The Summation Sign and Related Notation
Sometimes we’ll want to sum up values across a number of individuals in a sample.
To
denote this, it’s convenient to use a standard notation and a mathematical operator known
as the summation sign.
Notation
N = number of individuals
X
= variable (can use other letters as well, we will use
Y
in this class)
X
1
to X
N
= values of variable
X
for individuals 1 to N (arbitrarily ordered)
X
i
= value of variable
X
for individual i, where i = 1 to N.
Summation
N
Σ
X
i
=
X
1
+ X
2
+ X
3
+ X
4
+ ... + X
N1
+ X
N
i=1
The Greek symbol sigma is used to denote summation.
The values below and above
Σ
set the limits of the summation.
In this instance, you start with the value of
X
for the 1st
individual (
X
1
) and sum all values through to the
X
value for the Nth individual (
X
N
).
Algebraic Properties of Summation
N
N
1.
Σ
cX
i
=
c
Σ
X
i
i=1
i=1
N
2.
Σ
c
=
c
N
i=1
N
N
N
3.
Σ
(
X
i
+ Y
i
)
=
Σ
X
i
+
Σ
Y
i
i=1
i=1
i=1
An Important Note About Summation
As with all algebraic operations, brackets determine what the summation operator applies
to and what it doesn’t apply to.
Thus,
N
N
N
N
Σ
X
i
+
4
is
not
the same as
Σ
(
X
i
+
4);
Σ
X
i
2
is
not
the same as (
Σ
X
i
)
2
i=1
i=1
i=1
i=1
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A Guide to the Summation Sign, Groceries, and Baby Wipes
When you go to the grocery store and pick up a bunch of items, how does the cashier
determine the total price of the items?
Naturally, simply by adding up the prices of the
individual items. He or she scans in the price of the first item grabbed, scans in the price
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 Spring '08
 Gangestad
 Addition, Grocery store, Summation, summation sign, N. Summation

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