Standard deviation of a discrete random variable

The standard deviation of a discrete random variable is denoted by σ and the formula to use to compute it is the one you see below.

$$ σ = \sqrt{Σ[(x-µ)^2 × P(x)]} \ or \ σ = \sqrt{Σ[x^2 × P(x)] - µ^2} $$

We can use the example in the previous lesson about the number of people going to the movie theater each week to look for the standard deviation. We will show you how to use both formulas above.

In the lesson about mean of a discrete random variable we have this probability distribution table.

x P(x)
0 0.5
1  0.25
2  0.15
3  0.09
4  0.01
   ΣP(x) = 1

We have already looked for the mean and E(x) = 0.86

Here is a table showing how to compute the standard deviation using the formula.

$$ σ = \sqrt{Σ[(x-µ)^2 × P(x)]} $$
x x - μ (x - μ)2 P(x) (x - μ)2× P(x)
0  -0.86  0.7396  0.5  0.3698
1  0.14  0.0196  0.25  0.0049
2  1.14  1.2996  0.15  0.19494
3  2.14  4.5796  0.09  0.412164
4  3.14  9.8596  0.01  0.098596
        ∑[(x - μ)2× P(x)] = 1.0804

1.0804 was of course found by adding all the numbers in the last column. This number is called variance.

Variance =  ∑[(x - μ)2× P(x)] = 1.0804

σ = √(1.0804) = 1.039422

Here is a table showing how to compute the standard deviation using the formula.

$$ σ = \sqrt{Σ[x^2 × P(x)] - µ^2} $$
x x2 P(x) x2 × P(x)
0  0 0.5  0
1  1 0.25  0.25
2  4 0.15  0.6
3  9 0.09  0.81
4  16 0.01  0.16
      ∑[(x2 × P(x)] = 1.82

1.82 is found by adding all the numbers in the fourth column.

μ = 0.86, so μ2 = 0.7396

∑[(x2 × P(x)] - μ2 = 1.82 - 0.7396 = 1.0804

Again, this number is the variance. Just get the square root of this number to get the standard deviation.

Since the number is the same as the one before, the answer is the same.

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