Interpreting Chi Square

Contingency Tables

and the Chi Square

Statistic

Interpreting Computer

Printouts and

Constructing Tables

Contingency Tables/Chi Square

Statistics

• What are they?

A contingency table is a table that shows the

relationship between two categorical variables.

The Chi-square statistic reflects the strength of this

relationship. All else equal, the greater the chi-square

statistic, the stronger the relationship. The chi square

statistic is usually reported at the bottom of a

contingency table.

The probability associated with the chi-square statistic

indicates the probability that you would be incorrect if

you were to assert that there is a relationship between

these same two variables in the population from which

you drew your sample.


Statistics

• Why should you be able to interpret a contingency table and chi-square statistic?

If you are submitting an article for publication in a peer

reviewed journal you may use a contingency table. You want to interpret it accurately.

Before you implement a program, you want to review the literature and read about the same or similar programs. These articles may contain contingency tables. You want to be a wise consumer of information.

If you are evaluating your own program, it is likely that you will use contingency tables. Contingency tables can not only help you discern if your program is working, but how effective it is and/or if the effectiveness varies by such factors as gender, race, etc.

Before you make recommendations or engage in political advocacy, you want empirical evidence to substantiate your claims. Contingency tables can be used to provide this evidence.


Statistics

• When would you use them?

• Contingency tables and the corresponding chi-square

statistic is very useful for those implementing programs, and should be used BEFORE important decisions are made.

• Any responsible extension employee would use correctly implemented evaluation research to evaluate their programs and measure their objectives. Properly interpreted Chi square statistics can be an important part of that evaluation.

• When applying for grants to support existing programs, it is a good idea to use statistics to validate claims that the program “works.” Chi square can be one of these statistics.

• When justifying grant monies spent on programs, evaluation research should be used. Responsible administrators should use accurate statistics to provide evidence that their objectives were met.

Examples of When to Use Them

• You should use contingency tables and chi

square statistics:

• BEFORE you use any Intervention, so you can determine if

there is evidence that it will work and for whom it will work.

• For instance – You should be able to interpret contingency

tables reported in the literature that have been used to

evaluate similar programs. You want to know if there is

evidence that a program like yours will work, and what factors

could influence its effectiveness.

• BEFORE implementing and/or continuing any program, so you

will know if it’s worth your continued time and effort.

• For instance - Is the percent of students who received an A in

school greater for those who belonged to your 4-H program

relative to those who did not?

• For instance - What factors are most important when

attempting to increase the effectiveness of this intervention?

• For instance - What percent of those involved in intervention 1

saw the program as effective compared to the percent

involved in intervention 2.

Examples of When to Use Them

• BEFORE you report the results of your evaluation and/or make

recommendations, so you can accurately explain the risks and

probabilities.

• For instance – If having access to computers related to scores

on standardized tests, then (1) Are scores on standardized

tests related to academic achievement, and (2) How many

computers must you purchase to see a significant gain in

standardized test scores for this group?

• For instance - What percent of your clients experienced

increased economic stress after paying to participate in your

program? Does this economic stress outweigh benefits?

• For instance, Can you identify the most important factors in

your program? If you were forced to curtail activities which

activities could you eliminate without significantly decreasing

positive outcomes for your participants?

Assumptions of Chi Square

The statistics generated by the computer for chi-

square are only valid if the data meet the following

qualifications:

Both the independent and dependent variables are

categorical.*

Researchers used a random sample to collect data.

Researchers had an adequate sample size.

Generally the sample size should be at least 100.

The number of respondents in each cell should be at

least 5. If not, you can use a Fisher’s Exact or other

tests.

*Generally the number of categories is somewhere between 3 and 7.

More than 7 categories can be overwhelming and confusing to the

reader.


Statistics

• How do you interpret them?

• There are two parts of a contingency table that you must

correctly interpret:

• The chi square statistic

• The numbers in the table


Statistics


• The chi square statistic

• It is relatively easy to interpret a chi square statistic if you know three things

• First – all else equal, the greater the chi square number, the stronger the relationship between the dependent and independent variable

.

• Second – the lower the probability associated with a chi-square statistic, the stronger the relationship between the dependent and independent variable.

• Third – If your probability is .05 or less, then you can generalize from a random sample to a population, and claim the two variables are associated in the population.


Statistics


• The numbers in the contingency table

• There are a few simple rules that makes it easier to interpret a contingency table. These include:

• First – to avoid confusion, always put the independent variable on the side and the dependent variable on the top.

• Second – focus on the percent not on the frequency.

• Third – use the percent that totals to 100 percent for each independent variable (e.g., the row percent).

• Fourth – compare the percent for each category of the independent variable across the independent variables.

A contingency table reported in the literature should

look something like the one above.

When interpreting this table, you compare the

percent of each independent category. In this case, 10% of

White mothers, compared to 44.83% of Black mothers and

35.71% of Hispanic mothers received bad news.

The probability associated with the chi square

statistic of 9.205 is less than .01 indicating there is a strong

relationship between whether or not the mom received bad

news and her race.

Contingency Table. Mother’s Racial Identification by

Probability of Receiving Bad News about their Kindergarten

Child from the Child’s Teacher.

Computer Printouts

• How do you Interpret them?

