 0.6: 0.

3 minus 2 is 1.

Squared: 1.

Times 0.2: 0.2.

3 Add: the variance

0.2 plus 0 plus 0.2 is 0.4.

The variance, sigma squared, is 0.4.

σ² = 0.2 + 0 + 0.2 = 0.4

4 Square root

The SD is the square root of the variance: about 0.6325.

σ = √0.4 ≈ 0.6325

Careful!

0.4 is the variance.

The SD is its square root.

Careful: not σ = 0.4 ✗

5 Check

Let's check with the other formula.

Each x squared times P of x: 0.2 plus 2.4 plus 1.8 is 4.4.

x² × P(x): 0.2 + 2.4 + 1.8 = 4.4

Minus the mean squared: 4.4 minus 4 is 0.4, the variance again.

So it is right!

4.4 − 2² = 4.4 − 4 = 0.4

Let's look back.

First: the mean.

Then: distance from the mean.

Then: add: the variance.

Then: square root.

Last: check.

Answer