Page 94 - Statistics for Dummies

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Part II: Number-Crunching Basics
no sense. Finally, you take the square root of 6.67, to get 2.58. The standard
deviation for these four quiz scores is 2.58 points.
Because calculating the standard deviation involves many steps, in most
cases you have a computer calculate it for you. However, knowing how to cal-
culate the standard deviation helps you better interpret this statistic and can
help you figure out when the statistic may be wrong.
Statisticians divide by n – 1 instead of by n in the formula for s so the results
have nicer properties that operate on a theoretical plane that’s beyond the
scope of this book (not the Twilight Zone but close; trust me, that’s more than
you want to know about that!).
The standard deviation of an entire population of data is denoted with the
Greek letter σ. When I use the term standard deviation, I mean s, the sample
standard deviation. (When I refer to the population standard deviation, I let
you know.)
Interpreting standard deviation
Standard deviation can be difficult to interpret as a single number on its own.
Basically, a small standard deviation means that the values in the data set
are close to the mean of the data set, on average, and a large standard devia-
tion means that the values in the data set are farther away from the mean, on
average.
A small standard deviation can be a goal in certain situations where the results
are restricted, for example, in product manufacturing and quality control. A
particular type of car part that has to be 2 centimeters in diameter to fit prop-
erly had better not have a very big standard deviation during the manufactur-
ing process. A big standard deviation in this case would mean that lots of parts
end up in the trash because they don’t fit right; either that or the cars will have
problems down the road.
But in situations where you just observe and record data, a large standard
deviation isn’t necessarily a bad thing; it just reflects a large amount of varia-
tion in the group that is being studied. For example, if you look at salaries for
everyone in a certain company, including everyone from the student intern to
the CEO, the standard deviation may be very large. On the other hand, if you
narrow the group down by looking only at the student interns, the standard
deviation is smaller, because the individuals within this group have salaries
that are less variable. The second data set isn’t better, it’s just less variable.
Similar to the mean, outliers affect the standard deviation (after all, the for-
mula for standard deviation includes the mean). In the NBA salaries example,
the salaries of the L.A. Lakers in the 2009–2010 season (shown in Table 5-2)
range from the highest, $23,034,375 (Kobe Bryant) down to $959,111 (Didier
Ilunga-Mbenga and Josh Powell). Lots of variation, to be sure! The standard

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