Chi square

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The CHI SQUARE Statistic
Tests for Goodness of Fit
and Independence

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Preview
 Color is known to affect human moods and
emotion. Sitting in a pale-blue room is more
calming than sitting in a bright-red room
 Based on the known influence of color, Hill
and Barton (2005) hypothesized that the
color of uniform may influence the outcome
of physical sports contest
 The study does not produce a numerical
score for each participant. Each participant
is simply classified into two categories
(winning or losing)

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Preview
 The data consist of frequencies or proportions
describing how many individuals are in each
category
 This study want to use a hypothesis test to
evaluate data. The null hypothesis would state
that color has no effect on the outcome of the
contest
 Statistical technique have been developed
specifically to analyze and interpret data
consisting of frequencies or proportions 
CHI SQUARE

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PARAMETRIC AND NONPARAMETRIC
STATISTICAL TESTS
 The tests that concern parameter and
require assumptions about parameter are
called parametric tests
 Another general characteristic of parametric
tests is that they require a numerical score
for each individual in the sample. In terms
of measurement scales, parametric tests
require data from an interval or a ratio scale

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PARAMETRIC AND NONPARAMETRIC
STATISTICAL TESTS
 Often, researcher are confronted with
experimental situation that do not conform to
the requirements of parametric tests. In this
situations, it may not be appropriate to use a
parametric test because may lead to an
erroneous interpretation of the data
 Fortunately, there are several hypothesis
testing techniques that provide alternatives to
parametric test that called nonparametric tests

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NONPARAMETRIC TEST
 Nonparametric tests sometimes are called
distribution free tests
 One of the most obvious differences
between parametric and nonparametric
tests is the type of data they use
 All the parametric tests required numerical
scores. For nonparametric, the subjects are
usually just classified into categories

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NONPARAMETRIC TEST
 Notice that these classification involve
measurement on nominal or ordinal scales,
and they do not produce numerical values
that can be used to calculate mean and
variance
 Nonparametric tests generally are not as
sensitive as parametric test; nonparametric
tests are more likely to fail in detecting a
real difference between two treatments

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THE CHI SQUARE TEST FOR GOODNESS OF FIT
… uses sample data to test hypotheses about
the shape or proportions of a population
distribution. The test determines how well the
obtained sample proportions fit the
population proportions specified by the null
hypothesis

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THE NULL HYPOTHESIS FOR THE GOODNESS OF FIT
 For the chi-square test of goodness of fit, the
null hypothesis specifies the proportion (or
percentage) of the population in each category
 Generally H0 will fall into one of the following
categories:
○ No preference
H0 states that the population is divided equally
among the categories
○ No difference from a Known population
H0 states that the proportion for one population are
not different from the proportion that are known to
exist for another population

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THE DATA FOR THE GOODNESS OF FIT TEST
 Select a sample of n individuals and count how
many are in each category
 The resulting values are called observed
frequency (fo)
 A sample of n = 40 participants was given a
personality questionnaire and classified into
one of three personality categories: A, B, or C
Category A Category B Category C
15 19 6

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EXPECTED FREQUENCIES
 The general goal of the chi-square test for goodness
of fit is to compare the data (the observed
frequencies) with the null hypothesis
 The problem is to determine how well the data fit the
distribution specified in H0 – hence name goodness of
fit
 Suppose, for example, the null hypothesis states that
the population is distributed into three categories
with the following proportion
Category A Category B Category C
25% 50% 25%

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EXPECTED FREQUENCIES
To find the exact frequency expected for each
category, multiply the same size (n) by the
proportion (or percentage) from the null
hypothesis
25% of 40 = 10 individual in category A
50% of 40 = 20 individual in category B
25% of 40 = 10 individual in category C

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THE CHI-SQUARE STATISTIC
 The general purpose of any hypothesis test
is to determine whether the sample data
support or refute a hypothesis about
population
 In the chi-square test for goodness of fit, the
sample expressed as a set of observe
frequencies (fovalues) and the null
hypothesis is used to generate a set of
expected frequencies (fe values)

