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Chi Square Test of Independence Explained

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Ken Plummer
Ken Plummer

This document discusses the chi-square test of independence and how it can be used to determine if two variables are independent of each other. It provides an example using a contingency table to analyze the relationship between gender and voter turnout. The expected values for each cell are calculated based on the marginal totals. The chi-square statistic is then used to determine if the differences between observed and expected values are statistically significant, which would indicate the variables are dependent rather than independent.

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Chi Square Test of Independence Conceptual
Questions of Independence
Questions of independence are actually the flip side of questions of relationship. If a variable is independent of another variable, then functions in one will not be accompanied by functions in the other.
Like the higher the temperature the stronger people are.
Like the higher the temperature the stronger people are.
Questions of Independence are often posed as questions of bias.
For example, the question, “Are admissions decisions at a local community college fair?” can reasonably be interpreted as a question of independence (or bias).
If fairness is taken to mean that there is proportional representation of minority and majority students that mirrors the local proportions, then a test of independence can estimate whether admissions are “fair”.
The question becomes “Are admissions decisions independent of majority/minority status?”
Assuming that majority students are similar in their preparation and motivation as minority students and they apply to the community college in proportionally similar numbers as minority students, then a fair admissions process should be independent of majority status and render proportions of admissions that are similar to proportions of majority and minority students in the local populations
INDEPENDENT EXAMPLE: If you are a minority you are neither more likely nor less likely to be admitted.
Failure to be independent would indicate bias.
Failure to be independent would indicate bias. BIAS EXAMPLE: If you are a minority you are more likely to be admitted.
Failure to be independent would indicate bias. BIAS EXAMPLE: If you are a minority you are more likely to be admitted. BIAS EXAMPLE: If you are a minority you less likely to be admitted
Failure to be independent would indicate bias. BIAS EXAMPLE: If you are a minority you are more likely to be admitted. BIAS EXAMPLE: If you are a minority you less likely to be admitted. You will use certain statistical methods (like the chi square test of independence) to determine if independence is significant or not.
Here is an example taken from http://omega.albany.edu:8008/mat108dir/chi2i ndependence/chi2in-m2h.html:
Here is an example taken from http://omega.albany.edu:8008/mat108dir/chi2i ndependence/chi2in-m2h.html: In a certain town, there are about one million eligible voters. A simple random sample of 10,000 eligible voters was chosen to study the relationship between gender and participation in the last election.
Here is an example taken from http://omega.albany.edu:8008/mat108dir/chi2i ndependence/chi2in-m2h.html: In a certain town, there are about one million eligible voters. A simple random sample of 10,000 eligible voters was chosen to study the relationship between gender and participation in the last election. The results are summarized in the following 2X2 (read two by two) contingency table:
In a certain town, there are about one million eligible voters. A simple random sample of 10,000 eligible voters was chosen to study the relationship between gender and participation in the last election. The results are summarized in the following 2X2 (read two by two) contingency table: Men Women __________________________ Voted 2792 3591 Didn't vote 1486 2131
We want to check whether being a man or a woman (columns) is independent of having voted in the last election (rows). In other words is “gender and voting independent”? Men Women __________________________ Voted 2792 3591 Didn't vote 1486 2131
Solution:
Solution: In order to answer the question we need to build a test of hypothesis. We have
Solution: In order to answer the question we need to build a test of hypothesis. We have Null Hypothesis = ‘Gender is independent of Voting’
Solution: In order to answer the question we need to build a test of hypothesis. We have Null Hypothesis = ‘Gender is independent of Voting’ Alternative Hypothesis = ‘Gender and Voting are dependent’
Solution: In order to answer the question we need to build a test of hypothesis. We have Null Hypothesis = ‘Gender is independent of Voting’ Alternative Hypothesis = ‘Gender and Voting are dependent’ After specifying the Null Hypothesis, we need to compute the expected table under the assumption that rows and columns are in fact independent.
As you can see we have the observed table below:
As you can see we have the observed table below: Men Women __________________________ Voted 2792 3591 Didn't vote 1486 2131 We need to create an expected table and then determine if the difference between the observed and expected are significant:
As you can see we have the observed table below: Men Women __________________________ Voted 2792 3591 Didn't vote 1486 2131 We need to create an expected table and then determine if the difference between the observed and expected are significant:
As you can see we have the observed table below: Men Women __________________________ Voted 2792 3591 Didn't vote 1486 2131 We need to create an expected table and then determine if the difference between the observed and expected are significant: Observed Numbers Expected Numbers Difference
Remember that the smaller the DIFFERENCE, the better the fit which in this case would favor INDEPENDENCE between gender and voting tendencies.
Remember that the smaller the DIFFERENCE, the better the fit which in this case would favor INDEPENDENCE between gender and voting tendencies. Observed Numbers Expected Numbers Difference
Inversely, the larger the DIFFERENCE the worse the fit which in this case would indicate that gender and voting tendencies are dependent upon one another.
Inversely, the larger the DIFFERENCE the worse the fit which in this case would indicate that gender and voting tendencies are dependent upon one another. Observed Numbers Expected Numbers Difference
We use Chi-Square distribution to determine if that difference is significant or not.
We use Chi-Square distribution to determine if that difference is significant or not. We will now show you how to compute the chi-square statistic for a test of independence.
