x: 0, 1, 2, 3; P(x): 0.1, 0.3, 0.4, 0.2, find the standard deviation Step by Step

Step by step

The question

Question: x can be 0, 1, 2, 3, with probabilities 0.1, 0.3, 0.4, 0.2: find the standard deviation.

x: 0, 1, 2, 3; P(x): 0.1, 0.3, 0.4, 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 0.1 is 0.

1 times 0.3 is 0.3.

2 times 0.4 is 0.8.

3 times 0.2 is 0.6.

0 plus 0.3 plus 0.8 plus 0.6 is 1.7.

μ = 0 + 0.3 + 0.8 + 0.6 = 1.7

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 1.7 is −1.7.

Squared: 2.89.

Times 0.1: 0.289.

1 minus 1.7 is −0.7.

Squared: 0.49.

Times 0.3: 0.147.

2 minus 1.7 is 0.3.

Squared: 0.09.

Times 0.4: 0.036.

3 minus 1.7 is 1.3.

Squared: 1.69.

Times 0.2: 0.338.

3 Add: the variance

0.289 plus 0.147 plus 0.036 plus 0.338 is 0.81.

The variance, sigma squared, is 0.81.

σ² = 0.289 + 0.147 + 0.036 + 0.338 = 0.81

4 Square root

The SD is the square root of the variance: 0.9.

σ = √0.81 = 0.9

Careful!

0.81 is the variance.

The SD is its square root.

Careful: not σ = 0.81 ✗

5 Check

Let's check with the other formula.

Each x squared times P of x: 0 plus 0.3 plus 1.6 plus 1.8 is 3.7.

x² × P(x): 0 + 0.3 + 1.6 + 1.8 = 3.7

Minus the mean squared: 3.7 minus 2.89 is 0.81, the variance again.

So it is right!

3.7 − 1.7² = 3.7 − 2.89 = 0.81

Let's look back.

First: the mean.

Then: distance from the mean.

Then: add: the variance.

Then: square root.

Last: check.

Answer