# Chi-Square Test: Your Secret Weapon for Statistical Significance Do you want to know if your data is statistically significant and working, especially on categorical data, then Chi-Square Test is your secret weapon.

The Chi-Square statistical method is commonly used to test the independence between categorical variables, and it can help you uncover patterns, identify relationships, and make informed decisions.

But what exactly is the Chi-Square Test, and how does it work?

In this post, we'll provide a comprehensive overview of this powerful tool and show you how to use it to determine whether your data supports or rejects your hypothesis.

What exactly is the Chi-Square Test

We'll cover the assumptions of the test, the different types of Chi-Square Tests, and the steps involved in conducting the test.

Whether you're a student, researcher, or data analyst, the Chi-Square Test is essential in your statistical toolkit.

By the end of this post, you'll have a solid understanding of how to use the Chi-Square Test to analyze categorical data, interpret your results, and make sound decisions based on your findings.

## What Is Chi-Square Test

Let’s understand about chi-square test in laymen’s terms with a example.

Suppose you have a box of different coloured candies and want to know if the candies are distributed equally or not. To find out, you count the number of candies of each colour and write down the numbers. Now you have two sets of information, the actual numbers of each candy colour and the expected numbers if the candies were distributed equally.

To compare these two sets of information, you can use something called a chi-square test. It works by calculating a number that tells you how different the actual numbers are from the expected numbers.

If the actual numbers are very different from the expected numbers, then the chi-square will be large. On the other hand, if the actual numbers are similar to the expected numbers, then the chi-square number will be small.

By looking at the chi-square number, you can decide whether the candies are distributed equally or not.

If the chi-square number is small, then the candies are distributed equally.
If the chi-square number is large, then you can conclude that the candies are not distributed equally.

## Basic Terminologies

### Categorical Data

The chi-square test is used to analyze categorical data, which consists of data that can be placed into categories or groups like gender, nationality etc.

### Hypothesis Testing

The chi-square test is a hypothesis testing method that is used to determine if there is a significant association between two categorical variables.

### Null and Alternative Hypotheses

The null hypothesis is the hypothesis that there is no significant association between the two variables. In contrast, the alternative hypothesis is the hypothesis that there is a significant association.

## How Chi-square Test Used In Data Science Projects?

The chi-square test is a commonly used statistical method in data science and statistics to determine whether there is a significant association between two categorical variables. It is used to compare the observed data with the expected data and to determine if the differences between them are due to chance or if there is a real relationship between the two variables.

The chi-square test is used in a variety of fields, including medical research, social sciences, and business analytics. Some common examples of its application include

### Independence of Categorical Variables

For example, in a survey, we may want to test whether there is a relationship between gender and a person's preference for a particular brand.

### Goodness-of-fit Test

This tests whether the observed data fit a particular distribution. For example, if we have data on the number of people who prefer different types of music, we may want to test if the data fits a normal distribution.

### Homogeneity Test

This test is used to compare the proportions of different groups. For example, we can compare the proportions of people who prefer different political parties in two different regions.

## Key Assumptions of Chi-Square Test to Remember

The chi-square test has several assumptions that must be met in order for the results to be valid. Here are the main assumptions of the chi-square test. ### Independence

The observations in the sample must be independent. This means that the values in one category do not depend on those in another. If the observations are not independent, then the chi-square test may give misleading results.

### Sample Size

The sample size should be large enough for the expected frequencies to be greater than or equal to 5. If the expected frequencies are too small, then the chi-square test may not be reliable.

### Random Sampling

The sample should be drawn randomly from the population of interest. If the sample does not represent the population, the chi-square test may give misleading results.

### Measurement Scale

The chi-square test is appropriate for categorical data, which can be nominal or ordinal in nature. If the data is continuous or interval, other tests such as t-tests or ANOVA may be more appropriate.

### No Significant Outliers

The presence of outliers can affect the chi-square test results. It is important to check for outliers before conducting the test.

### Non-zero Expected Frequencies

The expected frequency for each category should not be zero. The chi-square test cannot be performed if any expected frequency is zero.

It is important to note that violating these assumptions may lead to biased or incorrect conclusions.

Therefore, assessing the assumptions before conducting the chi-square test is essential. If any assumptions are not met, then a different statistical test may be more appropriate.

