The standard deviation of a discrete random variable is denoted by σ and the formula to calculate it is as 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 who go to the movies each week to find the standard deviation. We’ll show you how to use both formulas above.

In the lesson on the 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
4th 0.01
ΣP (x) = 1

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

Here is a table that shows how to calculate 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
4th 3.14 9.8596 0.01 0.098596
Σ[(x – μ)2× P(x)] = 1.0804

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

Variance = ∑[(x – μ)2× P(x)] = 1.0804

= √ (1.0804) = 1.039422

Here is a table that shows how to calculate 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 4th 0.15 0.6
3 9 0.09 0.81
4th 16 0.01 0.16
Σ[(x2 × P(x)] = 1.82

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

μ = 0.86, i.e. μ2 = 0.7396

Σ[(x2 × P(x)] – μ2 = 1.82 – 0.7396 = 1.0804

This number is also the variance. Just calculate the square root of that number to get the standard deviation.

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



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