Chi-squared Test for Independence (DP IB Applications & Interpretation (AI)): Revision Note

Chi-Squared Test for Independence

What is a chi-squared test for independence?

• A chi-squared () test for independence is a hypothesis test used to test whether two variables are independent of each other

  • This is sometimes called a  two-way test

• This is an example of a goodness of fit test

  • We are testing whether the data fits the model that the variables are independent

• The chi-squared () distribution is used for this test

• You will use a contingency table

  • This is a two-way table that shows the observed frequencies for the different combinations of the two variables

    • For example: if the two variables are hair colour and eye colour then the contingency table will show the frequencies of the different combinations

Why might I have to combine rows or columns?

• The observed values are used to calculate expected values

  • These are the expected frequencies for each combination assuming that the variables are independent

    • Your GDC can calculate these for you after you input the observed frequencies

• The expected values must all be bigger than 5

• If one of the expected values is less than 5 then you will have to combine the corresponding row or column in the matrix of observed values with the adjacent row or column

  • The decision between row or column will be based on which seems the most appropriate

    • For example: if the two variables are age and favourite TV genre then it is more appropriate to combine age groups than types of genre

What are the degrees of freedom?

• There will be a minimum number of expected values you would need to know in order to be able to calculate all the expected values

• This minimum number is called the degrees of freedom and is often denoted by

• For a test for independence with an m × n contingency table

  •  

  • For example: If there are 5 rows and 3 columns then you only need to know 2 of the values in 4 of the rows as the rest can be calculated using the totals

What are the steps for a chi-squared test for independence?

• STEP 1: Write the hypotheses

  • H₀ : Variable X is independent of variable Y

  • H₁ : Variable X is not independent of variable Y

    • Make sure you clearly write what the variables are and don’t just call them X and Y

• STEP 2: Calculate the degrees of freedom for the test

  • For an m × n contingency table

  • Degrees of freedom is

• STEP 3: Enter your observed frequencies into your GDC using the option for a 2-way test

  • Enter these as a matrix

  • Your GDC will give you a matrix of the expected values (assuming the variables are independent)

    • If any values are 5 or less then combine rows/columns and repeat step 2

  • Your GDC will also give you the χ² statistic and its p-value

  • The χ² statistic is denoted as 

• STEP 4: Decide whether there is evidence to reject the null hypothesis

  • EITHER compare the χ² statistic with the given critical value

    • If χ² statistic > critical value then reject H₀

    • If χ² statistic < critical value then accept H₀

  • OR compare the p-value with the given significance level

    • If p-value < significance level then reject H₀

    • If p-value > significance level then accept H₀

• STEP 5: Write your conclusion

  • If you reject H₀

    • There is sufficient evidence to suggest that variable X is not independent of variable Y

    • Therefore this suggests they are associated

  • If you accept H₀

    • There is insufficient evidence to suggest that variable X is not independent of variable Y

    • Therefore this suggests they are independent

How do I calculate the chi-squared statistic?

• You are expected to be able to use your GDC to calculate the χ² statistic by inputting the matrix of the observed frequencies

• Seeing how it is done by hand might deepen your understanding but you are not expected to use this method

• STEP 1: For each observed frequency O_i calculate its expected frequency E_i

  • Assuming the variables are independent

    • E_i = P(X = x) × P(Y = y) × Total

    • Which simplifies to 

• STEP 2: Calculate the χ² statistic using the formula

  • 

  • You do not need to learn this formula as your GDC calculates it for you

• To calculate the p-value you would find the probability of a value being bigger than your χ² statistic using a χ² distribution with ν degrees of freedom