# Chi Square (χ2) Test Statistic - Simply Psychology (2023) Tag: h0 chi2

By Benjamin Frimodig, published 2020

## What Is a Chi-Square Statistic?

Chi-square (χ2) is used to test hypotheses about the distribution of observations into categories, with no inherent ranking.

The Chi square test (pronounced Kai) looks at the pattern of observations, and will tell us if certain combinations of the categories occur more frequently than we would expect by chance, given the total number of times each category occurred.

It looks for an association between the variables. We are not able to use a correlation coefficient to look for the patterns in this data because the categories often do not form a continuum.

There are three main types of Chi-square tests, tests of goodness of fit, the test of independence, and the test for homogeneity. All three tests rely on the same formula to compute a test statistic.

These tests function by deciphering relationships between observed sets of data and theoretical or “expected” sets of data that align with the null hypothesis.

## What is a Contingency Table?

Contingency table (also known as two-way tables) are grids in which Chi-square data is organized and displayed. They provide a basic picture of the interrelation between two variables and can help find interactions between them.

In contingency tables, one variable and each of its categories are listed vertically and the other variable and each of its categories are listed horizontally.

Additionally, including column and row totals, also known as “marginal frequencies”, will help facilitate the Chi-square testing process.

In order for the Chi-square test to be considered trustworthy, each cell of your expected contingency table must have a value of at least five.

Each Chi-square test will have one contingency table representing observed counts (see Fig. 1) and one contingency table representing expected counts (see Fig. 2). Figure 1. Observed table (which contains the observed counts).

To obtain the expected frequencies for any cell in any cross- tabulation in which the two variables are assumed independent, multiply the row and column totals for that cell and divide the product by the total number of cases in the table. Figure 2. Expected table (what we expect the two-way table to look like if the two categorical variables are independent).

To decide if our calculated value for χ2 is significant, we also need to work out the degrees of freedom for our contingency table using the following formula;df= (rows - 1) x (columns – 1).

(Video) Chi Square Test

## Chi Square Formula Calculation Calculate the chi square statistic (χ2) by completing the following steps:

1. Calculate the expected frequencies and the observed frequencies.
2. For each observed number in the table subtract the corresponding expected number (O — E).
3. Square the difference (O —E)².
4. Divide the squares obtained for each cell in the table by the expected number for that cell (O - E)² / E.
5. Sum all the values for (O - E)² / E. This is the chi square statistic.
6. Calculate the degrees of freedom for the contingency table using the following formula; df= (rows - 1) x (columns – 1).

Once we have calculated the defrees of freedom (df) and the chi squared value (χ2), we can use the χ2 table (often at the back of a statistics book) to check if our value for χ2 is higher than the critical value given in the table. If it is, then our result is significant at the level given.

## Intepreting the Chi Square Statistic

The chi-square statistic tells you how much difference exists between the observed count in each table cell to the counts you would expect if there were no relationship at all in the population.

A very small chi square test statistic means means there is a high correlation between the observed and expected values. Therefore, the sample data is a good fit for what would be expected in the general population.

In theory, if the observed and expected values were equal (no difference) then the chi-square statistic would be zero — but this is unlikely to happen in real life.

A very large chi square test statistic means that the sample data (observed values) does not fit the population data (expected values) very well. In other words, there isn't a relationship.

## How to Report a Chi Square Test Result (APA)?

To report a chi square output in an APA style results section, always rely on the following template:

χ2 (degrees of freedom, N = sample size) = chi-square statistic value, p = p value. In the case of the above example, the results would be written as follows:

A chi-square test of independence showed that there was a significant association between gender and post graduation education plans, χ2 (4, N = 101) = 54.50, p < .001.

### APA Style Rules

• Do not use a zero before a decimal when thestatistic cannot be greater than 1 (proportion,correlation, level of statistical significance).
• Report exact p values to two or three decimals (e.g., p = .006, p = .03).
• However, report p values less than .001 as “p < .001.”
• Put a space before and after a mathematicaloperator (e.g., minus, plus, greater than, lessthan, equals sign).
• Do not repeat statistics in both the text and atable or figure.

## How is the p-value Interpreted?

You test whether a given χ2 is statistically significant by testing it against a table of chi-square distributions, according to the number of degrees of freedom for your sample, which is the number of categories minus 1. The chi-square assumes that you have at least 5 observations per category.

If you are using SPSS then you will have an expect p-value.