We use Chi-Square distribution to determine if that difference is significant or not. We will now show you how to compute the chi-square statistic for a test of independence. First, we compute the row and column totals along with the grand total.
Men Women ________________________________________ Voted 2792 3591 Didn't vote 1486 2131
Total Who Voted Men Women ________________________________________ Voted 2792 + 3591 = 6386 Didn't vote 1486 2131
Men Women ________________________________________ Voted 2792 3591 6386 Didn't vote 1486 + 2131 = 3617 Total Who Did Not Vote
Men Women ________________________________________ Voted 2792 3591 6386 Didn't vote + 1486 2131 3617 = 4278 Total Men
Men Women ________________________________________ Voted 2792 3591 6386 Didn't vote 1486 + 2131 3617 4278 = 5722 Total Women
Total Men & Women or Total Voted/Not Voted Men Women ________________________________________ Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000
Now we have the information we need to create an expected table. Here is the equation for calculating the expected value for the cell “Men who Voted”:
Now we have the information we need to create an expected table. Here is the equation for calculating the expected value for the cell “Men who Voted”: Expected Value(Men who voted) = (Number (all who voted) * Number (all men)) Number(total number)
Observed Men Women _ Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Men Who Voted
OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who voted) = (6386 (all who voted) * Number (all men)) Number (total number)
OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who voted) = (6386 (all who voted) * 4278 (all men) ) Number (total number)
OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who voted) = (6386 (all who voted) * 4278 (all men) ) 10000 (total number)
OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who voted) = (27306474 (all who voted * all men)) 10000 (total number)
OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who voted) = 2730.6474 ((all who voted * all men)/total number)
OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who voted) = 2731 ((all who voted * all men)/total number)
EXPECTED Men Women _ TABLE Voted 2731 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who voted) = 2731 ((all who voted * all men)/total number)
EXPECTED Men Women _ TABLE Voted 2731 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 What is the expected value for Women who Voted?
Women who voted:
Women who voted: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000
Women who voted: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who voted) = (6386 (all who voted) * 5722 (all women) ) 10000 (total number)
Women who voted: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who voted) = (6386 (all who voted) * 5722 (all women) ) 10000 (total number)
Women who voted: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who voted) = (6386 (all who voted) * 5722 (all women) ) 10000 (total number)
Women who voted: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who voted) = (36523526 ((all who voted) * (all women)) ) 10000 (total number)
Women who voted: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who voted) = (3652.3526 ((all who voted) * (all women)))/total number
Women who voted: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who voted) = (3652 ((all who voted) * (all women)))/total number
Women who voted: EXPECTED Men Women _ TABLE Voted 2731 3652 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who voted) = (3652 ((all who voted) * (all women)))/total number
Women who voted: EXPECTED Men Women _ TABLE Voted 2731 3652 6386 Didn't vote 1486 2131 3617 4278 5722 10000 What is the expected value for Men who Didn’t Vote?
Men who didn’t vote:
Men who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000
Men who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who didn’t vote) = (3617 (all who didn’t vote) * 4278 (all men) ) 10000 (total number)
Men who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who didn’t vote) = (3617 (all who didn’t vote) * 4278 (all men) ) 10000 (total number)
Men who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who didn’t vote) = (3617 (all who didn’t vote) * 4278 (all men) ) 10000 (total number)
Men who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who didn’t vote) = (15473526 ((all who didn’t vote) * (all men)) ) 10000 (total number)
Men who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who didn’t vote) = (1547.3526 ((all who didn’t vote) * (all men)) / (total number))
Men who didn’t vote: EXPECTED Men Women _ TABLE Voted 2731 3652 6386 Didn't vote 1547 2131 3617 4278 5722 10000 Expected Value (Men who didn’t vote) = (1547 ((all who didn’t vote) * (all men)) / (total number))
Men who didn’t vote: EXPECTED Men Women _ TABLE Voted 2731 3652 6386 Didn't vote 1547 2131 3617 4278 5722 10000 What is the expected value for Women who Didn’t Vote?
Women who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000
Women who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who didn’t vote) = (3617 (all who didn’t vote) * 5722 (all women) ) 10000 (total number)
Women who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who didn’t vote) = (3617 (all who didn’t vote) * 5722 (all women) ) 10000 (total number)
Women who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who didn’t vote) = (3617 (all who didn’t vote) * 5722 (all women) ) 10000 (total number)
Women who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who didn’t vote) = (20696474 (all who didn’t vote) * (all women) ) 10000 (total number)
Women who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who didn’t vote) = (2069.6474 (all who didn’t vote) * (all women)) /(total number)
Women who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who didn’t vote) = (2070 (all who didn’t vote) * (all women)) /(total number)
Men who didn’t vote: EXPECTED Men Women _ TABLE Voted 2731 3652 6386 Didn't vote 1547 2070 3617 4278 5722 10000
OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 4278 5722 10000 EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 10000
OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 4278 5722 10000 EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 10000 With the information above, we can now plug in the numbers using the Chi-square independence test.
OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 4278 5722 10000 EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 10000 With the information above, we can now plug in the numbers using the Chi-square independence test. Note – this is the same equation that is used with the Chi-square goodness of fit test:
OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 4278 5722 10000 EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 10000 With the information above, we can now plug in the numbers using the Chi-square independence test. Note – this is the same equation that is used with the Chi-square goodness of fit test: 푥2 = Σ (푂 − 퐸)2 퐸
푥2 = 횺 (푂 − 퐸)2 퐸
푥2 = 횺 (푂 − 퐸)2 퐸
푥2 = 횺 (푂 − 퐸)2 퐸 Or in this case:
푥2 = 횺 (푂 − 퐸)2 퐸 Or in this case: 푥2 = (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸
푥2 = 횺 (푂 − 퐸)2 퐸 Or in this case: 푥2 = (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸
OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 4278 5722 10000 EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 10000
OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 4278 5722 Voting Men 10000 EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 10000 푥2 = (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 Voting Men
OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 푥2 = (2792 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 Voting Men
OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 푥2 = (2792 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 Voting Men
EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 4278 5722 OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men - = Difference 푥2 = (2792 − 2731)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 Voting Men
OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 푥2 = (2792 − 2731)2 2731 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 Voting Men
OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 4278 5722 Voting Men - = Difference 푥2 = (2792 − 2731)2 2731 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 Voting Women
EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 4278 5722 OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men - = Difference 푥2 = (2792 − 2731)2 2731 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 Voting Women
EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 4278 5722 OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men - = Difference 푥2 = (2792 − 2731)2 2731 + (3591 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 Voting Women
OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 Voting Women 푥2 = (2792 − 2731)2 2731 + (3591 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸
OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 Voting Women 푥2 = (2792 − 2731)2 2731 + (3591 − 3652)2 3652 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸
OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 Non-Voting Men 푥2 = (2792 − 2731)2 2731 + (3591 − 3652)2 3652 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸
OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 Non-Voting Men 푥2 = (2792 − 2731)2 2731 + (3591 − 3652)2 3652 + (1486 − 퐸)2 퐸 + (푂 − 퐸)2 퐸
EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 4278 5722 OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men - = Difference Non-Voting Men 푥2 = (2792 − 2731)2 2731 + (3591 − 3652)2 3652 + (1486 − 퐸)2 퐸 + (푂 − 퐸)2 퐸
EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 4278 5722 OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men - = Difference Non-Voting Men 푥2 = (2792 − 2731)2 2731 + (3591 − 3652)2 3652 + (1486 − 1547)2 1547 + (푂 − 퐸)2 퐸
OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 Non-Voting Women 푥2 = (2792 − 2731)2 2731 + (3591 − 3652)2 3652 + (1486 − 1547)2 1547 + (푂 − 퐸)2 퐸
OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 Non-Voting Women 푥2 = (2792 − 2731)2 2731 + (3591 − 3652)2 3652 + (1486 − 1547)2 1547 + (2131 − 퐸)2 퐸
EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 4278 5722 OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men - = Difference Non-Voting Women 푥2 = (2792 − 2731)2 2731 + (3591 − 3652)2 3652 + (1486 − 1547)2 1547 + (2131 − 2070)2 2070
Time to Calculate:
Time to Calculate: 푥2 = (2792 − 2731)2 2731 + (3591 − 3652)2 3652 + (1486 − 1547)2 1547 + (2131 − 2070)2 2070
Time to Calculate: 푥2 = (61)2 2731 + (3591 − 3652)2 3652 + (1486 − 1547)2 1547 + (2131 − 2070)2 2070
Time to Calculate: 푥2 = 3721 2731 + (3591 − 3652)2 3652 + (1486 − 1547)2 1547 + (2131 − 2070)2 2070
Time to Calculate: 푥2 = 1.4 + (3591 − 3652)2 3652 + (1486 − 1547)2 1547 + (2131 − 2070)2 2070
Time to Calculate: 푥2 = 1.4 + (−61)2 3652 + (1486 − 1547)2 1547 + (2131 − 2070)2 2070
Time to Calculate: 푥2 = 1.4 + 3721 3652 + (1486 − 1547)2 1547 + (2131 − 2070)2 2070
Time to Calculate: 푥2 = 1.4 + 1.0 + (1486 − 1547)2 1547 + (2131 − 2070)2 2070
Time to Calculate: 푥2 = 1.4 + 1.0 + (61)2 1547 + (2131 − 2070)2 2070
Time to Calculate: 푥2 = 1.4 + 1.0 + 3721 1547 + (2131 − 2070)2 2070
Time to Calculate: 푥2 = 1.4 + 1.0 + 2.4 + (2131 − 2070)2 2070
Time to Calculate: 푥2 = 1.4 + 1.0 + 2.4 + (61)2 2070
Time to Calculate: 푥2 = 1.4 + 1.0 + 2.4 + 3721 2070
Time to Calculate: 푥2 = 1.4 + 1.0 + 2.4 + 1.8
Time to Calculate: 푥2 = 6.6
Now we determine if a 푥2of 6.6 exceeds the critical 푥2 for terms.
To calculate the 푥2 critical we first must determine the degrees of freedom as well as set the probability level.
To calculate the 푥2 critical we first must determine the degrees of freedom as well as set the probability level. The probability or alpha level means the probability of a type 1 error we are willing to live with (i.e., this is the probability of being wrong when we reject the null hypothesis). Generally this value is .05 which is like saying we are willing to be wrong 5 out of 100 times (.05) before we will reject the null-hypothesis.
Degrees of Freedom are calculated by taking the number rows and subtracting them by 1 and then multiplying the result by taking the number of columns and subtracting them by 1.
