Introductory Statistics Lectures
Summation Notation
Compact notation for sums.
Anthony Tanbakuchi
Department of Mathematics
Pima Community College
Redistribution of this material is prohibited
without written permission of the author
2009
(Compile date: Tue May 19 14:48:06 2009)
Contents
1 Summation Notation
1.1 Introduction . . . . . . .
1.2 Summation notation . .
1.3 Notation: Greek alphabet & symbols . . . . .
1.4
1.5
1.6
1
1
2
Notation for sets of data
Summation with R . . .
Summary . . . . . . . .
Summation Notation
1.1
Introduction
In statistics, we often need to sum sets of numbers.
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
+ 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20
+ 21 + 22 + 23 + 2 + 25 + 26 + 27 + 28 + 29 + 30
+ 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40
+ 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50
Writing out a sum can be tedious!
A simpler way to write the above expression would be:
50
X
i=1
2
4
5
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1.2
1.2 Summation notation
Summation notation
Summation notation.
Definition 1.1
P
The summation sign appears as the greek symbol
(capitol sigma)
and indicates a sequence of sums.
n
X
f (i) =
i=1
n
X
(expression involving i)
(1)
i=1
i = 1 Indicates that the index variable is i and starts at 1.
n The index variable stops at n.
The index variable is always incremented by 1.
Example 1.
5
X
i=1+2+3+4+5
i=1
Question 1. Expand the following expression:
8
X
i2
i=5
1.3
Notation: Greek alphabet & symbols
Commonly used greek letters in statistics
, , , , , , , , ,
Hat notation for estimates
If we estimate x, we denote it as x
(x-hat). A hat over a variable indicates it
is an estimate.
1.4
Notation for sets of data
In statistics we often deal with sets of data. For example, if we have a class of
5 students we can write their ages as:
x = {21, 25, 22, 21, 23}
Anthony Tanbakuchi
MAT167
Summation Notation
A
B
E
Z
H
I
K
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Alpha
Beta
Gamma
Delta
Epsilon
Zeta
Eta
Theta
Iota
Kappa
Lambda
Mu
Nu
Xi
Omicron
Pi
Rho
Sigma
Tau
Upsilon
Phi
Chi
Psi
Omega
Table 1: Upper and lower case greek letters.
Where:
x1 = 21
x2 = 25
x3 = 22
x4 = 21
x5 = 23
Summing a set of data
We can write the sum of the data set x = {21, 25, 22, 21, 23} as
5
X
xi = x1 + x2 + x3 + x4 + x5
i=1
If the data set is known, then we can simplify the notation:
X
xi = 21 + 25 + 22 + 21 + 23
Expand the following expressions if y = {3, 4, 2, 1}
Question 2.
X
Anthony Tanbakuchi
(yi 1)2
MAT167
4 of 5
1.5 Summation with R
Question 3.
X
(yi 1)
2
Given y = {a, 3a, a}, show the left and right sides are equal by expanding
the summation notation and simplifying it. Assume that a is an unknown
constant.
Question 4.
X
1.5
R Command
yi2 11a = 11a(a 1)
Summation with R
Summation:
sum(x)
Where x is a vector.
P
P 2
CAUTION: sum(x^2) = x2i where sum(x)^2 = ( xi )
P
Example 2 (Summation in R). Given x = {2, 3, 7}, find
xi :
R: x = c ( 2 , 3 , 7 )
R: t o t a l = sum ( x )
R: t o t a l
[ 1 ] 12
Example 3 (Summation in R). Now find
P 2
(xi 2):
R: sum ( x 2 2 )
[ 1 ] 56
Anthony Tanbakuchi
MAT167
Summation Notation
1.6
5 of 5
Summary
n
X
(expression involving i)
i=1
i = 1 Indicates that the index variable is i and starts at 1.
n The index variable stops at n.
The sum of all the data in x is written as:
X
xi
Summations in R use the sum(x) function.
Anthony Tanbakuchi
MAT167
