Two Variable Statistics Limitations of the χ 2 Test

Correlation Finding the relationship between two quantitative variables without being able to infer causal relationships Correlation is a statistical technique used to determine the degree to which two variables are related

Limitations of the χ 2 Test  There are two situations in which the χ 2 test may be unreliable:  Any of the expected frequencies are less than 5. This can be resolved by combining data.  The degrees of freedom is 1. This can be resolved using Yates’ continuity correction.

These situations may arise in internal assessment tasks, but you will NOT be required to deal with them in the IB examinations. Reference to the IB

The χ 2 test may be unreliable if any of the expected frequency values are less than 5. COMBINING DATA Consider the contingency table alongside which shows observed values. For this contingency table, χ 2 ≈ For a 5% significance level and df = 3, the critical value is 7:81 Since χ 2 calc > 7:81, we would reject H0, and conclude that gender and television watching are dependent.

However, on inspecting the expected frequency table, there are two expected frequencies which are less than 5. This indicates that our conclusion may not be reliable. COMBINING DATA Values are less than 5…

We can improve the reliability of this test by combining rows or columns so that there are no cells with expected frequency less than 5. In this case we combine the often and very often columns to produce: COMBINING DATA BECOMES

The expected frequency table is now: COMBINING DATA Now χ 2 calc ≈ 4.18, and for a 5% level with df = 2, the critical value is 5:99. Since χ 2 calc < 5.99, we now conclude that the variables are independent. This is different from our original conclusion.

This will be covered when we work on examples of the internal assessment. Yates’ Continuity Correction

Classwork is on page 343 in the textbook exercise 11 E.3 numbers 1 and 2. Extra Practice on Combining…