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As you can see, this is a difficult function to work with. Historically, tables have been developed
with calculated areas, as the calculus was once quite difficult to do. In order to do this, it was
often necessary to first convert the desired range of values to -scores. Since every normal
distribution has a different mean and standard deviation, it would be impossible to create a table
for every possible combination. Instead, since each normal distribution is of the same shape, it
made sense to create just one table that represented a mean of and a standard deviation of
. That is, we can think about every distribution as the number of standard deviations each
score is from the mean. The mean is 0 standard deviations away from the mean (it is the mean!)
and each unit represents 1 standard deviation. We can think about any distribution this way!
Normal Distribution Expected Value and Variance
A normal probability distribution can be modeled by the function
( )
√
( )
where the
expected value is , defined as a standard mean,
∑
And variance is
, defined as a standard variance,
∑( )
IMPORTANT NOTE: and represent the population mean and variance. represents the
population size. Recall that the sample variance has a divisor of , so that it is an unbiased
estimator of the population variance.
Below is an example of what a typical table would look like. We call this a standard normal
table, since it requires that values between which we would like to know areas are
“standardized.” This means they are converted to scores prior to using the table:
Statistics for Decision-Making in Business © Milos Podmanik Page 173