Degrees of Freedom are calculated by taking the number rows and subtracting them by 1 and then multiplying the result by taking the number of columns and subtracting them by 1. (Two rows -1) or (2-1) X (2-1) or 1X1=1. Degrees of Freedom = 1.
We now have all of the information we need to determine the critical 푥2.
We now have all of the information we need to determine the critical 푥2. We go to the Chi-Square Distribution Table and locate the degrees of freedom:
We now have all of the information we need to determine the critical 푥2. We go to the Chi-Square Distribution Table and locate the degrees of freedom: df 0.100 0.050 0.025 1 2.71 3.84 5.02 2 4.61 5.99 7.38 3 6.25 7.82 9.35 4 7.78 9.49 11.14 5 9.24 11.07 12.83 6 10.64 12.59 14.45 7 12.02 14.07 16.10 8 13.36 15.51 17.54 9 14.68 16.92 19.20 … … … …
We now have all of the information we need to determine the critical 푥2. We go to the Chi-Square Distribution Table and locate the degrees of freedom: df 0.100 0.050 0.025 1 2.71 3.84 5.02 2 4.61 5.99 7.38 3 6.25 7.82 9.35 4 7.78 9.49 11.14 5 9.24 11.07 12.83 6 10.64 12.59 14.45 7 12.02 14.07 16.10 8 13.36 15.51 17.54 9 14.68 16.92 19.20 … … … … And then we locate the probability or alpha level:
We now have all of the information we need to determine the critical 푥2. We go to the Chi-Square Distribution Table and locate the degrees of freedom: df 0.100 0.050 0.025 1 2.71 3.84 5.02 2 4.61 5.99 7.38 3 6.25 7.82 9.35 4 7.78 9.49 11.14 5 9.24 11.07 12.83 6 10.64 12.59 14.45 7 12.02 14.07 16.10 8 13.36 15.51 17.54 9 14.68 16.92 19.20 … … … … And then we locate the probability or alpha level: Where these two values intersect in the table we find the critical 푥2.
Since the chi-square goodness of fit value (6.6) exceeds the critical 푥2 (3.84) we will reject the null-hypothesis.
Since the chi-square goodness of fit value (6.6) exceeds the critical 푥2 (3.84) we will reject the null-hypothesis. Voting patterns and gender status are not statistically significantly dependent on one another.
Since the chi-square goodness of fit value (6.6) exceeds the critical 푥2 (3.84) we will reject the null-hypothesis. Voting patterns and gender status are not statistically significantly dependent on one another.
Since the chi-square goodness of fit value (6.6) exceeds the critical 푥2 (3.84) we will reject the null-hypothesis. Voting patterns and gender status are not statistically significantly dependent on one another. There actually is a significant difference.
So what is the difference between chi-square test of goodness of fit and test of independence?
A goodness-of-fit test is a one variable Chi-square test.
A goodness-of-fit test is a one variable Chi-square test. In this example, a department chair wants to know if the enrollments across three professors are equally distributed.
A goodness-of-fit test is a one variable Chi-square test. In this example, a department chair wants to know if the enrollments across three professors are equally distributed. Here is the actual, or observed, data:
A goodness-of-fit test is a one variable Chi-square test. In this example, a department chair wants to know if the enrollments across three professors are equally distributed. Here is the actual, or observed, data: OBSERVED TABLE Prof A’s Class Prof B’s Class Prof C’s Class Students enrolled 31 25 10
A goodness-of-fit test is a one variable Chi-square test. OBSERVED TABLE Prof A’s Class Prof B’s Class Prof C’s Class Students enrolled 31 25 10
A goodness-of-fit test is a one variable Chi-square test. OBSERVED TABLE Prof A’s Class Prof B’s Class Prof C’s Class Students enrolled 31 25 10
A test of independence is a two variable Chi-square test.
A test of independence is a two variable Chi-square test. For example, a department chair wants to know if women and men enrollments are equally distributed across three professor classes.
A test of independence is a two variable Chi-square test. For example, a department chair wants to know if women and men enrollments are equally distributed across three professor classes. OBSERVED TABLE Prof A’s Class Prof B’s Class Prof C’s Class Men 21 7 7 Women 10 18 3
A test of independence is a two variable (gender) Chi-square test. OBSERVED TABLE Prof A’s Class Prof B’s Class Prof C’s Class Men 21 7 7 Women 10 18 3

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Chi Square Test of Independence Explained

  • 1. Chi Square Test of Independence Conceptual
  • 3. Questions of independence are actually the flip side of questions of relationship. If a variable is independent of another variable, then functions in one will not be accompanied by functions in the other.
  • 4. Like the higher the temperature the stronger people are.
  • 5. Like the higher the temperature the stronger people are.
  • 6. Questions of Independence are often posed as questions of bias.
  • 7. For example, the question, “Are admissions decisions at a local community college fair?” can reasonably be interpreted as a question of independence (or bias).
  • 8. If fairness is taken to mean that there is proportional representation of minority and majority students that mirrors the local proportions, then a test of independence can estimate whether admissions are “fair”.
  • 9. The question becomes “Are admissions decisions independent of majority/minority status?”
