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Chi- square test
x
2
Chi Square test
• Symbolized by Greek
• pronounced “Ki square”
• A Test of STATISTICAL
x
2
SIGNIFICANCE for TABLE data
What do tests of statistical
significance tell us?
• Are OBSERVED RESULTS
• SIGNIFICANTLY
DIFFERENT than would be expected
• BY CHANCE
• Criteria
ά < .05
Testing Hypothesis
Chi- square test
• Evaluates whether observed frequencies
for a qualitative variable (or variables)
are adequately described by hypothesized
or expected frequencies.
– Qualitative (or categorical) data is a set of
observations where any single observation is
a word or code that represents a class or
category.
Testing Hypothesis
Chi- square test
•X2 - test for
– Test of deviation from expected frequencies:
Test whether the observed frequencies
deviate from expected frequencies (e.g. using
a dice, there is an a priori chance of 16.67%
for each number)
– Test of association: Finding relationship
between two or more independent variables
(e.g. test relation between gender and the use
of high or low accents?)
Chi- square distribution
 ( fo  fe )
   
fe

2
2




Testing Hypothesis
Chi- square test
• In the test of significance of mean we are comparing
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the mean of one sample with the hypothesized
population mean. In the test of significance of
difference between two means, we are comparing the
means of two samples.
In chi-square test, we can check the equality of more
than two population parameters (like proportions,
means).
If we classify a population into several categories with
respect to two attributes (such age & job performance)
we can then use a chi-square test to determine whether
the two attributes are independent of each other.
Testing Hypothesis
Chi- square test
•Using the Chi- square test:
• Chi square test enables to test
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for the equality of
several proportions.
Chi square is test of statistics used to test a
hypothesis that provides a set of theoretical
frequencies with which observed frequencies are
computed.
Its really just a comparison between expected
frequencies and observed frequencies among the
cells in a crosstabulation table.
Testing Hypothesis
Chi- square test
Conditions for a the application of Chi- square test (x2):
• All raw data for X2 must be frequencies / actual
numbers (not percentages & proportions)
• The expected frequency of cell should be more
than 5.
• The chi square test work only when the sample
size is large enough (n > 50).
• The observation drawn need to be random and
independent.
• A constraint must be linear.
Chi- square test
• Properties of Chi- square test:
• As t-distribution there is different chi- square
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distribution for each different number of degree
of freedom.
For very small number of degree of freedom, the
x2 distribution is severely skewed to the right. As
the number of degree of freedom increases, the
curve rapidly becomes more symmetrical until
the number reaches large values, at which point
the distribution can approximated by the normal.
The chi-square distribution is a probability
distribution.
The chi-square distribution involves squared
observations and hence it is always positive .
Chi-Square
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 ( fo  fe )2
   
fe

2




Cannot be negative because all discrepancies are squared.
Will be zero only in the unusual event that each observed
frequency exactly equals the corresponding expected
frequency.
Other things being equal, the larger the discrepancy between
the expected frequencies and their corresponding observed
frequencies, the larger the observed value of chi-square.
It is not the size of the discrepancy alone that accounts for a
contribution to the value of chi-square, but the size of the
discrepancy relative to the magnitude of the expected
frequency.
The value of chi-square depends on the number of
discrepancies involved in its calculation.
Chi- square as a test of Goodness of fit
• Chi- square test developed by Karl Pearson in 1990.
• Chi- square as a test of Goodness of fit, which is
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used to test whether or not the observed
frequency results support a particular hypothesis.
The test can be used to identify whether the
deviation between the observed and estimated
values can be because of a chance.
In some situations researchers would like to see
how well the observed frequency pattern will fit
in to the expected frequency pattern. In such
cases the chi square test is used to test whether the
fit between the observed distribution and the
expected distribution is good.
Testing Hypothesis
Steps in Chi- square test
An Example:1. Chi- square test
• Ex: Suppose that 60 children were asked as to
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which ice -cream flavour they liked out of the
three flavours of vanilla, strawberry & chocolate.
The answer are recorded as follows. (ά = 0.05)
Flavours
Numbers
Vanilla
17
Strawberry
24
Chocolate
19
• x2 =
∑ (Fo – Fe)
Fe
An Example:2. Chi- square test
• The following table depicts the expected
sales (Fe) and actual sales (Fo) of television
sates for company. Test whether there is a
substantial difference between the
observed and expected values. Using Chisquare test. (ά = 0.05)
(Fo)
57
69
51
83
44
48
35
37
(Fe)
59
76
55
75
39
53
30
48