Abhiraj Suresh — Published On November 27, 2019 and Last Modified On May 22nd, 2023

“Science is advanced by proposing and testing a hypothesis, not by declaring questions unsolvable”

– Nick Matzke

Statistical analysis is a key tool for making sense of data and drawing meaningful conclusions. The chi-square test is a statistical method commonly used in data analysis to determine if there is a significant association between two categorical variables. By comparing observed frequencies to expected frequencies, the chi-square test can determine if there is a significant relationship between the variables. So let’s dive into the article to understand all about the chi-square test, what it is, how it works, and how we can implement it in R.

Learning Objectives:

• Understand what the chi-square test is and how it works
• Be able to calculate the chi-square value using the chi-square formula
• Learn about the different types of Chi-Square tests and where and when you should apply them
• Learn to implement a chi-square test in R

## What is Chi-Sqaure Test?

The chi-square test is a statistical test used to determine if there is a significant association between two categorical variables. It is a non-parametric test, meaning it does not make assumptions about the underlying distribution of the data.

It compares the observed frequencies of the categories in a contingency table with the expected frequencies that would occur under the assumption of independence between the variables. The test calculates a chi-square statistic, which measures the discrepancy between the observed and expected frequencies.

## When to Use the Chi-Square Test?

Let’s start with a case study. I want you to think of your favorite restaurant right now. Let’s say you can predict a certain number of people arriving for lunch five days a week. At the end of the week, you observe that the expected footfall was different from the actual footfall.

Sounds like a prime statistics problem? That’s the idea!

So, how will you check the statistical significance between the observed and the expected footfall values? Remember this is a categorical variable – ‘Days of the week’ – with 5 categories [Monday, Tuesday, Wednesday, Thursday, Friday].

One of the best ways to deal with this is by using the Chi-Square test.

We can always opt for z-tests, t-tests, or ANOVA when we’re dealing with continuous variables. But the situation becomes tricky when working with categorical features (as most data scientists will attest to!). I’ve found the chi-square test to be quite helpful in my own projects.

## What are Categorical Variables?

I’m sure you’ve encountered categorial variables before, even if you might not have intuitively recognized them. They can be tricky to deal with in the data science world, so let’s first define them.

Categorical variables fall into a particular category of those variables that can be divided into finite categories. These categories are generally names or labels. These variables are also called qualitative variables as they depict the quality or characteristics of that particular variable.

For example, the category “Movie Genre” in a list of movies could contain the categorical variables – “Action”, “Fantasy”, “Comedy”, “Romance”, etc.

There are broadly two types of categorical variables:

1. Nominal Variable: A nominal variable has no natural ordering to its categories. They have two or more categories. For example, Marital Status (Single, Married, Divorcee), Gender (Male, Female, Transgender), etc.
2. Ordinal Variable: A variable for which the categories can be placed in an order. For example, Customer Satisfaction (Excellent, Very Good, Good, Average, Bad), and so on

When the data we want to analyze contains this type of variable, we turn to the chi-square test, denoted by χ², to test our hypothesis.

## Why Do We Use It? [Explained with Example]

Let’s learn the use of chi-square with an intuitive example.

#### Problem Statement

A research scholar is interested in the relationship between the placement of students in the statistics department of a reputed University and their C.G.P.A (their final assessment score).

He obtains the placement records of the past five years from the placement cell database (at random). He records how many students who got placed fell into each of the following C.G.P.A. categories – 9-10, 8-9, 7-8, 6-7, and below 6.

Suppose there is no relationship between the placement rate and the C.G.P.A.. In that case, the placed students should be equally spread across the different C.G.P.A. categories (i.e., there should be similar numbers of placed students in each category).

However, if students having C.G.P.A more than 8 are more likely to get placed, then there would be a large number of placed students in the higher C.G.P.A. categories as compared to the lower C.G.P.A. categories. In this case, the data collected would make up the observed frequencies.

So the question is, are these frequencies being observed by chance, or do they follow some pattern?

#### Chi Square Test Solution

Here enters the chi-square test! The chi-square test helps us answer the above question by comparing the observed frequencies to the frequencies that we might expect to obtain purely by chance.

Chi-square test in hypothesis testing is used to test the hypothesis about the chi-square distribution of observations/frequencies in different categories.

