Let's use the same experiment/random variable as in the previous example, namely the number of heads one observes after flipping two coins. Recall we have the following table for our discrete probability distribution

Outcome \(X\)	Probability \(P(X)\)
0	0.25
1	0.50
2	0.25

To calculated the mean in another example which was \(\mu=1\), so to calculate the variance we need only square each outcome \(X\) then multiply with it's corresponding probability \(P(X)\) then sum all of these values and finally subtract the mean squared.

First we square

\(X^2\)

\(0^2=0\)

\(1^2=1\)

\(2^2=4\)

Now we multiply

\(X\cdot P(X)\)

\(0\cdot 0.25=0\)

\(1\cdot 0.50=0.50\)

\(4\cdot 0.25=1\)

Now we sum all these, and subtract the mean squared \(\mu^2=1^2=1\)

\[0+0.50+1-1=0.5\]

Thus our variance is 0.5.

Now to find our standard deviation we just take the square root of the variance \(\sigma=\sqrt{\sigma^2}\)

Thus our standard deviation is \(\sqrt{0.5}\approx 0.707\)