AP Syllabus focus:
‘Determine the p-value for the chi-square statistic using chi-square distribution tables or computer software. The p-value indicates the probability of observing a chi-square statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.’
Determining the p-value
A p-value in a chi-square test quantifies how surprising the observed discrepancy between observed and expected counts would be if the null hypothesis were true, guiding evidence-based statistical decisions.
Understanding the Role of the p-value in Chi-Square Inference
The p-value is central to interpreting chi-square test results because it expresses the likelihood of obtaining a chi-square statistic at least as large as the calculated value under the assumption that the null hypothesis is correct. A chi-square test evaluates categorical data by comparing observed patterns with those predicted under a specified distribution. When the test statistic is unusually large, it suggests stronger evidence that the observed data differ from what would be expected by chance alone.
When a chi-square statistic is computed, its position within a chi-square distribution determines the p-value. The chi-square distribution is right-skewed and consists only of positive values, and its exact shape depends on the degrees of freedom. These characteristics influence how probabilities accumulate in the upper tail of the distribution, where the p-value is found.
How the Chi-Square Distribution Is Used to Obtain a p-value
A chi-square distribution models the expected behavior of chi-square statistics generated under repeated sampling when the null hypothesis is true. Each distribution corresponds to a specific number of degrees of freedom determined by the structure of the categorical test. Because chi-square tests measure deviations between observed and expected counts, only unusually large deviations are considered evidence against the null hypothesis, making the p-value a one-tailed upper-tail probability.
Key Features of Chi-Square Distributions
They are right-skewed, especially when degrees of freedom are small.
As degrees of freedom increase, the distribution becomes more symmetric.
They assign greater probability weight to moderately sized chi-square statistics and rapidly decreasing weight to large ones, allowing the p-value to reflect extremeness.
This graph shows the inverse cumulative distribution function for the chi-square distribution, illustrating how each χ² value corresponds to an upper-tail p-value. The curve demonstrates that larger chi-square statistics yield smaller p-values. The full curve provides additional detail beyond the syllabus by showing continuous probabilities across a range of χ² values. Source.
Computing the p-value Using Technology or Tables
AP Statistics emphasizes two acceptable methods for determining p-values: chi-square distribution tables and statistical software. Both approaches rely on interpreting the magnitude of the chi-square statistic relative to its distribution.
Using Chi-Square Distribution Tables
Tables provide critical values for various significance levels and degrees of freedom. To locate an approximate p-value:
Identify the row corresponding to the test’s degrees of freedom.
Compare the computed chi-square statistic to the listed critical values.
Determine the interval of p-values in which the statistic falls.
Because tables do not list every possible probability, they give only a range rather than an exact value. However, this range is sufficient for inference decisions at standard significance levels.
Using Statistical Software
Software produces precise p-values by integrating the chi-square distribution’s density function from the test statistic to infinity. This method avoids approximation and is commonly used in modern applied statistics.
Why the p-value Reflects Evidence Against the Null Hypothesis
A chi-square statistic measures how far observed counts deviate from expected counts under the null hypothesis. The p-value quantifies the rareness of that deviation assuming the null hypothesis is correct. Thus, a small p-value signals that the observed pattern would occur infrequently if the null hypothesis were true.
When interpreting this value, the direction of extremeness matters: chi-square tests focus exclusively on unusually large test statistics because only large discrepancies indicate lack of fit or potential association.
Definition of the p-value in the Context of Chi-Square Tests
p-value: The probability, assuming the null hypothesis is true, of obtaining a chi-square statistic as extreme as or more extreme than the observed statistic.
A small p-value provides stronger evidence against the null hypothesis, while a large p-value suggests that the observed deviations could reasonably occur by chance.
A single sentence clarifies the importance of understanding this probability measure when interpreting categorical data patterns.
Interpreting the Extremeness of the Chi-Square Statistic
Because the chi-square statistic accumulates squared standardized residuals, even moderate discrepancies contribute to larger values. Therefore, the upper tail of the distribution represents outcomes with increasing lack of agreement between observed and expected counts.
Interpreting Large vs. Small p-values
Small p-values indicate that the observed data are unlikely under the null model, encouraging rejection of the null hypothesis.