We are almost at the implementing aspect of chi-square tests, but there’s one more thing we need to learn before we get there. In order to fully understand the distribution of a variable, both descriptive statistics and a chi-square test are essential tools. Descriptive statistics provide a snapshot of the data, while a chi-square test can reveal important relationships between categorical variables.

The normal distribution is a fundamental concept in statistics and is often used to model variables in experiments. A chi-square test can be used to determine if a set of observations follows a normal distribution.

## Properties of Chi-Square Test

The chi-square test possesses several important properties that make it a valuable statistical tool:

1. Non-parametric Test: The chi-square test is a non-parametric test, which means it does not require any assumptions about the underlying distribution of the data. It applies to small and large sample sizes and can be used with categorical data.
2. Test for Independence: The chi-square test of independence examines the association between two categorical variables. It determines whether there is a significant relationship or dependency between the variables rather than measuring the strength or direction of the association.
3. Goodness of Fit Test: The chi-square goodness of fit test assesses how well-observed data fit an expected distribution. It allows researchers to compare observed frequencies with those expected under a hypothesized distribution.
4. Chi-Square Statistic: The chi-square statistic compares the observed and expected frequencies in a contingency table. It measures the discrepancy between the observed and expected values, indicating the extent of the association or goodness of fit.
5. Degrees of Freedom: The degrees of freedom (df) in the chi-square test depends on the number of categories in the analyzed variables. It determines the critical values from the chi-square distribution table and affects the interpretation of the test result.
6. Null and Alternative Hypotheses: The chi-square test involves formulating null and alternative hypotheses. The null hypothesis assumes no association or no significant difference, while the alternative hypothesis suggests the presence of an association or difference.
7. Test Statistic and P-value: The chi-square test produces a test statistic and a corresponding p-value. The test statistic is compared to a critical value from the chi-square distribution to determine statistical significance. The p-value indicates the probability of obtaining the observed results under the assumption of the null hypothesis.
8. Interpretation: The null hypothesis is rejected if the calculated chi-square test statistic exceeds the critical value or if the p-value is less than the chosen significance level (e.g., α = 0.05). This implies evidence of a significant association or deviation from the expected distribution.

## Assumptions of the Chi-Square Test

The chi-square test uses the sampling distribution to calculate the likelihood of obtaining the observed results by chance and to determine whether the observed and expected frequencies are significantly different. Just like any other statistical test, the chi-square test comes with a few assumptions of its own:

• A large sample size is crucial for a reliable outcome in a chi-square test, as it helps ensure that the data distribution is representative of the population.
• The χ2 assumes that the data for the study is obtained through random selection, i.e., they are randomly picked from the population.
• The categories are mutually exclusive, i.e., each subject fits into only one category. For e.g., from our above example – the number of people who lunched in your restaurant on Monday can’t be filled in the Tuesday category.
• The data should be in the form of frequencies or counts of a particular category and not in percentages.
• The data should not consist of paired samples or groups, or we can say the observations should be independent of each other.
• When more than 20% of the expected counts (frequencies) have a value of less than 5, then Chi-square cannot be used.  To tackle this problem: Either one should combine the categories only if it is relevant or obtain more data.

The total number of observations is a crucial component in determining the validity of the chi-square test. The larger the number of observations, the more accurate the chi-square test results will be. The Yates correction is an adjustment used in the chi-square test to account for the expected counts (frequencies) being close to zero. The Yates correction ensures the validity of the chi-square test results when applied to small sample sizes.

## Types of Chi-Square Tests

In this section, we will see the different types of chi-square tests and study them by manual calculations and with their implementation in R.

There are three main types of chi-square tests commonly used in statistics:

1. Pearson’s Chi-Square Test: This test is used to determine if there is a significant association between two categorical variables in a single population. It compares the observed frequencies in a contingency table with the expected frequencies assuming independence between the variables.
2. Chi-Square Goodness of Fit Test: This test is used to assess whether observed categorical data follows an expected distribution. It compares the observed frequencies with the expected frequencies specified by a hypothesized distribution.
3. Chi-Square Test of Independence: This test is used to examine if there is a significant association between two categorical variables in a sample from a population. It compares the observed frequencies in a contingency table with the expected frequencies assuming independence between the variables.