  • 10. Assuming that majority students are similar in their preparation and motivation as minority students and they apply to the community college in proportionally similar numbers as minority students, then a fair admissions process should be independent of majority status and render proportions of admissions that are similar to proportions of majority and minority students in the local populations
  • 11. INDEPENDENT EXAMPLE: If you are a minority you are neither more likely nor less likely to be admitted.
  • 12. Failure to be independent would indicate bias.
  • 13. Failure to be independent would indicate bias. BIAS EXAMPLE: If you are a minority you are more likely to be admitted.
  • 14. Failure to be independent would indicate bias. BIAS EXAMPLE: If you are a minority you are more likely to be admitted. BIAS EXAMPLE: If you are a minority you less likely to be admitted
  • 15. Failure to be independent would indicate bias. BIAS EXAMPLE: If you are a minority you are more likely to be admitted. BIAS EXAMPLE: If you are a minority you less likely to be admitted. You will use certain statistical methods (like the chi square test of independence) to determine if independence is significant or not.
  • 16. Here is an example taken from http://omega.albany.edu:8008/mat108dir/chi2i ndependence/chi2in-m2h.html:
  • 17. Here is an example taken from http://omega.albany.edu:8008/mat108dir/chi2i ndependence/chi2in-m2h.html: In a certain town, there are about one million eligible voters. A simple random sample of 10,000 eligible voters was chosen to study the relationship between gender and participation in the last election.
  • 18. Here is an example taken from http://omega.albany.edu:8008/mat108dir/chi2i ndependence/chi2in-m2h.html: In a certain town, there are about one million eligible voters. A simple random sample of 10,000 eligible voters was chosen to study the relationship between gender and participation in the last election. The results are summarized in the following 2X2 (read two by two) contingency table:
  • 19. In a certain town, there are about one million eligible voters. A simple random sample of 10,000 eligible voters was chosen to study the relationship between gender and participation in the last election. The results are summarized in the following 2X2 (read two by two) contingency table: Men Women __________________________ Voted 2792 3591 Didn't vote 1486 2131
  • 20. We want to check whether being a man or a woman (columns) is independent of having voted in the last election (rows). In other words is “gender and voting independent”? Men Women __________________________ Voted 2792 3591 Didn't vote 1486 2131
  • 22. Solution: In order to answer the question we need to build a test of hypothesis. We have
  • 23. Solution: In order to answer the question we need to build a test of hypothesis. We have Null Hypothesis = ‘Gender is independent of Voting’
  • 24. Solution: In order to answer the question we need to build a test of hypothesis. We have Null Hypothesis = ‘Gender is independent of Voting’ Alternative Hypothesis = ‘Gender and Voting are dependent’
  • 25. Solution: In order to answer the question we need to build a test of hypothesis. We have Null Hypothesis = ‘Gender is independent of Voting’ Alternative Hypothesis = ‘Gender and Voting are dependent’ After specifying the Null Hypothesis, we need to compute the expected table under the assumption that rows and columns are in fact independent.
  • 26. As you can see we have the observed table below:
  • 27. As you can see we have the observed table below: Men Women __________________________ Voted 2792 3591 Didn't vote 1486 2131 We need to create an expected table and then determine if the difference between the observed and expected are significant:
  • 28. As you can see we have the observed table below: Men Women __________________________ Voted 2792 3591 Didn't vote 1486 2131 We need to create an expected table and then determine if the difference between the observed and expected are significant:
  • 29. As you can see we have the observed table below: Men Women __________________________ Voted 2792 3591 Didn't vote 1486 2131 We need to create an expected table and then determine if the difference between the observed and expected are significant: Observed Numbers Expected Numbers Difference
  • 30. Remember that the smaller the DIFFERENCE, the better the fit which in this case would favor INDEPENDENCE between gender and voting tendencies.
  • 31. Remember that the smaller the DIFFERENCE, the better the fit which in this case would favor INDEPENDENCE between gender and voting tendencies. Observed Numbers Expected Numbers Difference
  • 32. Inversely, the larger the DIFFERENCE the worse the fit which in this case would indicate that gender and voting tendencies are dependent upon one another.
  • 33. Inversely, the larger the DIFFERENCE the worse the fit which in this case would indicate that gender and voting tendencies are dependent upon one another. Observed Numbers Expected Numbers Difference
  • 34. We use Chi-Square distribution to determine if that difference is significant or not.
  • 35. We use Chi-Square distribution to determine if that difference is significant or not. We will now show you how to compute the chi-square statistic for a test of independence.
  • 36. We use Chi-Square distribution to determine if that difference is significant or not. We will now show you how to compute the chi-square statistic for a test of independence. First, we compute the row and column totals along with the grand total.