For a chi-square test, a p-value that is less than or equal to the .05 significance level indicates that the observed values are different to the expected values.

Thus, low p-values (p< .05) indicate a likely difference between the theoretical population and the collected sample. You can conclude that a relationship exists between the categorical variables.

(Video) Simple Explanation of Chi-Squared

Remember that p-values do not indicate the odds that the null hypothesis is true, but rather provides the probability that one would obtain the sample distribution observed (or a more extreme distribution) if the null hypothesis was in fact true.

A level of confidence necessary to accept the null hypothesis can never be reached. Therefore, conclusions must choose to either fail to reject the null or accept the alternative hypothesis, depending on the calculated p-value.

## Using SPSS to Perform a Chi-Square Test

The four steps below show you how to analyse your data using a chi-square goodness-of-fit test in SPSS (when you have hypothesised that you have equal expected proportions).

Step 1: Analyze > Nonparametric Tests > Legacy Dialogs > Chi-square... on the top menu as shown below:

Step 2: Move the variable indicatingcategories into the “Test Variable List:” box.

Step 3: If you want to test the hypothesis that allcategories are equally likely, click “OK.”

Step 4: Specify the expected count foreach category by first clicking the “Values” button under “Expected Values.”

Step 5: Then, inthe box to the right of “Values,” enter the expected count for category 1 and click the“Add” button. Now enter the expected count for category 2 and click “Add.” Continuein this way until all expected counts have been entered.

Step 6: Then click “OK.”

The four steps below show you how to analyse your data using a chi-square test of independence in SPSS Statistics.

Step 1: Open the Crosstabs dialog (Analyze > Descriptive Statistics > Crosstabs).

Step 2: Select the variables you want to compare using the chi square test. Click one variable in the left window and then click the arrow at the top to move the variable. Select the row variable, and the column variable.

Step 3: Click Statistics (a new pop up window will appear). Check Chi-square, then click Continue.

Step 4: (Optional) Check the box for Display clustered bar charts.

Step 5: Click OK.

## Goodness-of-Fit Test

The Chi-square goodness of fit test is used to compare a randomly collected sample containing a single, categorical variable to a larger population. This test is most commonly used to compare a random sample to the population from which it was potentially collected.

(Video) Chi-Square Tutorial Psychology Statistics

The test begins with the creation of a null and alternative hypothesis. In this case, the hypotheses are as follows:

Null Hypothesis (Ho): The null hypothesis (Ho) is that the observed frequencies are the same (except for chance variation) as the expected frequencies. The collected data is consistent with the population distribution.

Alternative Hypothesis(Ha): The collected data is not consistent with the population distribution.

The next step is to create a contingency table that represents how the data would be distributed if the null hypothesis were exactly correct.

The sample’s overall deviation from this theoretical/expected data will allow us to draw a conclusion, with more severe deviation resulting in smaller p-values.

## Test for Independence

The Chi-square test for independence looks for an association between two categorical variables within the same population. Unlike the goodness of fit test, the test for independence does not compare a single observed variable to a theoretical population, but rather two variables within a sample set to one another.

The hypotheses for a Chi-square test of independence are as follows:

Null Hypothesis (Ho): There is no association between the two categorical variables in the population of interest.

Alternative Hypothesis(Ha): There is no association between the two categorical variables in the population of interest.

The next step is to create a contingency table of expected values that reflects how a data set that perfectly aligns the null hypothesis would appear.

The simplest way to do this is to calculate the marginal frequencies of each row and column; the expected frequency of each cell is equal to the marginal frequency of the row and column that corresponds to a given cell in the observed contingency table divided by the total sample size.

## Chi-Square Test for Homogeneity

The Chi-square test for homogeneity is organized and executed exactly the same as the test for independence. The main difference to remember between the two is that the test for independence looks for an association between two categorical variables within the same population, while the test for homogeneity determines if the distribution of a variable is the same in each of several populations (thus allocating population itself as the second categorical variable).

The hypotheses for a Chi-square test of independence are as follows:

Null Hypothesis (Ho): There is no difference in the distribution of a categorical variable for several populations or treatments.

(Video) Statistics made easy ! ! ! Learn about the t-test, the chi square test, the p value and more

Alternative Hypothesis(Ha): There is a difference in the distribution of a categorical variable for several populations or treatments.

The difference between these two tests can be a bit tricky to determine especially in practical applications of a Chi-square test. A reliable rule of thumb is to determine how the data was collected.

