322 UNCERTAINTY IN FUTURE EVENTS
Thus, buying insurance lowers the expected cost of an accidentby $375. To evaluatewhether
we should buy insurance, we must also account for the cost of the insurance.Thus, these expected
costs are combined with the $0 for self-insuring (total $411) and the $800 for insuring (total')$836). Thus self-insuring has an expefted value cos,tthat is $425 less per year;($836 - $411).
This is not surprising, since the premiums collected must cover both.the costs of operating the
insurance company and the expected value of the payouts.
This is also an exampleofexpectedv(llues alone not determining the decision.Buying
iIlsuranc,ehas an expected costtb,atis $425perye~ highet, but tb,atinsunll1ce}imitsthe maxiInutn
loss to $500 rather thaI)$13,000.:rbe$425may be worth spendi,Ilgto avoid thatpsk.Risk
Risk can be thought of as the chance of getting an outcomeother than the expected value-
with an emphasis on something negative. One common measure of risk is the probability
of a loss (see Example 10-6).The other common measure is thestandard deviation(a),
which measures the dispersion of outcomes about the expected value. For example, many
students have used the normal distribution in other classes. The normal distribution has
68% of its probable outcomes within ::1:1stand~d deviation of the mean and 95% within
::1:2standard deviations of the mean.
Mathematically, the standard deviation is defined as the square root of the variance.
This term is defined as the weighted averageof the squareddifferencebetweenthe outc.omes
of the random variable X and its mean. Thus the larger the difference,between the mean
and the values, the larger are the standard deviation and the variance.This is Equation 10-6:Standard deviation(a) =V[EV(X - mean)2] (10-6)
Squaring the differences between individual outcomes roWthe EV ensures that positive. -.'
and negativedeviationsreceivepositiveweights. Consequently,negativevaluesfor the stan-
dard deviation are impossible, and they instantly indicatearithmeticmistakes. The standard
deviation equals. 0 if only one outcome is possible. Otherwise, the standard deviation is
positive.
This is not the standard deviation formula built into most calculators, just as the
weighted average is not the simple average built into most calculators. The calculator
formulas are forNequally likely data points from a randomly drawn sample, so that each
probability is 1/N.In economic analysis we will use a weighted average for the squared
deviations since the outcomes are not equally likely.
The second difference is that for calculations (by hand or the calculator), it is easier
to use Equation 10-7, which is shown to be equivalent to Equation 10-6 in introductory
probability and statistics texts.Standard deviation(a) =V{EV(X2) - [EV(X)]2} (10-7)
- {Outcomei. xpeA)+ Outcome~ xPCB)+... - expected value2} (10-7')
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