## Chi-Square Goodness of Fit Test

This is a non-parametric test. We typically use it to find how the observed value of a given event is significantly different from the expected value. In this case, we have categorical data for one independent variable, and we want to check whether the data distribution is similar or different from the expected distribution.

Let’s consider the above example where the research scholar was interested in the relationship between the placement of students in a reputed university’s statistics department and their C.G.P.A.

In this case, the independent variable is C.G.P.A with the categories 9-10, 8-9, 7-8, 6-7, and below 6.

The statistical question here is: whether or not the observed counts (frequencies) of placed students are equally distributed for different C.G.P.A categories (so that our theoretical frequency distribution contains the same number of students in each of the C.G.P.A categories).

We will arrange this data by using the contingency table, which will consist of both the observed and expected values as below:

After constructing the contingency table, the next task is to compute the value of the chi-square statistic. The formula for chi-square is given as:

where,

• χ 2 = Chi-Square value
• Oi = Observed frequency
• Ei = Expected frequency

#### Let us look at the step-by-step approach to calculate the chi-square value

For example, for the C.G.P.A category 10-9, it will be “30-20 = 10”. Apply similar operations for all the categories.

#### Step 2: Square each value obtained in step 1, i.e. (O-E)2.

For example: for the C.G.P.A category 10-9, the value obtained in step 1 is 10. It becomes 100 on squaring. Apply similar operations for all the categories.

For example: for the C.G.P.A category 10-9, the value obtained in step 2 is 100. On dividing it by the related expected frequency, which is 20, it becomes 5. Apply similar operations for all the categories.

#### Step 4: Add all the values obtained in step 3 to get the chi-square test statistic value.

In this case, the chi-square statistic value comes out to be 32.5.

#### Step 5: Once we have calculated the chi-square value, we will compare it with the critical chi-square statistic value.

Then the number of degrees of freedom represents the number of values in the data set that are free to vary and contribute to the test statistic. The number of degrees of freedom in the test affects the critical value and the level of significance, helping to determine if the differences are due to chance or are statistically significant. We can find the critical chi-square value in the below chi-square table against the number of degrees of freedom (number of categories – 1) and the significance level:

In this case, the degrees of freedom are 5-1 = 4. So, the critical value at a 5% significance level is 9.49. The significance level in a chi-square test determines the threshold for rejecting the null hypothesis and accepting the alternative hypothesis. A significance level of 0.05 means a 5% chance of making a Type I error or falsely rejecting the null hypothesis.

Our obtained value of 32.5 is much larger than the critical value of 9.49. Therefore, we can say that the observed frequencies from the sample data are significantly different from the expected frequencies. In other words, C.G.P.A is related to the number of placements that occur in the department of statistics.

Let’s further solidify our understanding by performing the Chi-Square test in R.

### The Chi-Square Goodness of Fit Test in R

Let’s implement the chi-square goodness of fit test in R. Time to fire up RStudio!

Problem Statement

Let’s understand the problem statement before we dive into R.

An organization claims that the experience of the employees of different departments is distributed in the following categories:

• 11 – 20 Years = 20%
• 21 – 40 Years = 17%
• 6 – 10 Years = 41% and
• Up to 5 Years = 22%

A random sample of 1470 employees is collected. Does this random sample provide evidence against the organization’s claim?

Setting up the hypothesis

• Null hypothesis: The true proportions of the experience of the employees of different departments are distributed in the following categories: 11 – 20 Years = 20%, 21 – 40 Years = 17%, 6 – 10 Years = 41%, and up to 5 Years = 22%
• Alternative hypothesis: The distribution of experience of the employees of different departments differs from what the organization states

Let’s begin!

#### Step 2: Validate it for correctness in R:

Output:

``` #Count of Rows and columns

 1470 2
#View top 10 rows of the dataset

age.intervals Experience.intervals
1 41 - 50 6 - 10 Years
2 41 - 50 6 - 10 Years
3 31 - 40 6 - 10 Years
4 31 - 40 6 - 10 Years
5 18 - 30 6 - 10 Years
6 31 - 40 6 - 10 Years
7 51 - 60 11 - 20 Years
8 18 - 30 Upto 5 Years
9 31 - 40 6 - 10 Years
10 31 - 40 11 - 20 Years

```

#### Step 3: Create a proportion table for expected frequencies:

Output:

```11 - 20 Years 21 - 40 Years 6 - 10 Years Upto 5 Years
0.2312925 0.1408163 0.4129252 0.2149660```

Step – 4: Calculate the chi-square value:

Output:

```Chi-square test for given probabilities

data: table(data\$Experience.intervals)
X-squared = 14.762, df = 3, p-value = 0.002032```

The p-value here is less than 0.05. Therefore, we will reject our null hypothesis. Hence, the distribution of experience of the employees of different departments differs from what the organization states.

