Calculate standard deviation of a probability distribution

For a finite discrete probability distribution, standard deviation is the square root of the variance, $\sigma=\sqrt{\sum (x-\mu)^2P(x)}$ , where $\mu=\sum xP(x)$ ; it represents the distribution’s typical distance from its mean and is expressed in the same units as the outcomes. The calculation requires weighting squared deviations by their probabilities and is distinct from average absolute deviation; it excludes continuous-distribution integration and inferential sample-standard-deviation methods.

Detailed Explanation: Calculate standard deviation of a probability distribution

For a finite discrete probability distribution, the standard deviation measures the typical distance of the outcomes from the mean.

Use these steps:

Find the mean:

\mu=\sum xP(x)

Find each squared deviation, $(x-\mu)^2$ .
Weight each squared deviation by its probability.
Add the weighted values to find the variance.
Take the square root:

\sigma=\sqrt{\sum (x-\mu)^2P(x)}

Example

Suppose the random variable $X$ has this probability distribution:

$x$	$P(x)$
0	0.10
1	0.20
2	0.40
3	0.30

Step 1: Find the mean

Multiply each outcome by its probability and add:

\mu=(0)(0.10)+(1)(0.20)+(2)(0.40)+(3)(0.30)

\mu=0+0.20+0.80+0.90=1.90

So, the mean is $\mu=1.90$ .

Step 2: Find the weighted squared deviations

$x$	$P(x)$	$x-\mu$	$(x-\mu)^2$	$(x-\mu)^2P(x)$
0	0.10	$-1.90$	3.61	0.361
1	0.20	$-0.90$	0.81	0.162
2	0.40	0.10	0.01	0.004
3	0.30	1.10	1.21	0.363

Add the last column to find the variance:

\sigma^2=0.361+0.162+0.004+0.363=0.890

Step 3: Take the square root

\sigma=\sqrt{0.890}\approx 0.943

Therefore, the standard deviation is

\boxed{\sigma\approx 0.943}

The standard deviation is expressed in the same units as $X$ . It describes the typical distance of an outcome from the mean of $1.90$ .

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Statistics - Standard Deviation - Data Set to Standard Deviation

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