Expected counts form the foundation of chi-square goodness-of-fit testing, providing a structured way to compare observed categorical outcomes with theoretical predictions under the null hypothesis.

Calculating Expected Counts

Expected counts play a central role in chi-square inference because they represent what the data should look like if the null hypothesis is true. For a goodness-of-fit test, these counts are derived from the sample size and the hypothesized proportions for each category, making them an essential benchmark for determining whether discrepancies in the observed data are meaningful or simply due to random variation.

Understanding the Purpose of Expected Counts

Expected counts reflect the theoretical distribution of outcomes across categories based on the null model. When observed counts differ noticeably from expected values, the chi-square test evaluates whether those differences are statistically significant. This comparison is fundamental to categorical inference because it translates qualitative categories into quantitative evidence of model fit.

• Expected counts represent the benchmark for comparison against observed counts.

• They enforce the assumption that the observed distribution should follow the hypothesized proportions if the null model holds.

• The chi-square statistic assesses the magnitude of deviation between observed and expected values.

• Expected counts ensure that comparisons across categories remain fair, even when categories differ in probability.

Expected Counts Under the Null Hypothesis

In a chi-square goodness-of-fit test, the null hypothesis specifies the proportion of the population that falls into each category. Because these proportions describe the probability structure assumed to be true, they provide the basis for calculating expected counts. Expected counts scale these probabilities to the sample size, allowing the comparison to occur on the same measurement scale as the data.

To compute expected counts, each null proportion must be known or supplied by the context.

This table shows observed customer counts for each weekday in a chi-square goodness-of-fit context, illustrating the starting point before expected counts are computed from sample size and hypothesized proportions. Source.

These proportions may derive from theory, historical data, or an assumed equal distribution. Once the hypothesized distribution is established, expected values follow deterministically from the formula.

EQUATION

This equation formalizes the link between the hypothesized model and the observed data structure. Because expected counts arise directly from the null hypothesis, they represent the outcomes we anticipate if the sample data genuinely reflect the hypothesized distribution.

A sentence is required here to maintain spacing between structured blocks, ensuring conceptual continuity for students progressing through the material.

Key Properties of Expected Counts

Expected counts provide a structured reference distribution that allows for unbiased comparison across categories. They exhibit several important characteristics:

• Positive values only, since both sample size and proportions are nonnegative.

• Scale dependence, meaning larger samples produce larger expected counts even under identical proportions.

• Direct connection to probability, as they translate probabilistic expectations into numerical targets for the sample.

Expected counts also serve a diagnostic role. If certain categories yield expected counts that are too small, the chi-square test may not reliably approximate the chi-square distribution. This is due to the sensitivity of the chi-square approximation to sparse data, where very small expected values increase variability and distort the test statistic.

Expected Counts and the Structure of Categorical Data

Expected counts ensure that categorical data are analyzed in a way that accounts for each category’s relative likelihood under the null model. Because the chi-square statistic compares the squared deviations of observed and expected counts standardized by expected values, they must be sufficiently large to stabilize that ratio.

When categories have unequal hypothesized probabilities, expected counts reflect those differences proportionally. This preserves the relative weighting of categories within the null model, preventing categories with low probability from being overemphasized in the analysis.

• The calculation embeds assumptions about population structure directly into the test.

• Using expected counts prevents misleading conclusions that might arise from examining raw observed differences alone.

• They provide a consistent framework for evaluating deviations across heterogeneous categories.

Role of Expected Counts in Subsequent Inference

Expected counts do not stand alone as mere computational outputs; instead, they integrate directly into the inferential process for the chi-square statistic.

This software output displays observed and expected counts in a chi-square goodness-of-fit test, demonstrating how expected values appear directly in statistical output and contribute to the computation of the chi-square statistic. Source.

Their primary function is to provide the denominator standardization necessary for assessing proportional deviation. Without expected counts, the chi-square statistic would lack the capacity to meaningfully scale differences according to the likelihood of each category.

Expected counts also shape the conditions for inference, particularly the requirement that each expected value exceed 5 in typical AP Statistics applications. This guideline exists to ensure that:

• The sampling distribution of the chi-square statistic approximates the theoretical chi-square distribution.

• Random variability does not dominate the contribution of small categories.

• The inference remains statistically valid and interpretable at conventional significance levels.

Because expected counts embody the null model, they act as the anchor of the entire goodness-of-fit procedure, enabling the transition from descriptive observations to formal inferential statements.

Integrating Expected Counts Into the Testing Process

Although the calculation itself is straightforward, its importance cannot be overstated. Expected counts directly determine:

• The structure of the chi-square statistic.

• The sensitivity of the test to differences in category frequencies.

• The validity of the resulting p-value and inference.

Expected counts guide the analyst in understanding whether deviations observed in the data are consistent with theoretical expectations or whether they warrant further statistical investigation.