If the data consists of only one random sample with the observations classified according to two categorical variables, it is a test for independence. If the data consists of more than one independent random sample, it is a test for homogeneity.

Ben is a 2021 graduate of Harvard College, where he studied History of Science. He is interested in the intersection of psychology and public health, and hopes to pursue a career in medicine that will allow him to contribute to this growing field of research!

Further Information

APA (7th) Numbers and Statistics GuideChi Square Video (Kahn Academy) Statistics for PsychologyWhat a p-value Tells You About Statistical SignificanceReporting Statistics in PsychologyAPA Numbers and Statistics GuideStatistics for Psychology Book Download

Frimodig, B. (2020, Oct 20). Chi square test. Simply Psychology. www.simplypsychology.org/chi-square.html

Simply Psychology's content is for informational and educational purposes only. Our website is not intended to be a substitute for professional medical advice, diagnosis, or treatment.

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## FAQs

### What is the meaning of χ2 value in statistical chi-square analysis? ›

A chi-square (χ2) statistic is a measure of the difference between the observed and expected frequencies of the outcomes of a set of events or variables. Chi-square is useful for analyzing such differences in categorical variables, especially those nominal in nature.

What is the test statistic for the χ2 test of independence? ›

If you perform the Chi-square test of independence using this new data, the test statistic is 0.903. The Chi-square value is still 7.815 because the degrees of freedom are still three. You would fail to reject the idea of independence because 0.903 < 7.815.

What is chi-square test answer? ›

A chi-square test is a statistical test that is used to compare observed and expected results. The goal of this test is to identify whether a disparity between actual and predicted data is due to chance or to a link between the variables under consideration.

How do you solve chi-square questions? ›

To calculate the chi-square, we will take the square of the difference between the observed value O and expected value E values and further divide it by the expected value. Depending on the number of categories of the data, we end up with two or more values. Chi-square is the sum total of these values.

Is higher or lower chi-square better? ›

The larger the Chi-square value, the greater the probability that there really is a significant difference. There is a significant difference between the groups we are studying.

What does the P value mean in a chi-square test of independence? ›

P value. In a chi-square analysis, the p-value is the probability of obtaining a chi-square as large or larger than that in the current experiment and yet the data will still support the hypothesis. It is the probability of deviations from what was expected being due to mere chance.

What is a chi-square test quizlet? ›

It is a statistical test used to determine if a significant relationship is present between two variables such as the expected frequencies and observed frequencies of a population, assesses if these variables are independent from one another, and if sample size is adequate.

How do you use a chi-square test example? ›

How to perform a chi-square test
1. Create a table of the observed and expected frequencies. ...
2. Calculate the chi-square value from your observed and expected frequencies using the chi-square formula.
3. Find the critical chi-square value in a chi-square critical value table or using statistical software.
May 23, 2022

What is an example of a chi-square test research question? ›

An example research question that could be answered using a Chi-Square analysis would be: Is there a significant relationship between voter intent and political party membership? How does the Chi-Square statistic work?

What is a chi-square test for dummies? ›

The Chi-square test is intended to test how likely it is that an observed distribution is due to chance. It is also called a "goodness of fit" statistic, because it measures how well the observed distribution of data fits with the distribution that is expected if the variables are independent.

### How do you do a chi-square step by step? ›

How to perform a Chi-square test
2. Decide on the alpha value. ...
3. Check the data for errors.
4. Check the assumptions for the test. ...
5. Perform the test and draw your conclusion.

What does it mean if chi-square is less than 1? ›

A reduced chi-squared value much less than one means that the discrepancies are much smaller than you expected, based on your estimate of what the experimental errors will be. Therefore either the discrepancies really are too small (i.e. overfitting) or the experimental errors are smaller than you estimated.

What does it mean to be a good fit in a chi-square? ›

In Chi-Square goodness of fit test, the term goodness of fit is used to compare the observed sample distribution with the expected probability distribution. Chi-Square goodness of fit test determines how well theoretical distribution (such as normal, binomial, or Poisson) fits the empirical distribution.

What does it mean when your chi squared is higher than the critical value? ›

What does critical value mean? Basically, if the chi-square you calculated was bigger than the critical value in the table, then the data did not fit the model, which means you have to reject the null hypothesis.

