This variability can be measured. That is,
risk can be quantified! The most common measure
of risk is the standard deviation 
a numerical measure of the dispersion around the
expected value. Here is how the standard
deviation (designated by s ) is calculated for
our two dice probability distributions:
 compute the expected return;
 square the variance of each return from the
expected return;
 multiply this by the probability;
 sum these weighted values;
 calculate the square root of this sum (the
standard deviation).
This gives a numerical indication of how far
the returns are dispersed from the average.
There are various measures of risk: beta, standard
deviation, Rsquared. (See a comparison
of these three measures of risk.) cFor an excellent
primer on risk in securities markets, see Vanguard's
Investing
Primer. 
Return 
Square
of variance 
Probability 
Weighted
square 
$2 
(7 
2)^{2} = 25 
1/36
= .028 
0.694 
$3 
(7 
3)^{2} = 16 
2/36
= .056 
0.889 
$4 
(7 
4)^{2} = 9 
3/36
= .083 
0.750 
$5 
(7 
5)^{2} = 4 
4/36
= .111 
0.444 
$6 
(7 
6)^{2} = 1 
5/36
= .139 
0.139 
$7 
(7 
7)^{2} = 0 
6/36
= .167 
0.000 
$8 
(7 
8)^{2} = 1 
5/36
= .139 
0.139 
$9 
(7 
9)^{2} = 4 
4/36
= .111 
0.444 
$10 
(7 
10)^{2} = 9 
3/36
= .083 
0.750 
$11 
(7 
11)^{2} = 16 
2/36
= .056 
0.889 
$12 
(7 
12)^{2} = 25 
1/36
= .028 
0.694 
Statistics: 

Sum
of weighted squares 
5.833 
Standard
deviation
(square root of sum) 
2.415 
