The Mathematics of Money

(Darren Dugan) #1

628 Chapter 16 Business Statistics


values. To get rid of the positive/negative distinction, we next square the deviations (since
the squares of both positive and negative numbers come out positive):

Speed Deviation Squared Deviation

14  2 4
15  1 1
16 0 0
17 1 1
18 2 4

The next step is to add up these squared deviations and divide by n  1, where n is the
number of values in the data set. The idea here is more or less to “average” the squared
deviations, though we divide by one less than the number of values rather than the number
of values. (The reasons for using n  1 instead of n are technical and fall outside the scope
of this section.)^6
Summing the column of squared deviations gives:

Speed Deviation Squared Deviation

14  2 4

(^15)  1 1
16 0 0
17 1 1
18 2 4
To t a l 1 0
Dividing this by 5  1  4, we get 10/4  2.5. This value is known as the variance.
Our last step is intended to undo a side effect of how we eliminated the positives and
negatives. Squaring not only makes everything positive, it also changes the magnitude of
the numbers. So, as a last step to bring things back into proportion, we take the square root
of the variance to get:
Standard deviation  
——


2.5  1.58


Before discussing the interpretation of this value, let’s get a bit more practice with the
calculation.

Example 16.3.1 Calculate the standard deviation of the wind speeds for Location B.

We build a table just as we did for Location A:

Speed Deviation Squared Deviation

2  14 196
9  7 49
16 0 0
23 7 49
30 14 196

To t a l 490

Following through with the remaining steps, we get:

Variance  490/(5  1)  490/4  122.5
Standard deviation  11.07

(^6) Actually, the rule is that you divide by n when your data set represents the entire population being studied. In this
and in most other cases we are only working from some representative sample–the wind measured at 5 moments
rather than at every moment–and so in this book we will always divide by n  1.

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