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© Sven Thommesen 2011
Math: Summation and product operators (Needed in Ch 3)
[Edited 09/22/07]
Summation signs are used everywhere in statistics as a shorthand way to
describe the adding up of data values in a given data set. If you do not
understand summation notation, you cannot understand the formulas given
for various concepts in this and later chapters.
(The textbook briefly discusses this topic in Appendix C on p. 604.)
Here are a few examples of expressions using summation notation:
ii
px
7
3
()
i
i
xx
2
1
(
) (
)
n
i
x
x
y
y
The
operator is the capital Greek letter Sigma (= our „S‟).
Summation operators implicitly refer to a given data set; which data set that
is should be understood by the context. (It may be given to you on an exam
problem, for example.)
Let us use the following data set as our example:
i
X
i
Y
i

1
Bob
3
5
2
Alice
1
4
3
Xena
4
2
4
Alex
1
0
5
Joe
3
3
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This data set has 5
observations
, numbered from 1 to 5. We use the letter
“i” as an
index
to the data set; “i” stands for “observation number” or
“record number” (in data processing terms).
For each observation, we have 3 data
variables
: the name of the subject,
and two measurements which we refer to here only as X and Y. (In an actual
data set, we might choose to use the letter “X” to refer to someone‟s height,
or the letter Y to refer to someone‟s yearly income.)
We see that data about subject Xena are contained in observation #3 (i=3).
We refer to the Xmeasurement for Xena as X
3
(we have X
3
= 4) and the Y
measurement for Xena as Y
3
(we have Y
3
= 2).
Our sample data set has a total of 5 observations. By convention, we use the
letter N to designate the
total number of observations
; we use capital N
(N=5) if the data set represents a population, and a small n (n=5) if it is a
sample. [These are just conventions among statisticians.]
With those explanations taken care of, we can generalize the above
examples with an expression of the following kind:
()
b
i
ia
fx
where f(x
i
) is some function of x.
We read this expression as follows: “
The sum of f(x
i
) from i=a to i=b
.”
Operationally, that means:
First,
for each observation
, starting with #a and ending with #b,
calculate the value of the expression “f(x
i
)”.
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