x: 1, 2, 3, 4; P(x): 0.1, 0.2, 0.3, 0.4, find the standard deviation Step by Step

Step by step

The question

Question: x can be 1, 2, 3, 4, with probabilities 0.1, 0.2, 0.3, 0.4: find the standard deviation.

x: 1, 2, 3, 4; P(x): 0.1, 0.2, 0.3, 0.4. 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 0.1 is 0.1.

2 times 0.2 is 0.4.

3 times 0.3 is 0.9.

4 times 0.4 is 1.6.

0.1 plus 0.4 plus 0.9 plus 1.6 is 3.

μ = 0.1 + 0.4 + 0.9 + 1.6 = 3

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 3 is −2.

Squared: 4.

Times 0.1: 0.4.

2 minus 3 is −1.

Squared: 1.

Times 0.2: 0.2.

3 minus 3 is 0.

Squared: 0.

Times 0.3: 0.

4 minus 3 is 1.

Squared: 1.

Times 0.4: 0.4.

3 Add: the variance

0.4 plus 0.2 plus 0 plus 0.4 is 1.

The variance, sigma squared, is 1.

σ² = 0.4 + 0.2 + 0 + 0.4 = 1

4 Square root

The SD is the square root of the variance: 1.

σ = √1 = 1

Careful!

1 is the variance.

The SD is its square root.

Careful: not σ = 1 ✗

5 Check

Let's check with the other formula.

Each x squared times P of x: 0.1 plus 0.8 plus 2.7 plus 6.4 is 10.

x² × P(x): 0.1 + 0.8 + 2.7 + 6.4 = 10

Minus the mean squared: 10 minus 9 is 1, the variance again.

So it is right!

10 − 3² = 10 − 9 = 1

Let's look back.

First: the mean.

Then: distance from the mean.

Then: add: the variance.

Then: square root.

Last: check.

Answer