  • 37. Men Women ________________________________________ Voted 2792 3591 Didn't vote 1486 2131
  • 38. Total Who Voted Men Women ________________________________________ Voted 2792 + 3591 = 6386 Didn't vote 1486 2131
  • 39. Men Women ________________________________________ Voted 2792 3591 6386 Didn't vote 1486 + 2131 = 3617 Total Who Did Not Vote
  • 40. Men Women ________________________________________ Voted 2792 3591 6386 Didn't vote + 1486 2131 3617 = 4278 Total Men
  • 41. Men Women ________________________________________ Voted 2792 3591 6386 Didn't vote 1486 + 2131 3617 4278 = 5722 Total Women
  • 42. Total Men & Women or Total Voted/Not Voted Men Women ________________________________________ Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000
  • 43. Now we have the information we need to create an expected table. Here is the equation for calculating the expected value for the cell “Men who Voted”:
  • 44. Now we have the information we need to create an expected table. Here is the equation for calculating the expected value for the cell “Men who Voted”: Expected Value(Men who voted) = (Number (all who voted) * Number (all men)) Number(total number)
  • 45. Observed Men Women _ Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Men Who Voted
  • 46. OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who voted) = (6386 (all who voted) * Number (all men)) Number (total number)
  • 47. OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who voted) = (6386 (all who voted) * 4278 (all men) ) Number (total number)
  • 48. OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who voted) = (6386 (all who voted) * 4278 (all men) ) 10000 (total number)
  • 49. OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who voted) = (27306474 (all who voted * all men)) 10000 (total number)
  • 50. OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who voted) = 2730.6474 ((all who voted * all men)/total number)
  • 51. OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who voted) = 2731 ((all who voted * all men)/total number)
  • 52. EXPECTED Men Women _ TABLE Voted 2731 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who voted) = 2731 ((all who voted * all men)/total number)
  • 53. EXPECTED Men Women _ TABLE Voted 2731 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 What is the expected value for Women who Voted?
  • 55. Women who voted: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000
  • 56. Women who voted: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who voted) = (6386 (all who voted) * 5722 (all women) ) 10000 (total number)
  • 57. Women who voted: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who voted) = (6386 (all who voted) * 5722 (all women) ) 10000 (total number)
  • 58. Women who voted: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who voted) = (6386 (all who voted) * 5722 (all women) ) 10000 (total number)
  • 59. Women who voted: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who voted) = (36523526 ((all who voted) * (all women)) ) 10000 (total number)
  • 60. Women who voted: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who voted) = (3652.3526 ((all who voted) * (all women)))/total number
  • 61. Women who voted: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who voted) = (3652 ((all who voted) * (all women)))/total number
  • 62. Women who voted: EXPECTED Men Women _ TABLE Voted 2731 3652 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who voted) = (3652 ((all who voted) * (all women)))/total number
  • 63. Women who voted: EXPECTED Men Women _ TABLE Voted 2731 3652 6386 Didn't vote 1486 2131 3617 4278 5722 10000 What is the expected value for Men who Didn’t Vote?
  • 65. Men who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000
  • 66. Men who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who didn’t vote) = (3617 (all who didn’t vote) * 4278 (all men) ) 10000 (total number)
  • 67. Men who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who didn’t vote) = (3617 (all who didn’t vote) * 4278 (all men) ) 10000 (total number)
  • 68. Men who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who didn’t vote) = (3617 (all who didn’t vote) * 4278 (all men) ) 10000 (total number)
  • 69. Men who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who didn’t vote) = (15473526 ((all who didn’t vote) * (all men)) ) 10000 (total number)
  • 70. Men who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Men who didn’t vote) = (1547.3526 ((all who didn’t vote) * (all men)) / (total number))
  • 71. Men who didn’t vote: EXPECTED Men Women _ TABLE Voted 2731 3652 6386 Didn't vote 1547 2131 3617 4278 5722 10000 Expected Value (Men who didn’t vote) = (1547 ((all who didn’t vote) * (all men)) / (total number))
  • 72. Men who didn’t vote: EXPECTED Men Women _ TABLE Voted 2731 3652 6386 Didn't vote 1547 2131 3617 4278 5722 10000 What is the expected value for Women who Didn’t Vote?
  • 73. Women who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000
  • 74. Women who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who didn’t vote) = (3617 (all who didn’t vote) * 5722 (all women) ) 10000 (total number)
  • 75. Women who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who didn’t vote) = (3617 (all who didn’t vote) * 5722 (all women) ) 10000 (total number)
  • 76. Women who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who didn’t vote) = (3617 (all who didn’t vote) * 5722 (all women) ) 10000 (total number)
  • 77. Women who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who didn’t vote) = (20696474 (all who didn’t vote) * (all women) ) 10000 (total number)
  • 78. Women who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who didn’t vote) = (2069.6474 (all who didn’t vote) * (all women)) /(total number)
  • 79. Women who didn’t vote: OBSERVED Men Women _ TABLE Voted 2792 3591 6386 Didn't vote 1486 2131 3617 4278 5722 10000 Expected Value (Women who didn’t vote) = (2070 (all who didn’t vote) * (all women)) /(total number)
  • 80. Men who didn’t vote: EXPECTED Men Women _ TABLE Voted 2731 3652 6386 Didn't vote 1547 2070 3617 4278 5722 10000
  • 81. OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 4278 5722 10000 EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 10000
  • 82. OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 4278 5722 10000 EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 10000 With the information above, we can now plug in the numbers using the Chi-square independence test.
