Find definitions and interpretation guidance for every statistic and graph that is provided with the chi-square goodness-of-fit test.

This bar chart plots each category's contribution to the overall chi-square statistic. You can choose a chart that orders the categories by contribution, from largest contribution to smallest contribution.

Categories with a large difference between observed and expected values make a larger contribution to the overall chi-square statistic.

Use a bar chart that plots the observed and expected values for each category to determine whether there is a difference in a particular category.

If you determined that the difference between the observed and expected counts is statistically significant, you can use this bar chart to determine which categories have the largest difference between observed and expected values.

Use the individual category contributions to quantify how much of the total chi-square statistic is attributable to each category's difference between observed and expected values.

Minitab calculates each category's contribution to the chi-square statistic as the square of the difference between the observed and expected values for a category, divided by the expected value for that category. The chi-square statistic is the sum of these values for all the categories.

Categories with a large difference between observed and expected values make a larger contribution to the overall chi-square statistic.

In these results, the contribution values from each category sum to the overall chi-square statistic, which is 0.648. The largest contribution comes from size Medium shirts, and the smallest contribution comes from the Extra Large category.

The degrees of freedom for the chi-square goodness-of-fit test is the number of categories minus 1.

Minitab uses the degrees of freedom to determine the p-value. The more categories you have in your study, the more degrees of freedom you have.

In these results, the degrees of freedom (DF) is 3.

Chi-Square Test
N DF Chi-Sq P-Value
225 3 0.648148 0.885

N is the total sample size. N equals the sum of all the observed counts.

In these results, the total sample size (N) is 225.

Observed and Expected Counts
Test Contribution
Category Observed Proportion Expected to Chi-Square
Small 25 0.1 22.5 0.277778
Medium 41 0.2 45.0 0.355556
Large 91 0.4 90.0 0.011111
Extra Large 68 0.3 67.5 0.003704

Chi-Square Test
N DF Chi-Sq P-Value
225 3 0.648148 0.885

The observed values are the actual number of observations in a sample that belong to a category.

The expected values are the number of observations that you would expect to occur, on average, if the test proportions were true. Minitab calculates the expected counts by multiplying the test proportions from each category by the total sample size.

You can compare the observed values and the expected values by using the output table or the bar chart.

In these results, the expected counts seem very close to the observed counts for all categories.

The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.

Use the p-value to determine whether to reject or fail to reject the null hypothesis, which states that the population proportions in each category are consistent with the specified values in each category.

To determine whether the observed values from the sample and expected values from the specified distribution are statistically different, compare the p-value to the significance level. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of incorrectly rejecting the null hypothesis.

- P-value ≤ α: The observed data are statistically different from the expected values (Reject H
_{0}) - If the p-value is less than or equal to the significance level, you reject the null hypothesis and conclude that the data does not follow a distribution with certain proportions. Use your specialized knowledge to determine whether the difference is practically significant.
- P-value > α: You cannot conclude that the observed data are statistically different from the expected values (Fail to reject H
_{0}) - If the p-value is larger than the significance level, you fail to reject the null hypothesis because you do not have enough evidence to conclude that the data do not follow the distribution with specified proportions. However, you cannot conclude that the distributions are the same. A difference might exist, but your test might not have enough power to detect it.

In these results, the p-value is 0.885. Because the p-value is greater than the chosen α value of 0.05, you fail to reject the null hypothesis. Therefore, you cannot conclude that observed proportions are significantly different from the specified proportions.

Chi-Square Test
N DF Chi-Sq P-Value
225 3 0.648148 0.885