Large p-values indicate that observed deviations fall within the range of typical random variation, supporting retention of the null hypothesis.
The interpretation must always be tied to the context of the categorical variables and the study design.
The Connection Between Degrees of Freedom and p-value Interpretation
Degrees of freedom shape the chi-square distribution’s tail behavior. With lower degrees of freedom, the distribution spreads out more, making moderately large statistics more extreme. With higher degrees of freedom, greater variability is expected, so even large chi-square statistics may correspond to moderate p-values.
Understanding this relationship ensures that students evaluate chi-square results appropriately and recognize how the underlying distribution influences probability assessments.
This figure shows chi-square probability density functions for multiple degrees of freedom, demonstrating how distribution shape changes as df increases. The curves become less skewed and more spread out, illustrating why the extremeness of a test statistic depends on its degrees of freedom. The multiple overlaid curves include additional detail beyond the syllabus but effectively highlight differences in tail behavior. Source
FAQ
Minor rounding generally has little impact when using technology, as software computes the p-value using full precision.
When using tables, rounding can shift the statistic across a critical value, slightly altering the inferred p-value range.
If a value is close to a table boundary, it is safer to use technology or report the p-value interval rather than an exact value.
The chi-square statistic measures the total deviation between observed and expected counts, and large values indicate stronger evidence against the null hypothesis.
Because negative deviations cancel out when squared, only unusually large totals matter.
This makes the upper tail the appropriate region for assessing extremeness in chi-square tests.
With fewer degrees of freedom, the distribution is more heavily skewed, so moderate chi-square values may already be considered extreme, producing smaller p-values.
With more degrees of freedom, the distribution spreads out, meaning much larger chi-square values are required before the p-value becomes small.
This is why interpreting a chi-square statistic must always account for the correct degrees of freedom.
Uneven expected counts cause categories with larger expected values to influence the chi-square statistic more strongly.
This can result in:
A chi-square statistic that is dominated by a few categories
A p-value that reflects disproportionate contributions
While the test remains valid, interpretation should acknowledge that some categories drive the inference more than others.
Larger samples reduce random variation, making even small differences between observed and expected counts statistically detectable.
Consequently:
Small p-values may reflect minor deviations rather than meaningful effects
Practical significance should be considered alongside statistical significance
In smaller samples, only larger discrepancies will yield small p-values, as variability is naturally higher.
Practice Questions
Question 1 (1–3 marks)
A chi-square goodness-of-fit test produces a chi-square statistic of 12.4 with 4 degrees of freedom. Explain how the p-value for this test would be obtained and state what the p-value represents in context.
Question 1
Total: 3 marks
1 mark: States that the p-value is found by comparing the chi-square statistic (12.4) with the chi-square distribution for 4 degrees of freedom, using tables or software.
1 mark: Mentions that the p-value is the probability of obtaining a chi-square statistic as large as or larger than 12.4 if the null hypothesis is true.
1 mark: Provides a contextual reference (e.g., reflects how unlikely the observed differences between observed and expected counts are under the null model).
Question 2 (4–6 marks)
A researcher conducts a chi-square test for independence and obtains a chi-square statistic of 18.9 with 6 degrees of freedom.
(a) Describe how to determine the p-value using statistical tables.
(b) The researcher instead uses statistical software and finds the exact p-value is 0.0041. Interpret this p-value in the context of the test.
(c) Using a 5% significance level, state the conclusion of the test and justify your decision.
Question 2
Total: 6 marks
(a) 2 marks
1 mark: States that the chi-square statistic (18.9) is compared with critical values for 6 degrees of freedom in a chi-square table.
1 mark: States that the p-value corresponds to the upper-tail probability beyond the statistic’s value.
(b) 2 marks
1 mark: Interprets 0.0041 as the probability of observing a chi-square value of 18.9 or more extreme if the variables are truly independent.
1 mark: Clearly contextualises the interpretation (e.g., such a result would be very unlikely if there were no association).
(c) 2 marks
1 mark: Correct decision: reject the null hypothesis at the 5% significance level.
1 mark: Justification: the p-value (0.0041) is less than 0.05, providing sufficient evidence of an association between the variables.
