This document provides an overview of chi-square tests, including: - Chi-square tests determine if observed and expected frequencies differ significantly. They are used in fields like medical research. - There are tests for goodness of fit and independence. Goodness of fit compares data to a hypothesized distribution. Independence examines the relationship between two variables. - Examples demonstrate applying chi-square tests, calculating statistics, determining significance. The independence example found age impacts political party choice.

1. ACHARYA NARENDRA DEVA UNIVERSITY OF AGRICULTURE &
TECHNOLOGY, KUMARGANJ, AYODHYA (U.P.) 224229
Assignment
on
Chi-square test
Course No : STAT-502 4(3+1)
Course name : Statistical methods for applied sciences
Presented to : Presented by :
Dr. Vishal Mehta Vikas Yadav
Assistant Professor Id. No. A-11153/19/22
Department of Agril. Statistics Ph. D. 1st Semester
Soil Science and Agril. Chemistry

2. Content:
• Introduction
• Properties of Chi-square test
• Limitations of Chi-square test
• Type of Chi-square test
1. Chi-square test for goodness of fit
2. Chi-square test for independence
• Chi-square test for goodness of fit: Example
• Chi-square test for independence: Example
• References

3. Introduction:
• Chi-square (χ2) test is a statistical method used to
determine whether there is a significant difference
between the observed and expected frequencies in one
or more categories. It is commonly used in various
fields such as medical research, social sciences, and
business to test the goodness of fit, independence, and
association.

4. Properties
The chi-square test has the following significant properties:
1.If you multiply the number of degrees of freedom by two,
you will receive an answer that is equal to the variance.
2.The chi-square distribution curve approaches the data is
normally distributed as the degree of freedom increases.
3.The mean distribution is equal to the number of degrees of
freedom

5. Limitations of Chi-Square Test
There are two limitations to using the chi-square test that you
should be aware of.
• The chi-square test, for starters, is extremely sensitive to
sample size. Even insignificant relationships can appear
statistically significant when a large enough sample is used.
Keep in mind that "statistically significant" does not always
imply "meaningful" when using the chi-square test.

6. • Be mindful that the chi-square can only determine whether
two variables are related. It does not necessarily follow that
one variable has a causal relationship with the other. It would
require a more detailed analysis to establish causality

7. Types of chi-square test
1. Chi-square test for goodness of fit
2. Chi-square test for independence

8. Chi-square test for goodness of fit:
• Chi-square test for goodness of fit is used to determine
whether the observed data follow a certain distribution. For
example, if we want to know whether the observed data
follow a normal distribution or not, we can use the chi-square
test for goodness of fit. The test compares the observed
frequencies with the expected frequencies based on the null
hypothesis.

9. Chi-Square Goodness of Fit Test: Formula
• A Chi-Square goodness of fit test uses the following null and
alternative hypotheses
• H0: (null hypothesis) A variable follows a hypothesized
distribution.
• H1: (alternative hypothesis) A variable does not follow a
hypothesized distribution.

10. We use the following formula to calculate the Chi-Square test
statistic X2:
X2 = Σ(O-E)2 / E
where:
Σ: is a fancy symbol that means “sum”
O: observed value
E: expected value

11. Chi-Square test Goodness of Fit Test: Example
• A shop owner claims that an equal number of customers
come into his shop each weekday. To test this hypothesis, an
independent researcher records the number of customers that
come into the shop on a given week and finds the following:
• Monday: 50 customers
• Tuesday: 60 customers
• Wednesday: 40 customers
• Thursday: 47 customers
• Friday: 53 customers

12. Solution:
We will use the following steps to perform a Chi-Square goodness
of fit test to determine if the data is consistent with the shop
owner’s claim
Step 1: Define the hypotheses.
We will perform the Chi-Square goodness of fit test using the
following hypotheses:
• H0: An equal number of customers come into the shop each day.
• H1: An equal number of customers do not come into the shop
each day.

13. Step 2: Calculate (O-E)2 / E for each day.
There were a total of 250 customers that came into the shop
during the week. Thus, if we expected an equal amount to
come in each day then the expected value “E” for each day
would be 50
• Monday: (50-50)2 / 50 = 0
• Tuesday: (60-50)2 / 50 = 2
• Wednesday: (40-50)2 / 50 = 2
• Thursday: (47-50)2 / 50 = 0.18
• Friday: (53-50)2 / 50 = 0.18

14. • Step 3: Calculate the test statistic X2.
X2 = Σ(O-E)2 / E = 0 + 2 + 2 + 0.18 + 0.18 = 4.36
• Step 4: Calculate the p-value of the test statistic X2.
The p-value associated with X2 = 4.36 and n-1 = 5-1 = 4
degrees of freedom is 0.359472

15. Conclusion
• Since this p-value is not less than 0.05, we fail to reject the
null hypothesis. This means we do not have sufficient
evidence to say that the true distribution of customers is
different from the distribution that the shop owner claimed

16. Chi-square test for independence:
Chi-square test for independence is used to determine whether
there is a relationship between two variables. For example, if
we want to know whether there is a relationship between
smoking and lung cancer, we can use the chi-square test for
independence. The test compares the observed frequencies
with the expected frequencies based on the null hypothesis that
there is no relationship between the two variables.

17. Assumptions
• Both variables are CATEGORICAL
• Observations are INDEPENDENT
• The COUNT for each category is GREATER THAN 5
• Each count in a category is MUTUALLY EXCLUSIVE
• Data is chosen RANDOMLY

18. Chi-Square test for independence : Example
• We want to see if age has an impact on what political party
you vote for. We collect a random sample of 135 people and
display it in the following contingency table broken down by
age and political party.

20. Solution
Hypothesis
Lets start by stating our hypotheses:
• H_0: Age has no impact on the political party you vote for.
The two variables are independent.
• H_1: Age does have an impact on the political party. The two
variables are dependent.

21. Significance Level and Critical Value
For this example we will use a 5% significance level. As we
have 2 degrees of freedom (using the formula above):
v = (3 - 1) (2 - 1) = 2
Using the significance level, degrees of freedom and Chi-
Square probability table we find our critical value to be 5.991.
This means our Chi-Square statistic needs to be greater than
5.991 in order for us to reject the null hypothesis and the
variables to not be independent

22. • Calculating Expected Counts
We now need to determine the expected count frequency for each cell in
our contingency table. These are the expected values if the null
hypothesis is true and is calculated using the following formula:
Er,c = nr*nc /nT
Where n_r and n_c are the row and column totals for certain categories
and n_T is the total number of counts.
For example, the expected count for ages 18–30 who voted Liberals is:
E1,1 = 35*90/135 = 23.3
We can then populate the contingency table with these expected values
(in brackets):

24. Chi-Square Statistic
It is now time to calculate the Chi-Square statistic using the
formula above
Χ2
2=(10-23.3)2/23.3 + (30-30)2/30 + (50–36.7)2/36.7 +
(25–11.7)2/11.7 + (15–15)2/15 + (5–18.3)2/18.3
This equals 37.2
Therefore, our statistic is much greater than the critical value and
so we can reject the null hypothesis

25. Conclusion
In this article we have described and shown an example of the
Chi-Square test of independence. This test measures if two
categorical variables are dependent on each-other. This is used
in Data Science for Feature Selection where we only want
modelling features that have an effect on the target.

26. References
1. Wikipedia
2. www.towarddatascience.com
3. www.statology.org
4. Agresti, A. (2018). An introduction to categorical data
analysis. Wiley.
5. Kothari, C. R. (2004). Research methodology: methods and
techniques. New Age International.