Chi-square Tests & Calculating Degrees of Freedom.

Introduction

Let's learn Chi square tests with a twist. Are you a fan of Formula one? If yes then this will be a perfect article for you, if not it will help you learn a lot so don't be scared.

What's a chi square test?

From Wikipedia ,Chi-Square test is a non parametric statistical procedure for determining the difference between observed and expected data.
It can also be used to decide whether the data correlates with our categorical variables. Thus helps to determine whether a difference between two categorical variables is due to chance or a relationship between them.

It is one of the most widely used techniques for hypothesis testing.

Types of Chi-square tests

1. The chi-square goodness of fit test is used to test whether the frequency distribution of a categorical variable is different from your expectations.
2. The chi-square test of independence is used to test whether two categorical variables are related to each other.

The two types of Pearson’s chi-square tests test whether the observed frequency distribution of a categorical variable is significantly different from its expected frequency distribution.
A frequency distribution describes how observations are distributed between different groups.
Frequency distributions are often displayed using frequency distribution tables. A frequency distribution table shows the number of observations in each group.

When to use a chi-square test

1. Testing a hypothesis about one or more categorical variables. If one or more of your variables is quantitative, you should use a different statistical test.
2. The sample was randomly selected from the population.
3. There are a minimum of five observations expected in each group or combination of groups.

How to Solve Chi-Square Problems?

1. State the Hypotheses
   Null hypothesis (H0): There is no association between the variables
   Alternative hypothesis (H1): There is an association between the variables.

2. Calculate the Expected Frequencies
   Use the formula: E=(Row Total×Column Total)Grand TotalE = \frac{(Row \ Total \times Column \ Total)}{Grand \ Total}E=Grand Total(Row Total×Column Total)​

3. Compute the Chi-Square Statistic
   Use the formula: χ2=∑(O−E)2E\chi^2 = \sum \frac{(O - E)^2}{E}χ2=∑E(O−E)2, where O is the observed frequency and E is the expected frequency.

4. Determine the Degrees of Freedom (df)
   Use the formula: df=(number of rows−1)×(number of columns−1)df = (number \ of \ rows - 1) \times (number \ of \ columns - 1)df=(number of rows−1)×(number of columns−1)

5. Find the Critical Value and Compare
   Use the chi-square distribution table to find the critical value for the given df and significance level (usually 0.05).
   Compare the chi-square statistic to the critical value to decide whether to reject the null hypothesis.

Example of Use with Formula One

Why We Use the Chi-Square Test

1. To analyze categorical data - Many factors in Formula 1, such as tire choices, pit stop strategies and driver performance are categorical variables.
2. To test independence - The Chi-Square test can be used to determine if two categorical factors are related, such as whether race outcomes are dependent on specific tire strategies.
3. To compare expected vs. observed outcomes - This helps determine if trends in Formula 1 results occur by chance or if they are statistically significant.

When We Use the Chi-Square Test

• Team Performance vs. Engine Supplier: Testing if certain engine suppliers (e.g., Mercedes, Ferrari, Honda) correlate with better race performance.
• Pit Stop Strategy vs. Race Results: Checking if a team's pit stop strategy significantly affects their final position.
• Driver Nationality vs. Team Preference: Investigating if specific nationalities are more likely to be hired by particular teams.

Degrees of Freedom in Chi-Square Tests
Degrees of freedom play a crucial role in determining the validity of the Chi-Square test. In Formula 1, there are three main ways to calculate this:

1. Contingency Tables (df = (rows - 1) * (columns - 1))

   • Example: If we analyze tire choices (soft, medium, hard) across three different teams, the degrees of freedom would be (3-1) * (3-1) = 4.

2. Goodness-of-Fit Test (df = categories - 1)

   • Example: If we test whether the distribution of podium finishes among 5 teams is as expected, the degrees of freedom would be 5-1 = 4.

3. Homogeneity Test (df = (number of groups - 1) * (number of categories - 1))

   • Example: If we test whether wet weather races impact finishing positions differently across four different seasons, we calculate df as (4-1) * (2-1) = 3.

Conclusion
The Chi-Square test is a valuable statistical tool for analyzing trends and associations in Formula 1. Whether it’s evaluating pit stop strategies or comparing team performance across different seasons, this method helps uncover patterns beyond random chance.