Math 141: Business Mathematics I
Fall 2015
§8.3 Variance and Standard Deviation
Instructor: Yeong-Chyuan Chung
In the previous section, we considered three different measures of central tendency, namely
the mean, median, and mode. However, knowledge of the central tendency of a probability
distribution is often not enough to give an accurate picture of the distribution.
Example. Consider the following two probability distributions:
x
1
P (X = x) .1
y
P (Y = y)
2 3 4
.1 .6 .1
1 2 3
.3 .1 .2
5
.1
4 5
.1 .3
Draw histograms representing each distribution. Compute E(X) and E(Y ).
Besides the measures of central tendency, we need measures of the degree of dispersion, or
spread, of a probability distribution about its mean.
The variance of a random variable X associated with the probability distribution is a
measure of the spread of the probability distribution about its mean. A distribution with a
larger spread about its mean will have a larger variance.
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§8.3 Variance and Standard Deviation
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Suppose that a random variable X has probability distribution
x
P (X = x)
x1
p1
x2
p2
x3
p3
···
···
xn
pn
and has expected value E(X) = µ. Then the variance of X is given by
V ar(X) = p1 (x1 − µ)2 + p2 (x2 − µ)2 + · · · + pn (xn − µ)2 .
To make sense of this formula, note that the differences x1 − µ, x2 − µ, . . . , xn − µ measure
the deviations of x1 , . . . , xn from the mean µ. We take the square of each of them so that
we do not get cancellations due to positive and negative contributions. Then multiplying by
the respective probabilities attaches the appropriate weights to each of these deviations to
give us the correct overall picture.
Suppose that X is a random variable whose values are measured in certain units, say ft.
Then due to the squaring in the formula above, we see that V ar(X) will be measured in ft2
rather than ft. To remedy this situation, we sometimes consider the standard deviation
of X, which is simply the square root of V ar(X). In other words, the standard deviation of
X is given by
p
σ = V ar(X).
Using the graphing calculator to compute means and standard deviations
If we know the probability distribution of a random variable X, then we can use the calculator
to compute the expected value and the standard deviation of the random variable as follows.
1. Enter the x-values into list L1, and enter the corresponding probabilities into list L2.
The lists can be accessed by pressing STAT and then selecting 1:Edit.
2. Return to the home screen, press STAT, and then select 1-Var Stats, and type L1,L2.
3. A list of values will be shown. x̄ is the expected value, and σx is the standard deviation.
We can find the standard deviation in this way, and then square it to get the variance.
Example. Find the variance of the two random variables in the first example. (Verify that
they reflect the difference in spread between the two distributions.)
§8.3 Variance and Standard Deviation
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Example (Exercise 12 in the text). The happiness score, by generation, conducted in April
2013 is given in the following table (percent indicates the top-two box scores on a five-point
happiness scale):
Generation
Score (%)
Average 18+ Millennials Generation X
58.6
54.1
55.7
Boomer Silent Generation
61.0
66.7
Find the average happiness score for the five generations. What is the standard deviation for
these data.
Example (Exercise 14 in the text). Paul Hunt is considering two business ventures. The
anticipated returns (in thousands of dollars) of each venture are described by the following
probability distributions:
Venture A
Earnings
Probability
-20
.3
40 50
.4 .3
Venture B
Earnings
Probability
-15
.2
30 40
.5 .3
(a) Compute the mean and variance for each venture.
(b) Which investment would provide Paul with the higher expected return (the greater
mean)?
(c) In which investment would the element of risk be less (that is, which probability distribution has the smaller variance)?