AP Syllabus focus:
‘Demonstrate how to apply expected counts in the calculation of the chi-square statistic for tests of independence, including the importance of ensuring that expected counts meet the condition (e.g., all expected counts should be greater than 5) for the chi-square test to be valid. Illustrate the process with examples of two-way tables from real-world scenarios.’
Understanding how expected counts are applied in chi-square tests for independence is essential for measuring discrepancies between observed and anticipated values and determining whether variables may be statistically associated.
Applying Expected Counts in Chi-Square Tests for Independence
Expected counts play a foundational role in evaluating whether two categorical variables may exhibit an association in a population. When a two-way table is constructed from sampled data, the expected counts for each cell represent what would occur if the variables were truly independent, meaning the distribution of one variable does not differ across the levels of the other.
A two-way table summarizing survey responses for cola preference by demographic groups, illustrating the structure used to assess potential associations between categorical variables. Source.
Determining Expected Counts
To apply expected counts within the chi-square procedure, students must first identify the structure of the two-way table. Each cell of the table contains an observed count, which is compared against an expected count, derived solely under the assumption of independence.
EQUATION
= Sum of counts across the row
= Sum of counts down the column
= Total number of observations
These expected counts provide the benchmark against which observed values are compared to assess discrepancies attributable to more than random variation. The chi-square test for independence evaluates whether these differences are large enough to challenge the assumption of independence.
After identifying the expected counts, students must ensure that the assumptions for the chi-square test are satisfied before continuing with analysis.
Importance of Meeting Conditions for Validity
The syllabus emphasizes that expected counts should be greater than 5 for the test to be considered statistically reliable. This large-count condition supports the accuracy of the chi-square distribution as an approximation for the sampling distribution of the chi-square statistic. Cells with expected counts lower than this threshold risk producing misleadingly large test statistics or inaccurate p-values, undermining the validity of any inference. While the independence test accommodates a variety of categorical contexts, it remains important to verify that data collection occurred through an appropriate sampling method or experimental design ensuring independence within and between observations.
Applying Expected Counts in the Chi-Square Statistic
Once all expected counts are calculated and conditions verified, the next step is computing the chi-square statistic, which summarizes the overall discrepancy between observed and expected values.
This measure aggregates deviations across all cells of the two-way table.
EQUATION
= Recorded sample count in a cell
= Anticipated count when variables are independent
This statistic quantifies how far the sample data depart from what would be anticipated if independence held true. Larger deviations contribute more heavily to the chi-square statistic, especially when they occur in cells with moderately sized expected counts.
Before determining the significance of the result, students must compute the appropriate degrees of freedom for the chi-square distribution associated with the test.
Degrees of Freedom and Interpretation Framework
The chi-square distribution used to evaluate independence depends on the structure of the two-way table. Degrees of freedom reflect the number of values that can vary once marginal totals are fixed.
EQUATION
= Number of rows in the two-way table
= Number of columns in the two-way table
After computing the degrees of freedom, the chi-square value is compared to a reference distribution to obtain a p-value.
A graph of a chi-square distribution showing how larger chi-square statistics fall farther into the right tail, resulting in smaller p-values used to assess statistical significance in tests of independence. Source.
A small p-value indicates that the observed pattern of counts is unlikely under the assumption of independence, providing evidence that the variables in the table may be associated.
Process Summary for Applying Expected Counts in Independence Testing
Students should follow a structured sequence when applying expected counts in a chi-square test for independence:
• Construct a two-way table of observed counts from sampled categorical data.
• Compute expected counts for each cell under the assumption of independence using row totals, column totals, and the table total.
• Verify conditions, especially that all expected counts exceed 5 and observations were collected appropriately.
• Calculate the chi-square statistic to quantify discrepancies between observed and expected counts.
• Determine degrees of freedom and obtain the p-value from the chi-square distribution.
• Evaluate whether variables may be associated, based on whether the discrepancy between observed and expected counts is statistically significant.
These steps demonstrate how expected counts play an essential interpretive and computational role in the application of chi-square tests for independence, supporting statistical reasoning about relationships between categorical variables.
FAQ
Expected counts reflect what the table would look like if there were no association between variables. This allows the chi-square statistic to measure how far the observed table deviates from this independence pattern.
Using the observed distribution would obscure whether variables relate to each other, as the test relies on comparing reality with a clearly defined null structure.
Larger sample sizes increase expected counts across cells, improving the accuracy of the chi-square approximation and strengthening the test’s reliability.
However, very large samples can produce statistically significant results even when associations are extremely weak, so context remains important.
Several strategies may be appropriate:
• Combine categories logically where possible.
• Consider whether the sample size is too small for this test.
• Use Fisher’s exact test for 2x2 tables, though this is typically outside AP scope.
The key goal is ensuring expected counts are sufficiently large for the chi-square method to function as intended.
The chi-square contribution depends not only on the difference between observed and expected counts but also on the expected count itself.
When expected counts are large, a given difference produces a smaller contribution. When expected counts are small, the same difference produces a relatively larger contribution, making the test more sensitive to deviations where expected frequencies are lower.
Uneven marginal totals do not invalidate a chi-square test for independence, but they can create highly unbalanced expected counts. This sometimes means that a few cells dominate the chi-square statistic because their expected values are comparatively large.
If the imbalance leads to expected counts falling below 5 in small categories, researchers may consider combining similar categories, provided doing so still aligns with the context and preserves interpretive meaning.
Practice Questions
Question 1 (1–3 marks)
A researcher collects data on two categorical variables: type of snack chosen (crisps, fruit, biscuits) and time of day (morning, afternoon). The observed counts are organised in a two-way table. Explain why expected counts are needed when carrying out a chi-square test for independence.
Question 1
• 1 mark: States that expected counts represent what would be anticipated if the variables were independent.
• 1 mark: Mentions that expected counts provide a benchmark against which observed counts are compared.
• 1 mark: Explains that comparing observed and expected counts allows the chi-square statistic to measure discrepancies.
Question 2 (4–6 marks)
A school surveys students about their preferred study location (library, classroom, home) and whether they study alone or with others. The results are summarised in a two-way table.
(a) Describe how to calculate the expected count for a cell in this table under the assumption that the two variables are independent.
(b) Explain why verifying expected counts greater than 5 is important before proceeding with the chi-square test for independence.
(c) Briefly describe how the chi-square statistic uses expected counts to determine whether study location and study group preference may be associated.
Question 2
(a)
• 1 mark: States that expected counts are calculated using row total multiplied by column total.
• 1 mark: States that the product is divided by the overall table total.
(b)
• 1 mark: States that expected counts should be greater than 5 for validity.
• 1 mark: Explains that this ensures the chi-square approximation is accurate or reliable.
(c)
• 1 mark: States that the chi-square statistic measures the difference between observed and expected counts.
• 1 mark: Explains that larger discrepancies lead to a larger chi-square value, providing evidence of an association if sufficiently large.