## Chi-Square Test of Association

The second type of chi-square test is the Pearson’s chi-square test of association. This test is used when we have categorical data for two independent variables and want to see if there is any relationship between the variables.

Let’s take another example to understand this. A teacher wants to know whether the outcome of a mathematics test is related to the gender of the person taking the test. Or in other words, she wants to know if males show a different pattern of pass/fail rates than females.

So, here are two categorical variables: Gender (Male and Female) and mathematics test outcome (Pass or Fail). Let us now look at the contingency table:

Looking at the above contingency table, we can see that girls have a comparatively higher pass rate than boys. However, to test whether or not this observed difference is significant, we will carry out the chi-square test.

#### Step 2: Calculate the expected frequency for each individual cell by multiplying row sum by column sum and dividing by total number:

Expected Frequency = (Row Total x Column Total)/Grand Total

For the first cell, the expected frequency would be (37*25)/50 = 18.5. Now, write them below the observed frequencies in brackets:

#### Step 3: Calculate the value of chi-square using the formula:

Calculate the right-hand side part of each cell. For example, for the first cell, ((17-18.5)^2)/18.5 = 0.1216.

#### Step 4: Then, add all the values obtained for each cell. In this case, the values are:

0.1216+0.1216+0.3461+0.3461 = 0.9354

#### Step 5: Calculate the degrees of freedom, i.e (Number of rows-1)*(Number of columns-1) = 1*1 = 1

The next task is to compare it with the critical chi-square value from the above table.

The Chi-Square calculated value is 0.9354, which is less than the critical value of 3.84. So, in this case, we fail to reject the null hypothesis. This means there is no significant association between the two variables, i.e., boys and girls have a statistically similar pattern of pass/fail rates on their mathematics tests.

Let’s further solidify our understanding by performing the chi-square test in R.

## Chi-Square Test for Independence in R

Problem Statement

A Human Resources department of an organization wants to check whether the employees’ age and experience depend on each other. For this purpose, a random sample of 1470 employees is collected with their age and experience. You can download the data here.

Setting up the hypothesis:

• Null hypothesis: Age and Experience are two independent variables
• Alternative hypothesis: Age and Experience are two dependent variables

Let’s begin!

Step 1: First, import the data

Step 2: Validate it for correctness in R:

Output:

``` #Count of Rows and columns

 1470 2
> #View top 10 rows of the dataset

age.intervals Experience.intervals
1 41 - 50 6 - 10 Years
2 41 - 50 6 - 10 Years
3 31 - 40 6 - 10 Years
4 31 - 40 6 - 10 Years
5 18 - 30 6 - 10 Years
6 31 - 40 6 - 10 Years
7 51 - 60 11 - 20 Years
8 18 - 30 Upto 5 Years
9 31 - 40 6 - 10 Years
10 31 - 40 11 - 20 Years

```

Step 3: Construct a contingency table and calculate the chi-square value:

Output:

```ct<-table(data\$age.intervals,data\$Experience.intervals)
> ct

11 - 20 Years 21 - 40 Years 6 - 10 Years Upto 5 Years
18 - 30 22 0 172 192
31 - 40 190 20 308 101
41 - 50 85 112 110 15
51 - 60 43 75 17 8
> chisq.test(ct)

Pearson's Chi-squared test

data: ct
X-squared = 679.97, df = 9, p-value < 2.2e-16

```

The p-value here is less than 0.05. Therefore, we will reject our null hypothesis. We can conclude that age and experience are two dependent variables, aka, as the experience increases, the age also increases (and vice versa).