## What Is the Range of Values In the Chi-Square Test?

The range of values in a chi-square test depends on the degrees of freedom (df) associated with the test. The degrees of freedom are calculated based on the number of categories in the variables being compared. In general, the chi-square distribution is a continuous probability distribution that takes only non-negative values. As the degrees of freedom increase, the distribution becomes less skewed, and the range of values increases.

Degrees of freedom: The degrees of freedom for the chi-square test are calculated as

Degrees of Freedom = (r - 1)(c - 1)

Where

• r is the number of rows,
• c is the number of columns in the contingency table

For example, if we have two categories in each variable being compared (i.e., a 2x2 contingency table), then the degrees of freedom would be 1, and the range of values for the chi-square statistic would be from 0 to positive infinity.

If we have more than two categories in each variable being compared, the degrees of freedom would be higher, and the range of values for the chi-square statistic would also increase.

For example, if we have a 3x3 contingency table, the degrees of freedom would be 4, and the range of values for the chi-square statistic would be from 0 to positive infinity.

It's important to note that the chi-square test only tells us if there is a significant association between the two categorical variables being compared.

It doesn't provide any information about the strength or direction of the relationship. The magnitude of the chi-square statistic and its associated p-value is used to determine whether to reject or fail to reject the null hypothesis, which states that there is no association between the variables.

## What Is the Effect Size In the Chi-Square Test?

Effect size in the chi-square test refers to the strength or magnitude of the association between the two categorical variables.

It is a measure of the practical significance of the observed association, in contrast to the statistical significance that is determined by the p-value.

One commonly used effect size measure for the chi-square test is Cramer's V. Cramer's V is a statistic that ranges from 0 to 1.

Where 0 indicates no association between the variables and 1 indicates a perfect association.

Cramer's V is calculated using the following formula:

V = sqrt(X^2 / (n * (min(r, c) - 1)))

Where

• X^2 is the chi-square statistic,
• n is the total number of observations,
• r is the number of rows in the contingency table,
• c is the number of columns in the contingency table.

Cramer's V is interpreted as follows:

• V = 0: No association between the variables
• V < 0.1: Weak association
• 0.1 <= V < 0.3: Moderate association
• 0.3 <= V < 0.5: Strong association
• V >= 0.5: Very strong association

Cramer's V is a helpful effect size measure because it is normalized and easy to interpret. However, it should be used with the p-value to understand the chi-square test results fully.

## Understanding With a Mathematical Example

Suppose we conduct a survey of 200 people to determine whether there is a relationship between gender and the preference for coffee or tea. The results of the survey are shown in the table below:

Gender

Coffee

Tea

Male

60

40

Female

90

10

To perform a chi-square test, we first need to calculate the expected frequencies. We would expect to see these frequencies in each cell if there was no relationship between gender and beverage preference.

We can calculate the expected frequency for each cell by multiplying the row total by the column total and dividing by the total number of observations:

Gender

Coffee

Tea

Total

Male

60

40

100

Female

90

10

100

Total

150

50

200

• Expected frequency for Male/Coffee cell: (100/200) * 150 = 75
• Expected frequency for Male/Tea cell: (100/200) * 50 = 25
• Expected frequency for Female/Coffee cell: (100/200) * 150 = 75
• Expected frequency for Female/Tea cell: (100/200) * 50 = 25

Using these expected frequencies, and we can calculate the chi-square statistic using the following formula:

χ² = Σ [ (O - E)² / E ]

Where

• O is the observed frequency,
• E is the expected frequency.

Plugging in the values from our table, we get:

χ² = [(60-75)²/75] + [(40-25)²/25] + [(90-75)²/75] + [(10-25)²/25]

= 5 + 15 + 5 + 45

= 70

The degrees of freedom (df) for the chi-square test is calculated as

(number of rows - 1) * (number of columns - 1)

Which in this case is (2-1)*(2-1) = 1.

To determine whether the chi-square statistic is significant, we compare it to the critical value from the chi-square distribution table with 1 degree of freedom and a given level of significance.

For example, at a 0.05 level of significance, the critical value is 3.84.

Since our calculated chi-square statistic of 70 is much larger than the critical value of 3.84, we reject the null hypothesis and conclude that there is a significant relationship between gender and beverage preference.

## When to Use Chi Square Test

Chi-squared test is a statistical hypothesis test that is used to determine whether there is a significant association or independence between two categorical variables. It is commonly used in data science to assess the relationship between two categorical variables and determine whether the relationship is statistically significant.

