How to Perform a Chi-Square Goodness of Fit Test in Python

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Introduction

The Chi-Square Goodness of Fit Test is a statistical technique used to determine whether the observed frequencies of a categorical variable match the expected frequencies. It’s particularly useful in scenarios where you want to see if the empirical data conforms to an expected distribution. This article provides a comprehensive guide on performing the Chi-Square Goodness of Fit Test in Python.

Background and Significance of Chi-Square Goodness of Fit Test

In practice, you often want to test if the distribution of a categorical variable follows a hypothesized distribution. For instance, you might want to check if the number of students in a school who have brown, black, blonde, and red hair follows an expected ratio. The Chi-Square Goodness of Fit Test is well-suited to this kind of problem.

How to Perform a Chi-Square Goodness of Fit Test in Python

Introduction

The Chi-Square Goodness of Fit Test is a statistical technique used to determine whether the observed frequencies of a categorical variable match the expected frequencies. It’s particularly useful in scenarios where you want to see if the empirical data conforms to an expected distribution. This article provides a comprehensive guide on performing the Chi-Square Goodness of Fit Test in Python.

Table of Contents

  1. Background and Significance of Chi-Square Goodness of Fit Test
  2. Understanding Chi-Square Goodness of Fit Test a. Hypotheses b. Test Statistic c. Assumptions d. Applications
  3. Python Environment Setup
  4. Loading and Preparing Data
  5. Performing Chi-Square Goodness of Fit Test in Python a. Using scipy.stats b. Interpreting the Results
  6. Practical Example
  7. Conclusion

Background and Significance of Chi-Square Goodness of Fit Test

In practice, you often want to test if the distribution of a categorical variable follows a hypothesized distribution. For instance, you might want to check if the number of students in a school who have brown, black, blonde, and red hair follows an expected ratio. The Chi-Square Goodness of Fit Test is well-suited to this kind of problem.

Understanding Chi-Square Goodness of Fit Test

a. Hypotheses

The null and alternative hypotheses for the Chi-Square Goodness of Fit Test are as follows:

  • Null Hypothesis (H0): The observed frequencies follow the expected distribution.
  • Alternative Hypothesis (H1): The observed frequencies do not follow the expected distribution.

b. Test Statistic

The test statistic is calculated based on the squared differences between observed and expected frequencies, scaled by the expected frequencies.

c. Assumptions

  • The samples are random and independent.
  • The categories are mutually exclusive.
  • The expected frequency for each category should be at least 5 for the chi-square approximation to be valid.

d. Applications

  • Checking the fairness of a die by comparing the frequencies of each face with the expected frequencies.
  • Testing whether the observed frequencies of a categorical variable, like blood type, follow the expected frequencies in a population.

Loading and Preparing Data

Before you can perform the Chi-Square Goodness of Fit Test, you need to have some data. Load your data from a CSV file, excel, SQL database, or any other source. The pandas library is useful for loading and managing data.

Example:

import pandas as pd

# Load data from a CSV file
data = pd.read_csv('your-data-file.csv')

Performing Chi-Square Goodness of Fit Test in Python

a. Using scipy.stats

The scipy library provides the chisquare function for performing the Chi-Square Goodness of Fit Test.

from scipy.stats import chisquare

# Observed frequencies
observed = [16, 18, 16, 14, 12, 12]

# Expected frequencies
expected = [16, 16, 16, 16, 16, 8]

# Perform Chi-Square Goodness of Fit Test
chi_statistic, p_value = chisquare(observed, f_exp=expected)

# Output the results
print(f"Chi-Square Statistic: {chi_statistic}")
print(f"P-value: {p_value}")

b. Interpreting the Results

The p-value tells you whether or not the differences between the observed and expected frequencies are statistically significant. If the p-value is below a threshold, usually 0.05, you can reject the null hypothesis and conclude that the observed frequencies do not follow the expected distribution.

Practical Example

Let’s consider a practical example where you have data on the number of individuals with blood types A, B, AB, and O. You want to test if the observed frequencies match the expected frequencies in a general population.

from scipy.stats import chisquare

# Observed frequencies [Type A, Type B, Type AB, Type O]
observed = [95, 60, 45, 120]

# Expected frequencies based on general population ratios
expected = [90, 60, 50, 120]

# Perform Chi-Square Goodness of Fit Test
chi_statistic, p_value = chisquare(observed, f_exp=expected)

# Output the results
print(f"Chi-Square Statistic: {chi_statistic}")
print(f"P-value: {p_value}")

# Interpret the results
alpha = 0.05
if p_value < alpha:
    print("Reject the null hypothesis - The observed frequencies do not follow the expected distribution.")
else:
    print("Fail to reject the null hypothesis - The observed frequencies follow the expected distribution.")

Conclusion

The Chi-Square Goodness of Fit Test is a powerful tool for comparing the distribution of categorical variables against expected frequencies. Python’s scipy library makes it easy and efficient to perform this test. Being able to correctly interpret the results in the context of the data is crucial. This test has broad applications in fields like market research, quality control, and any area where distributional analysis is needed.

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