Transcript Greek and Roman letters

Deciphering Math Notation
Billy Skorupski
Associate Professor, School of Education
Agenda
• General overview of data, variables
• Greek and Roman characters in math and statistics
• Parameters vs. Statistics
• Common operators and how they work
• Particular focus on Summation and Product operators,
and the associated use of SUBSCRIPTS
• Miscellaneous statistical symbols and terminology
• List a few common symbols from Set Theory
(to be discussed in Probability)
An Example Data Set
“i” is an “indexing”
variable: i = 1, 2, …, N
N=9
Let’s say “X” contains
measurements for a
numeric variable
Let’s say “Y” indicates
designations for a
categorical variable
i
1
2
3
4
5
6
7
8
9
X
15
27
32
11
23
21
9
44
26
Y
1
1
1
1
2
2
2
3
3
An Example Data Set
i
1
“X” and “Y” are column
2
vectors of length 9.
3
Together, “X” and “Y”
4
make a 9 x 2 matrix
5
In most cases, your data 6
will have N rows, one for 7
every subject, and one 8
column per variable.
9
X
15
27
32
11
23
21
9
44
26
Y
1
1
1
1
2
2
2
3
3
Repeated Measures
i
1
“X” has been observed
on 3 occasions (e.g., at 2
Time1, Time2, Time3).
3
4
We call this “wide
5
format”
6
“Long format” would
7
need 27 rows: N people
8
x 3 observations per.
9
Y
1
1
1
1
2
2
2
3
3
X1
12
17
30
11
13
15
11
39
25
X2
15
27
32
11
23
21
9
44
26
X3
25
32
27
25
33
29
11
46
32
Repeated Measures
i
1
“long format” (only first
1
3 subjects)
1
“Y” is repeated 3 times 2
for each subject. “T”
2
indicates which
2
observation of “X”
3
The 9 “X” values are the
3
first 3 rows of “X” from
3
the previous slide
Y
1
1
1
1
1
1
1
1
1
T
1
2
3
1
2
3
1
2
3
X
12
15
25
17
27
32
30
32
27
Greek and Roman letters
•The purpose of most (all?) data analysis is to
make an inference about population
PARAMETERS that exist as part of the
POPULATION. We can’t directly observe
them, so we make educated guesses by
collecting a SAMPLE of data and calculating
STATISTICS.
Greek and Roman letters
• Parameters are almost always indicated as Greek
letters. Corresponding statistics (parameter
estimates) are indicated in one of two ways:
1. Using the Roman letter that
corresponds to the Greek :   b
2. Using a " hat" over the Greek letter :   ˆ
Greek and Roman letters
•So, Greek letters (e.g., a, , g, d, …) are used
to indicate population parameters, fixed
constants out there in the world (things we
are trying to estimate). Parameter estimates
come from samples, (that’s the job of
inferential statistics) and are indicated by
Roman letters or Greek letters with hats
Greek and Roman letters
• Articles using such symbols will either adopt
standard practice (e.g., use 0, 1, … , p as
population regression coefficients), or they will
establish the notation to be used in the paper.
• For example, if more than one regression model is
presented, one model may use a0, a1, … , ap as
coefficients, the next may use 0, 1, … , p, and the
next may use g0, g1, … , gp, and so on.
Greek and Roman letters
st
1
Check out the table
in the Handout…
Operators
•Check out the 2nd and 3rd tables…
•Most symbols are quite familiar, but S and
P as operators can be confusing at first...
•S (a Greek upper-case “S”) is for
“Summation” (Add them up)
•P (a Greek upper-case “P”) is for
“Product” (Multiply them)
Subscripts
• Subscripts are variables that index other
variables. For example, the variable “i” in our
example data set, whose only meaning is the
serial position of the subjects in the data set.
N
X
X
i 1
N
i
When you see “S” it means, “add up the variables that
appear to the right.” The “i = 1” at the bottom of S and
the “N” at the top are instructions. “i” will be an
indexing variable that starts at 1 and goes to N.
Subscripts
• Often, if the instructions are to add up all N of
the values, the summation will be presented in a
shorter form without subscripts:
N
X
X
i 1
N
i
or
X

X
N
Another Example…
• Population and Sample Variance
N
 
2
X
(X
i 1
i
N
 )
N
2
s 
2
X
(X
i 1
i
 X)
2
N 1
X i has a subscript to indicate each X value
 and X have no subscripts...why?
ANOVA Example
• Let’s say we’ve conducted an experiment after randomly
assigning participants to one of three treatment
conditions. For each subject in each group, we measure
the dependent variable, “X”
• Each person’s score can be notated as Xij (or sometimes
X[i,j]), the score for person “i” in group “j”.
• “i” will go from 1 to nj while “j” goes from 1 to M, the
number of groups (M=3, in this case)
A one-way ANOVA table
Source
Between
SS (Sum of
Squares)
M
n (X
j 1
M
Within
j
j
 X)
nj
 ( X
j 1 i 1
ij
2
 X j)
2
df
MS
SS
M-1
df B
M
N-M
=
SSW
dfW
 (n
j 1
B
j
 1)
FMS
B
MSW
p
Sig.
N
Total
2
(
X

X
)
 i
N-1
i 1
MS = “Mean Square” which is just another name for “variance”
One more (trickier) example…
•Let’s say I am describing the
population Variance-Covariance
matrix, S, among “P” variables
(P=5). The elements are referred to
as ij. What if I want to add up just
the elements in the lower triangle?
Population Variance-Covariance Matrix, S
11
12
13
14
15
21
22
23
24
25
31
32
33
34
35
41
42
43
44
45
51
52
53
54
55
Sum down the rows, across the columns…
• Say “i” are the rows, and “j” are the columns:
P
Sum of lower triangle    ij
i 1 j i
Set Theory
•The notation presented in the final table on
Set Theory will be very useful for various
probability statements. This notation will
also sometimes appear in Summation and
Product notation when creating subsets of
members for aggregating data.
Thanks!
Any Questions, Discussion?