• How do you construct a table from a computer

printout?

• How do you construct and interpret a bar chart

from a contingency table?

Chi- Square/Contingency Table

• Purpose

• Examine relationship between two categorical variables

• Determine if the dependent variable is contingent on the

independent variable

• Definition of Terms

• Cells

• Squares that make up chi-square printout (sometimes

referred to as table)

• Each cell contains 4 numbers. These are referred to

as: • Frequency (raw numbers or count)

• Percent

• Row Percent

• Column Percent

• Tabulates the number of times each possible

combination of the values of the independent and

dependent variables occur (in your sample)

• Rows (r)

• Cells that are attached horizontally are referred to as

a row

• Row percents make up the numbers in the cells that

percentage to 100 across these rows.

• Columns (c)

• Cells that are attached vertically are referred to as a column

• Column percents make up the numbers in the cells that percentage to 100 down these columns.

• Degrees of Freedom

• A number that you would use to find the critical value of a chi square statistic using a chi square table.

• The formula you use to compute a chi square statistic is (r – 1) (c – 1) = df

• Critical Value of Chi-Square Statistic

• The value of the chi-square statistic associated with a .05 probability of making an error if you reject the null hypothesis. Your chi square statistic must be greater than this if your relationship is significant. This is not reported in the printout, but the reported value must be greater than this if you are to reject the null hypothesis.

• Computed Chi-Statistic

• The computed value of the chi-square statistic.

• Generally the greater the chi square statistic, the lower the probability that you make a mistake if you reject the null hypothesis.

• Generally the greater the chi square statistic, the

stronger the relationship between the independent

and dependent variable.

Chi Square

Statistic

reported here

Probability

reported

here

Research

Question:

Is preferred

ice cream

flavor

related to

race?

*Please note….The data is

fabricated and to be used

only for illustration.

Research Question: Is race related to ice cream flavor preference?

*Please note….The data is

fabricated and to be used only

for illustration.

One Interpretation:

65% of Blacks,

compared to

35.96% of

Hispanics and

34.18% of Whites

prefer Butter

Pecan. Blacks

were almost twice

as likely to prefer

Butter Pecan.

The relationship

between race and

ice cream

preference is

significant at the

.0001 level.

Here is an

example of a table

(above) that was

constructed from a

printout (below).

The table reports

the row percent,

the frequency in

each cell and the

chi square statistic.

Chi-square results can be presented in a

contingency table. In this table you report, the frequency in

each cell. If the independent (causal) variable is on the side,

you report the row percent. If the independent (causal)

variable is on the top, you report the column percent.

When interpreting this table, you compare the

percent of each independent category. In this case, 10% of

White mothers, compared to 44.83% of Black mothers and

35.71% of Hispanics mothers received bad news.

Contingency Table. Mother’s Racial Identification by

Probability of Receiving Bad News about their Kindergarten

Child from the Child’s Teacher.

Using data from the previous

slide, you could construct the

following graph:

0

10

20

30

40

50

60

70

80

90

White Black Hispanic

No

Yes

Or you could use the same

data to construct this graph:

0

10

20

30

40

50

60

70

80

90

Yes No

Black

White

Hispanic

CHART 5

Middle/High School Students’ Perception of How Good

They are at Finding Resources by Whether or Not They

Made Decisions about Service Learning Projects

11%7%

28%

7%13%

44%

17%

43%

9%

24%

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

Strongly

Disagree

Disagree Undecided Agree Strongly

Agree

D id

D id N ot

(p <0.0001)

Contingency tables are often used to construct two-variable bar charts.

In this case, the extent to which students agree that they are good at

finding resources is contingent on whether or not they made decisions

about their Service Learning Project. Assuming the independent

variable is on the side of the computer printout, the row percent is used

to construct the bar chart. The probability is less than .0001 indicating a

strong relationship between these two variables. All else equal, the

less the probability, the stronger the relationship.

CHART 5

Middle/High School Students’ Perception of How Good

They are at Finding Resources by Whether or Not They

Made Decisions about Service Learning Projects

11%7%

28%

7%13%

44%

17%

43%

9%

24%

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

Strongly

Disagree


Agree

D id

D id N ot

(p <0.0001)

Interpreting the Bar Chart

Interpret the most striking comparison for your audience. For

instance, “Twenty-eight percent of those who made decisions

about their Service Learning Project agreed that they were good

at finding resources, compared to only seventeen percent of

those who did not make decisions about their projects.”

Interpreting Bar Charts

CHART 7

Middle/High School Students’ Perception of the Level to

Which They Agree Teachers Listen to Their Ideas by

Whether or Not They Made Decisions about Service

Learning Projects

8%3%

27%

9%

21%

35%

20%

39%

15%22%

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

Strongly

Disagree


Agree

D id

D id N ot

(p <0.0001)

In this case, the extent to which students feel teachers

listen to them is contingent on whether or not they made

decisions about their Service Learning Project. 8% of

those who did make decisions about their Service Learning

Project felt their teachers listened to them compared to

only 3% of those who did not make decisions.

Contact Information

• Dr. Carol Albrecht

• Assessment Specialist

• USU Extension

• [email protected]

• 979-777-2421

mailto:[email protected]

Interpreting Chi Square

Documents