## Limitations of Chi Square Test

While the chi-square test is a useful statistical tool, it does have some limitations that should be considered:

1. Assumptions of Independence: The chi-square test assumes that observations are independent and that each observation contributes independently to the expected cell frequencies. Violations of this assumption, such as dependence or correlation between observations, can affect the validity of the test results.
2. Sample Size Requirements: The chi-square test may not be reliable with small sample sizes, particularly when the expected frequencies in cells are low. It is generally recommended to have an expected frequency of at least 5 in each cell to ensure the validity of the test.
3. Sensitivity to Sample Composition: The chi-square test may be sensitive to the composition of the sample or the categories being analyzed. If the categories have imbalanced frequencies or if there are empty cells, the test results may be biased or inaccurate.
4. Categorical Variables Only: The chi-square test is specifically designed for categorical variables and is not applicable to continuous or ordinal variables. It cannot directly assess relationships or differences between continuous variables.
5. Lack of Directionality or Magnitude: The chi-square test determines whether an association exists between variables but does not provide information about the strength, direction, or magnitude of the relationship. It only indicates whether the association is statistically significant or not.
6. Type of Association: The chi-square test can detect associations or dependencies between variables but does not differentiate between different types of associations. It cannot determine cause-and-effect relationships or provide insights into the nature of the association.
7. Large Sample Bias: In large samples, the chi-square test may detect even small, practically insignificant deviations from independence, leading to statistically significant results that may not have meaningful implications.
8. Multiple Comparisons: Conducting multiple chi-square tests on the same data increases the likelihood of finding a statistically significant result by chance alone. Adjustments, such as Bonferroni correction, should be considered to address the issue of multiple comparisons.
9. Potential Interpretation Issues: The chi-square test results should be interpreted cautiously, taking into account the context and the specific research question. A significant chi-square test does not necessarily imply a meaningful or practically important association.

It is important to consider these limitations and ensure that the chi-square test is appropriate for the specific research question and data characteristics.

## Conclusion

In this article, we learned how to analyze the significant difference between data that contains categorical measures with the help of chi-square tests. We enhanced our knowledge of the use of chi-square, the assumptions involved in carrying out the test, and how to conduct different types of chi-square tests both manually and in R.

If you are new to statistics, want to cover your basics, and also want to get a start in data science, I recommend taking the Introduction to Data Science course. It gives you a comprehensive overview of both descriptive and inferential statistics before diving into data science techniques.

Did you find this article useful? Can you think of any other applications of the chi-square test? Let me know in the comments section below, and we can come up with more ideas!

Key Takeaways

• The Chi-square test is a hypothesis testing method used to compare observed data with expected data.
• The chi-square value, calculated using the chi-square formula, tells us the extent of similarity or difference between the categories of data being considered.
• There are two types of Chi-Square Tests: the Chi-Square Goodness of Fit Test and the Chi-Square Test of Independence/Association.

Q1. What is chi-square test and how is it calculated?

A. The Chi-square test is a hypothesis testing method used to test the hypothesis about the chi-square distribution of observations/frequencies in different categories.
Here’s how it is calculated. We first find the difference between the observed (o) and expected (e) values. We then take the square of that number and divide it by the expected value. Finally, we add all of these calculated values from the various categories to get the chi-square.

Q2. What is a chi-square test used for?

A. A chi-square test is used to predict the probability of observations, assuming the null hypothesis to be true. It is often used to determine if a set of observations follows a normal distribution. It can also be used to find the relationship between the categorical data for two independent variables.

Q3. What does it mean when the chi-square value is large?

A. If the chi-square value is larger than the critical value, it means that there is a significant difference between the categories of data in consideration. The larger the chi-square value, the greater the probability of a significant difference.

Q4. What is the p-value in chi-square?

In the context of the chi-square test, the p-value is the probability of obtaining a test statistic as extreme or more extreme than the observed one, assuming the null hypothesis is true. The test statistic in the chi-square test follows a chi-square distribution, and the p-value is calculated by comparing the observed test statistic to the critical value of the chi-square distribution at the specified level of significance (typically 0.05 or 0.01).
A small p-value (typically less than the chosen level of significance) indicates that the observed data are unlikely to have occurred by chance if the null hypothesis is true, and thus the null hypothesis can be rejected in favor of the alternative hypothesis. Conversely, a large p-value indicates that the observed data are consistent with the null hypothesis, and there is insufficient evidence to reject it. ###### Abhiraj Suresh   