What does p 0.05 mean in chi-square? ›

P > 0.05 is the probability that the null hypothesis is true. 1 minus the P value is the probability that the alternative hypothesis is true. A statistically significant test result (P ≤ 0.05) means that the test hypothesis is false or should be rejected. A P value greater than 0.05 means that no effect was observed.

What does using a P value of 0.05 in a chi-square test mean? ›

A significance level of 0.05 indicates a 5% risk of concluding that an association between the variables exists when there is no actual association.

What does the chi-square p-value of less than 0.05 usually suggest? ›

A p-value of less than or equal to 0.05 is regarded as evidence of a statistically significant result, and in these cases, the null hypothesis should be rejected in favor of the alternative hypothesis.

What 3 conditions must be met when using the chi-square test? ›

Your data must meet the following requirements: Two categorical variables. Two or more categories (groups) for each variable. Independence of observations.

What conditions must be met before completing a chi-square test? ›

The chi-square goodness of fit test is appropriate when the following conditions are met: The sampling method is simple random sampling. The variable under study is categorical. The expected value of the number of sample observations in each level of the variable is at least 5.

Which of the following is an assumption required by the χ2 test? ›

The required assumption of the chi-squared test of association is that "There are at least 5 observations in most or all of the cells (say 75% or more)".

### How do you calculate test statistic? ›

Generally, the test statistic is calculated as the pattern in your data (i.e. the correlation between variables or difference between groups) divided by the variance in the data (i.e. the standard deviation).

What is chi-square test with real life example? ›

Chi-square test of independence can be used. Another example might be a study showing whether there is an association between income level and educational qualification. Another example of chi-square test of independence can be analyzing the correlation between gender and academic performance.

What is the best sample size for chi-square test? ›

Another consideration one must make is that the chi-square statistic is sensitive to sample size. Most recommend that chi-square not be used if the sample size is less than 50, or in this example, 50 F2 tomato plants. If you have a 2x2 table with fewer than 50 cases many recommend using Fisher's exact test.

Why is chi square test important? ›

Importance: Chi-square tests enable us to compare observed and expected frequencies objectively, since it is not always possible to tell just by looking at them whether they are "different enough" to be considered statistically significant.

What does chi-square 0.05 mean? ›

A significance level of 0.05 indicates a 5% risk of concluding that an association between the variables exists when there is no actual association.

Does a chi-square .05 mean that this is statistically significant? ›

Among statisticians a chi square of . 05 is a conventionally accepted threshold of statistical significance; values of less than . 05 are commonly referred to as "statistically significant." In practical terms, a chi square of less than .

What is chi-square χ2 test state the uses of chi-square χ2 test? ›

You can use a chi-square test of independence when you have two categorical variables. It allows you to test whether the two variables are related to each other. If two variables are independent (unrelated), the probability of belonging to a certain group of one variable isn't affected by the other variable.

What if p-value is greater than 0.05 in chi-square test? ›

A statistically significant test result (P ≤ 0.05) means that the test hypothesis is false or should be rejected. A P value greater than 0.05 means that no effect was observed.

When p-value is less than 0.05 chi-square? ›

A p-value of less than or equal to 0.05 is regarded as evidence of a statistically significant result, and in these cases, the null hypothesis should be rejected in favor of the alternative hypothesis.

What do you do when chi-square expected count is less than 5? ›

The conventional rule of thumb is that if all of the expected numbers are greater than 5, it's acceptable to use the chi-square or G–test; if an expected number is less than 5, you should use an alternative, such as an exact test of goodness-of-fit or a Fisher's exact test of independence.

### What significance does a chi-square allow you to determine? ›

A chi-square test is a statistical test used to compare observed results with expected results. The purpose of this test is to determine if a difference between observed data and expected data is due to chance, or if it is due to a relationship between the variables you are studying.

Why chi-square test is used for hypothesis testing? ›

You use a Chi-square test for hypothesis tests about whether your data is as expected. The basic idea behind the test is to compare the observed values in your data to the expected values that you would see if the null hypothesis is true.

What is the primary purpose of doing a chi-square test quizlet? ›

*The chi-squared test is typically used to analyze the relationship between two qualitative variables, however, it an also be applied when one or both variables are quantitative.

What does a chi-square test tell you quizlet? ›

What is a Chi squared test? It is a statistical test used to determine if a significant relationship is present between two variables such as the expected frequencies and observed frequencies of a population, assesses if these variables are independent from one another, and if sample size is adequate.

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