  • 83. OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 4278 5722 10000 EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 10000 With the information above, we can now plug in the numbers using the Chi-square independence test. Note – this is the same equation that is used with the Chi-square goodness of fit test:
  • 84. OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 4278 5722 10000 EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 10000 With the information above, we can now plug in the numbers using the Chi-square independence test. Note – this is the same equation that is used with the Chi-square goodness of fit test: 푥2 = Σ (푂 − 퐸)2 퐸
  • 85. 푥2 = 횺 (푂 − 퐸)2 퐸
  • 86. 푥2 = 횺 (푂 − 퐸)2 퐸
  • 87. 푥2 = 횺 (푂 − 퐸)2 퐸 Or in this case:
  • 88. 푥2 = 횺 (푂 − 퐸)2 퐸 Or in this case: 푥2 = (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸
  • 89. 푥2 = 횺 (푂 − 퐸)2 퐸 Or in this case: 푥2 = (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸
  • 90. OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 4278 5722 10000 EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 10000
  • 91. OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 4278 5722 Voting Men 10000 EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 10000 푥2 = (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 Voting Men
  • 92. OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 푥2 = (2792 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 Voting Men
  • 93. OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 푥2 = (2792 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 Voting Men
  • 94. EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 4278 5722 OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men - = Difference 푥2 = (2792 − 2731)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 Voting Men
  • 95. OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 푥2 = (2792 − 2731)2 2731 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 Voting Men
  • 96. OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 4278 5722 Voting Men - = Difference 푥2 = (2792 − 2731)2 2731 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 Voting Women
  • 97. EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 4278 5722 OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men - = Difference 푥2 = (2792 − 2731)2 2731 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 Voting Women
  • 98. EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 4278 5722 OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men - = Difference 푥2 = (2792 − 2731)2 2731 + (3591 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 Voting Women
  • 99. OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 Voting Women 푥2 = (2792 − 2731)2 2731 + (3591 − 퐸)2 퐸 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸
  • 100. OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 Voting Women 푥2 = (2792 − 2731)2 2731 + (3591 − 3652)2 3652 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸
  • 101. OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 Non-Voting Men 푥2 = (2792 − 2731)2 2731 + (3591 − 3652)2 3652 + (푂 − 퐸)2 퐸 + (푂 − 퐸)2 퐸
  • 102. OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 Non-Voting Men 푥2 = (2792 − 2731)2 2731 + (3591 − 3652)2 3652 + (1486 − 퐸)2 퐸 + (푂 − 퐸)2 퐸
  • 103. EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 4278 5722 OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men - = Difference Non-Voting Men 푥2 = (2792 − 2731)2 2731 + (3591 − 3652)2 3652 + (1486 − 퐸)2 퐸 + (푂 − 퐸)2 퐸
  • 104. EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 4278 5722 OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men - = Difference Non-Voting Men 푥2 = (2792 − 2731)2 2731 + (3591 − 3652)2 3652 + (1486 − 1547)2 1547 + (푂 − 퐸)2 퐸
  • 105. OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 Non-Voting Women 푥2 = (2792 − 2731)2 2731 + (3591 − 3652)2 3652 + (1486 − 1547)2 1547 + (푂 − 퐸)2 퐸
  • 106. OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 - = Difference 4278 5722 Non-Voting Women 푥2 = (2792 − 2731)2 2731 + (3591 − 3652)2 3652 + (1486 − 1547)2 1547 + (2131 − 퐸)2 퐸
  • 107. EXPECTED Men Women TABLE Voted 2731 3652 Didn't vote 1547 2070 4278 5722 OBSERVED Men Women TABLE Voted 2792 3591 Didn't vote 1486 2131 Voting Men - = Difference Non-Voting Women 푥2 = (2792 − 2731)2 2731 + (3591 − 3652)2 3652 + (1486 − 1547)2 1547 + (2131 − 2070)2 2070
  • 109. Time to Calculate: 푥2 = (2792 − 2731)2 2731 + (3591 − 3652)2 3652 + (1486 − 1547)2 1547 + (2131 − 2070)2 2070
  • 110. Time to Calculate: 푥2 = (61)2 2731 + (3591 − 3652)2 3652 + (1486 − 1547)2 1547 + (2131 − 2070)2 2070
  • 111. Time to Calculate: 푥2 = 3721 2731 + (3591 − 3652)2 3652 + (1486 − 1547)2 1547 + (2131 − 2070)2 2070
  • 112. Time to Calculate: 푥2 = 1.4 + (3591 − 3652)2 3652 + (1486 − 1547)2 1547 + (2131 − 2070)2 2070
  • 113. Time to Calculate: 푥2 = 1.4 + (−61)2 3652 + (1486 − 1547)2 1547 + (2131 − 2070)2 2070
  • 114. Time to Calculate: 푥2 = 1.4 + 3721 3652 + (1486 − 1547)2 1547 + (2131 − 2070)2 2070
  • 115. Time to Calculate: 푥2 = 1.4 + 1.0 + (1486 − 1547)2 1547 + (2131 − 2070)2 2070
  • 116. Time to Calculate: 푥2 = 1.4 + 1.0 + (61)2 1547 + (2131 − 2070)2 2070
  • 117. Time to Calculate: 푥2 = 1.4 + 1.0 + 3721 1547 + (2131 − 2070)2 2070
  • 118. Time to Calculate: 푥2 = 1.4 + 1.0 + 2.4 + (2131 − 2070)2 2070
  • 119. Time to Calculate: 푥2 = 1.4 + 1.0 + 2.4 + (61)2 2070
  • 120. Time to Calculate: 푥2 = 1.4 + 1.0 + 2.4 + 3721 2070
  • 121. Time to Calculate: 푥2 = 1.4 + 1.0 + 2.4 + 1.8
  • 122. Time to Calculate: 푥2 = 6.6
  • 123. Now we determine if a 푥2of 6.6 exceeds the critical 푥2 for terms.