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THE CHI-SQUARE STATISTIC
 The chi-square statistic simply measures ho
well the data (fo) fit the hypothesis (fe)
 The symbol for the chi-square statistic is χ2
 The formula for the chi-square statistic is
χ2
= ∑
(fo – fe)2
fe

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A researcher has developed three different
design for a computer keyboard. A sample of n
= 60 participants is obtained, and each
individual tests all three keyboard and identifies
his or her favorite.
The frequency distribution of preference is:
Design A = 23, Design B = 12, Design C = 25.
Use a chi-square test for goodness of fit with α
= .05 to determine whether there are significant
preferences among three design
LEARNING CHECK

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Dari https://twitter.com/#!/palangmerah
diketahui bahwa persentase golongan darah
di Indonesia adalah:
A : 25,48%,
B : 26,68%,
O : 40,77 %,
AB : 6,6 %
Golongan darah di kelas kita?
Apakah berbeda dengan data PMI?
LEARNING CHECK

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THE CHI-SQUARE TEST FOR INDEPENDENCE
 The chi-square may also be used to test
whether there is a relationship between two
variables
 For example, a group of students could be
classified in term of personality (introvert,
extrovert) and in terms of color preferences
(red, white, green, or blue).
RED WHITE GREEN BLUE ∑
INTRO 10 3 15 22 50
EXTRO 90 17 25 18 150
100 20 40 40 200

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OBSERVED AND EXPECTED FREQUENCIES
fo
RED WHITE GREEN BLUE ∑
INTRO 10 3 15 22 50
EXTRO 90 17 25 18 150
∑ 100 20 40 40 200
fe
RED WHITE GREEN BLUE ∑
INTRO 50
EXTRO 150
∑ 100 20 40 40 200

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OBSERVED AND EXPECTED FREQUENCIES
fo
RED WHITE GREEN BLUE ∑
INTRO 10 3 15 22 50
EXTRO 90 17 25 18 150
∑ 100 20 40 40 200
fe
RED WHITE GREEN BLUE ∑
INTRO 25 5 10 10 50
EXTRO 75 15 30 30 150
∑ 100 20 40 40 200

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OBSERVED AND EXPECTED FREQUENCIES
fo R W G B ∑
INTRO 10 3 15 22 50
EXTRO 90 17 25 18 150
∑ 100 20 40 40 200
(fo– fe)2
R W G B
INTRO
EXTRO
fe R W G B ∑
INTRO 25 5 10 10 50
EXTRO 75 15 30 30 150
∑ 100 20 40 40 200

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OBSERVED AND EXPECTED FREQUENCIES
fo R W G B ∑
INTRO 10 3 15 22 50
EXTRO 90 17 25 18 150
∑ 100 20 40 40 200
(fo– fe)2
R W G B
INTRO (-15)2
(-2)2
(5)2
(12)2
EXTRO (15)2
(-2)2
(-5)2
(-12)2
fe R W G B ∑
INTRO 25 5 10 10 50
EXTRO 75 15 30 30 150
∑ 100 20 40 40 200

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OBSERVED AND EXPECTED FREQUENCIES
(fo– fe)2
/fe R W G B
INTRO
EXTRO
fe R W G B
INTRO 25 5 10 10
EXTRO 75 15 30 30
(fo– fe)2
R W G B
INTRO 225 4 25 144
EXTRO 225 4 25 144

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OBSERVED AND EXPECTED FREQUENCIES
(fo– fe)2
/fe R W G B
INTRO 9 0,8 2,5 14,4
EXTRO 3 0,267 0,833 4,8
fe R W G B
INTRO 25 5 10 10
EXTRO 75 15 30 30
(fo– fe)2
R W G B
INTRO 225 4 25 144
EXTRO 225 4 25 144

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THE CHI-SQUARE STATISTIC
χ2
= ∑
(fo – fe)2
fe
χ2
= 35,6
df = (C-1) (R-1) = (3) (1) = 3
χ2
critical at α = .05 is 7,81