Copyright © 2008, The McGraw-Hill Companies, Inc.
Interpreting Standard Deviation
If we compare the standard deviation from Location A with the one from Location B, we can
get a sense of how to interpret the standard deviation. The measured speeds at Location B
were much more variable, and the standard deviation reflects that. More variation means
larger deviations, which makes for larger squared deviations, which makes for a larger total
of the squared deviations, which leads to a larger standard deviation. On the other hand, at
Location A, where there was not much variation in the wind speeds, the deviations were
small, and following through the process, we found these small deviations led to a small
standard deviation.
Speaking generally, we can thus say that the size of the standard deviation is an indicator
of the amount of variation in a set of data. The key to interpreting standard deviation is this:
the higher the standard deviation, the greater the variation; the lower the standard devia-
tion, the lower the variation.Example 16.3.2 Suppose that you are considering investing in two different mutual
funds. Over the past 10 years, the annual returns of the Hopewell American Growth
Fund have had a standard deviation of 11.25%. The annual returns of the Hopewell
Adrenaline Aggressive Growth Fund have had a standard deviation of 28.4%. Which
investment has seen the greatest variation in its annual rates of return?The Adrenaline Aggressive Growth fund has a larger standard deviation. We can conclude
from this that it has had a much greater degree of variation.We must be careful, though, when comparing standard deviations, that the items that
we are comparing are on similar scales. In Example 16.2.3 they were (since both were
percents) and in all of the remaining examples and exercises of this section this will
also be true. However, the Additional Exercises at the end of this section provide
an example where differing scales can make the comparison of standard deviations
misleading.
It is usually easiest to interpret standard deviations in comparisons. Interpreting the
standard deviation in absolute terms is a bit trickier. For example, suppose you are told that
the mean exam score in your business math class was 78.2 and the standard deviation was
8.3. How can you interpret this?
Interpreting a standard deviation by itself is not as clear as interpreting one in com-
parison. From everyday situations, we all have at least a rough sense of what an “aver-
age” is. We are less likely to have a good everyday sense of what a standard deviation
is, and unfortunately standard deviation does not possess any simple, commonsense
interpretation.
However, there are a few tools that may be helpful. One is Chebyshev’s Theorem,
which gives a minimum percent of the scores from any collection of data that must fall
within a certain number of standard deviations from the average. Chebyshev’s theorem
guarantees that for any collection of data, at least ¾ (75%) of the data must fall within
2 standard deviations of the mean, and at least^8 ⁄ 9 (88.9%) must fall within 3 standard
deviations. It does not put any minimum on how many values must fall within 1 standard
deviation, though.Example 16.3.3 If the mean is 78.2 and the standard deviation is 8.3, use
Chebyshev’s theorem to interpret what this tells you about where the class’s scores
fell.Two standard deviations would be 2(8.3) 16.6.Two standard deviations below the mean would be 78.2 16.6 61.6; two standard devia-
tions above the mean would be 78.2 16.6 94.8. Chebyshev’s theorem tells us that at
least ¾ of the class scored between 61.6 and 94.8.Following the same approach with three standard deviations, Chebyshev’s Theorem tells us
that^8 ⁄ 9 of the class scored between 53.3 and 103.1.16.3 Measures of Variation 629