ANATOMY OF SIGMA: summation notation
n
x
This is Sigma. It is the Greek symbol
for uppercase “S.” Sigma is not unique
to statistics, but is a notation used in
many areas of mathematics. When you
see this notation, you will take the sum
of whatever follows it. It is read as “the
sum of.” Here, in words, it is “the sum
as i goes from one to n of x sub i.”
i 1
This is the upper index
of the sum.
When x, and its subscript
are replaced by actual numbers,
it will denote a unique value in
a data set.
i
This is a
subscript (or index).
This is the lower index of the sum.
Two examples will prove very helpful here.
Let:
x1  6
x2  3
5
Then,
 xi  6  3  17  12  2  40
The expression to the left is read as, “Sum the x sub-i as i
goes from 1 (the first value) to n (the last value).”
i 1
x3  17
x4  12
x5  2
x 8
n
 (x
i 1
_
i
 x) 2
= (6 - 8)2 + (3 - 8)2 + (17 – 8)2 + (12 – 8)2 + 2 – 8)2
=
(-2)2
+
(-5)2
+
(9)2
+(4)2
= 4 + 25 + 81 + 16 + 36
= 162
+
(-6)2
The expression to the left
is read as, “Sum the
values obtained from the
x sub-i minus the mean,
quantity squared as i goes
from 1 to n.”