The chi-square test is appropriate when the data consists of frequencies or counts for categorical data.

In other words, when you have two categorical variables and want to investigate whether they are related, a chi-square test can help you determine whether the association between the two variables is significant.

For example, suppose you want to investigate whether there is an association between smoking and lung cancer. In that case, you could use a chi-square test to determine if there is a statistically significant relationship between the two variables.

## Difference Between Chi-Square Test and Anova

Both the Chi-Square test and ANOVA (Analysis of Variance) are statistical tests used to analyze data and test hypotheses. However, they are used for different types of data and research questions. The Chi-Square test is used for categorical data analysis, while ANOVA is used for continuous data analysis. The Chi-Square test compares observed data to expected data and determines if the differences between the two are significant.

It is used to test whether two categorical variables are independent or whether they are associated with each other.

For example, a researcher might use a Chi-Square test to determine if there is a relationship between smoking and lung cancer in a group of patients.

On the other hand, ANOVA is used to test for significant differences between two or more means.

It is typically used in experimental research to test the effect of an independent variable on a dependent variable. For example, a researcher might use ANOVA to test whether a new medication affects pain relief differently compared to a placebo.

Another difference between the Chi-Square test and ANOVA is the type of data required for each test.

The Chi-Square test requires categorical data, which means data that can be divided into categories or groups. ANOVA, on the other hand, requires continuous data, which means data that can take any value within a range.

In summary, the Chi-Square and ANOVA are useful statistical tests for analyzing data and testing hypotheses.

However, they are used for different types of data and research questions. The Chi-Square test is used for categorical data analysis, while ANOVA is used for continuous data analysis.

## Chi-Square Test Python Implementation We loaded the dataset, now let's see how we need to apply the chi-square test to the loaded dataset.

Let’s understand the above code in detail.

### Code Explanation

First, the necessary libraries - NumPy and SciPy - are imported.

• NumPy is a Python library for numerical computing,
• SciPy is a library for scientific computing that includes modules for statistics.

Then, the observed contingency table is created using the panda's crosstab() function, which computes a cross-tabulation of two factors.

The crosstab() function takes in the two variables to be cross-tabulated and returns a table showing the categories' counts for each variable.

In this case, the variables are "Gender" and "Preference".

Next, the expected contingency table is computed by multiplying the marginal totals of the rows and columns and dividing by the total number of observations. The expected table is used to compute the chi-square statistic.

The chi-square statistic is calculated using the scipy.stats.chi2_contingency() function takes in the observed contingency table and returns the chi-square statistic, p-value, degrees of freedom, and expected contingency table.

Finally, the p-value is compared to the significance level (alpha) to determine whether the null hypothesis should be rejected.

If the p-value is less than alpha, the null hypothesis is rejected, and we conclude that there is a significant association between the two variables.

Suppose the p-value is greater than the alpha. In that case, we fail to reject the null hypothesis and conclude there is insufficient evidence to suggest a significant association between the two variables.

Overall, this code demonstrates how the chi-square test can be implemented in Python to test the independence of two categorical variables.

Now let’s understand the output we got.

### Output Explanation

The chi-square statistic is 102.89, which is much larger than the critical value for a 2-degree of freedom test at the 0.05 level of significance (5.99), so we reject the null hypothesis and conclude that there is a significant relationship between survival and class.

The p-value is very small (4.55e-23), which supports the conclusion that the relationship between survival and class is significant.

The degree of freedom is 2, which corresponds to the number of categories in each variable minus 1.

The expected frequencies show the frequencies we would expect to see if there was no relationship between survival and class.

We can compare these expected frequencies to the observed frequencies in the contingency table to see where the differences are.

# Conclusion

In conclusion, the chi-square test is a statistical method used to determine whether there is a significant association between two categorical variables. It is commonly used in various fields, such as medical research, social sciences, and business analytics.

The chi-square test compares the observed data with the expected data. It determines if the differences between them are due to chance or if there is a real relationship between the two variables.

The chi-square test has several assumptions, such as independence, sample size, random sampling, and measurement scale, that must be met to ensure accurate results.

The range of values in the chi-square test depends on the degrees of freedom associated with the test.

However, the chi-square test only tells us if there is a significant association between two categorical variables, and it does not provide any information about the strength or direction of the relationship.

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