Greenwood press

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STANDARD DEVIATION
About 68 percent of the area under a normal distribution curve is between 
-1 and 1 standard deviations (z scores). 

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pressure, glucose content in urine, and cholesterol in blood can have different
distributions based on sex or age. Some electronic sensors have the statistics for
different population groups in memory. When a reading for a particular type of
patient is more than two standard deviations from the mean for his or her group,
the instrument will sound a tone, alerting nurse or doctor to the critical value. A
dynamic instrument that accounts for patient’s variables establishes a more pre-
cise diagnosis of medical problems.
When the standard deviation is computed from statistics on many samples,
such as a standard deviation of ACT composite school averages for many
schools, the standard deviation is called a standard error. Survey statistics in
newspapers are often reported as a range of values, such as in “our survey of 250
randomly selected adults showed that 62 percent of the residents oppose the new
highway. The margin of error was 6 percent.” In most cases, the margin of error
for a reported statistic is two standard errors. The report of the survey results
would be “62% ± 2• Standard Error.” This gives a range of values that is likely
(95 percent certain) to trap the percentage that would have been obtained had the
entire population been surveyed. So the newspaper would be saying, “If the
entire population of residents had been surveyed, there is a 95 percent chance
that the true proportion is between 56 percent and 68 percent.” In the weeks prior
to national and state elections, you will read about polls that indicate which can-
didate is ahead in the race, and whether the candidate has a clear lead. If candi-
dates are separated by two standard errors, the newspaper would project a win-
ner. The sampling of voters as they leave polling booths is a method that
television networks have used to make predictions of winners on their news pro-
gramming shortly after the polls close. However, as the networks found out in the
November 2000 presidential election, it is necessary that samples be carefully
designed to be representative of the population. Had the networks followed the
cautious recommendations of statisticians, they would not have had to make their
embarrassing switches of victory reports from George Bush to Al Gore based on
the controversial voting reports from the state of Florida. 
Statistics computed on samples establish the close connection between stan-
dard deviation and the normal curve. Although the numbers in an entire popula-
tion might not follow a normal distribution, the central limit theorem states that
means of samples from the population will be normally distributed. Further, the
standard deviation of the sample means (standard error of the mean) is the stan-
dard deviation of the population divided by the square root of the sample size.
The central limit theorem is the foundation for inferential statistics, the branch of
statistics that is used to determine whether a new drug is better than older treat-
ments, whether consumers really like the flavor of a new, improved toothpaste,
when an assembly line is producing too many defects, whether students in a
school are not doing well on a state test, and when a stock price is stabilizing.
Pollsters use the central limit theorem to determine how large their samples must
be to reach a desired level of accuracy.