Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

SECTION 1.4 | Statistical Notation 27
To compute ΣX 2 , the correct order of operations is to square each score and then find
the sum of the squared values. The computational table shows the original scores and the
results obtained from squaring (the first step in the calculation). The second step is to find
the sum of the squared values, so we simply add the numbers in the X 2 column:
ΣX 2 = 9 + 1 + 49 + 16 = 75
The final calculation, (ΣX) 2 , includes parentheses, so the first step is to perform the
calculation inside the parentheses. Thus, we first find ΣX and then square this sum. Earlier,
we computed ΣX = 15, so
(ΣX) 2 = (15) 2 = 225
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EXAMPLE 1.4
Use the same set of four scores from Example 1.3 and compute Σ(X – 1) and Σ(X – 1) 2 . The
following computational table will help demonstrate the calculations.
X (X 2 1) (X 2 1) 2
3 2 4
1 0 0
7 6 36
4 3 9
The first column lists the
original scores. A second
column lists the (X − 1)
values, and a third column
shows the (X − 1) 2 values.
To compute Σ(X – 1), the first step is to perform the operation inside the parentheses. Thus,
we begin by subtracting one point from each of the X values. The resulting values are listed
in the middle column of the table. The next step is to add the (X – 1) values, so we simply
add the values in the middle column.
Σ(X – 1) = 2 + 0 + 6 + 3 + = 11
The calculation of Σ(X + 1) 2 requires three steps. The first step (inside parentheses) is
to subtract 1 point from each X value. The results from this step are shown in the middle
column of the computational table. The second step is to square each of the (X – 1) values.
The results from this step are shown in the third column of the table. The final step is to add
the (X – 1) 2 values, so we add the values in the third column to obtain
Σ(X – 1) 2 = 4 + 0 + 36 + 9 = 49
Notice that this calculation requires squaring before adding. A common mistake is to
add the (X – 1) values and then square the total. Be careful!
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EXAMPLE 1.5
In both of the preceding examples, and in many other situations, the summation operation
is the last step in the calculation. According to the order of operations, parentheses,
exponents, and multiplication all come before summation. However, there are situations
in which extra addition and subtraction are completed after the summation. For this
example, use the same scores that appeared in the previous two examples, and compute
ΣX – 1.
With no parentheses, exponents, or multiplication, the first step is the summation. Thus,
we begin by computing ΣX. Earlier we found ΣX = 15. The next step is to subtract one
point from the total. For these data,
ΣX – 1 = 15 – 1 = 14
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