If
you take a variable with a normal distribution and square it, then
you get a chisquare distribution
Pearson
ChiSquare Test
Example:
You survey 80 students from a university
Event
A: 30 students are men
Event
B: 50 students are women
Mutually
exclusive – all events have to add up to the total
We
can test the hypothesis that men and women occur 50/50 in a group
Expected
value for men is 40
Expected
value for women is 40
Compute
the X^{2} statistic
Notation
O_{i}
is the observed data
E_{i}
is the expected
n
is the outcomes of each event
The
degrees of freedom are df = n – 1 = 2 – 1 = 1
Chisquare
test statistic is
Reject
the H_{0}
Test
statistic is calculated in Excel using =chiinv(a,
df)
Note
– Chisquares are onetail tests, because negative numbers are
converted to positive when they are squared
Contingency
tables
The
simplest is called a 2 X 2
Example:
Students took an exam
Occurrences 
Failed 
Passed 
Marginal Total 
Male 
20 
30 
50 
Female 
10 
20 
30 
Marginal Total 
30 
50 
80 
You
have to use the number of occurrences
Have
to calculate the expected occurrences from the marginals
Expected 
Failed 
Passed 
Marginal Total 
Male 


50 
Female 


30 
Marginal Total 
30 
50 
80 
The
degrees of freedom are df = (columns – 1)(rows – 1) = 1 (1) = 1
Chisquare
test statistic is
Fail
to reject the H_{0}
hypothesis and conclude males and females are equal when taking
the exam
Note
– There is a fast way to calculate X^{2}
for a 2 X 2 Contingency Table
Occurrences 
Failed 
Passed 
Marginal
Total 
Male 
a 
b 
a
+ b 
Female 
c 
d 
c
+ d 
Marginal Total 
a + c 
b + d 
a + b + c +d 
Redoing
the example using the fast method
Note
– You can build contingency tables with any dimensions
The
Chisquare test may be poor if
Grand
total is less than 100
Or
a cell total is less than 10
Then
you should use Yate’s Correction