  • 124. To calculate the 푥2 critical we first must determine the degrees of freedom as well as set the probability level.
  • 125. To calculate the 푥2 critical we first must determine the degrees of freedom as well as set the probability level. The probability or alpha level means the probability of a type 1 error we are willing to live with (i.e., this is the probability of being wrong when we reject the null hypothesis). Generally this value is .05 which is like saying we are willing to be wrong 5 out of 100 times (.05) before we will reject the null-hypothesis.
  • 126. Degrees of Freedom are calculated by taking the number rows and subtracting them by 1 and then multiplying the result by taking the number of columns and subtracting them by 1.
  • 127. Degrees of Freedom are calculated by taking the number rows and subtracting them by 1 and then multiplying the result by taking the number of columns and subtracting them by 1. (Two rows -1) or (2-1) X (2-1) or 1X1=1. Degrees of Freedom = 1.
  • 128. We now have all of the information we need to determine the critical 푥2.
  • 129. We now have all of the information we need to determine the critical 푥2. We go to the Chi-Square Distribution Table and locate the degrees of freedom:
  • 130. We now have all of the information we need to determine the critical 푥2. We go to the Chi-Square Distribution Table and locate the degrees of freedom: df 0.100 0.050 0.025 1 2.71 3.84 5.02 2 4.61 5.99 7.38 3 6.25 7.82 9.35 4 7.78 9.49 11.14 5 9.24 11.07 12.83 6 10.64 12.59 14.45 7 12.02 14.07 16.10 8 13.36 15.51 17.54 9 14.68 16.92 19.20 … … … …
  • 131. We now have all of the information we need to determine the critical 푥2. We go to the Chi-Square Distribution Table and locate the degrees of freedom: df 0.100 0.050 0.025 1 2.71 3.84 5.02 2 4.61 5.99 7.38 3 6.25 7.82 9.35 4 7.78 9.49 11.14 5 9.24 11.07 12.83 6 10.64 12.59 14.45 7 12.02 14.07 16.10 8 13.36 15.51 17.54 9 14.68 16.92 19.20 … … … … And then we locate the probability or alpha level:
  • 132. We now have all of the information we need to determine the critical 푥2. We go to the Chi-Square Distribution Table and locate the degrees of freedom: df 0.100 0.050 0.025 1 2.71 3.84 5.02 2 4.61 5.99 7.38 3 6.25 7.82 9.35 4 7.78 9.49 11.14 5 9.24 11.07 12.83 6 10.64 12.59 14.45 7 12.02 14.07 16.10 8 13.36 15.51 17.54 9 14.68 16.92 19.20 … … … … And then we locate the probability or alpha level: Where these two values intersect in the table we find the critical 푥2.
  • 133. Since the chi-square goodness of fit value (6.6) exceeds the critical 푥2 (3.84) we will reject the null-hypothesis.
  • 134. Since the chi-square goodness of fit value (6.6) exceeds the critical 푥2 (3.84) we will reject the null-hypothesis. Voting patterns and gender status are not statistically significantly dependent on one another.
  • 135. Since the chi-square goodness of fit value (6.6) exceeds the critical 푥2 (3.84) we will reject the null-hypothesis. Voting patterns and gender status are not statistically significantly dependent on one another.
  • 136. Since the chi-square goodness of fit value (6.6) exceeds the critical 푥2 (3.84) we will reject the null-hypothesis. Voting patterns and gender status are not statistically significantly dependent on one another. There actually is a significant difference.
  • 137. So what is the difference between chi-square test of goodness of fit and test of independence?
  • 138. A goodness-of-fit test is a one variable Chi-square test.
  • 139. A goodness-of-fit test is a one variable Chi-square test. In this example, a department chair wants to know if the enrollments across three professors are equally distributed.
  • 140. A goodness-of-fit test is a one variable Chi-square test. In this example, a department chair wants to know if the enrollments across three professors are equally distributed. Here is the actual, or observed, data:
  • 141. A goodness-of-fit test is a one variable Chi-square test. In this example, a department chair wants to know if the enrollments across three professors are equally distributed. Here is the actual, or observed, data: OBSERVED TABLE Prof A’s Class Prof B’s Class Prof C’s Class Students enrolled 31 25 10
  • 142. A goodness-of-fit test is a one variable Chi-square test. OBSERVED TABLE Prof A’s Class Prof B’s Class Prof C’s Class Students enrolled 31 25 10
  • 143. A goodness-of-fit test is a one variable Chi-square test. OBSERVED TABLE Prof A’s Class Prof B’s Class Prof C’s Class Students enrolled 31 25 10
  • 144. A test of independence is a two variable Chi-square test.
  • 145. A test of independence is a two variable Chi-square test. For example, a department chair wants to know if women and men enrollments are equally distributed across three professor classes.
  • 146. A test of independence is a two variable Chi-square test. For example, a department chair wants to know if women and men enrollments are equally distributed across three professor classes. OBSERVED TABLE Prof A’s Class Prof B’s Class Prof C’s Class Men 21 7 7 Women 10 18 3
  • 147. A test of independence is a two variable (gender) Chi-square test. OBSERVED TABLE Prof A’s Class Prof B’s Class Prof C’s Class Men 21 7 7 Women 10 18 3