Interpret all statistics for Chi-Square Test for Association

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

Observed and expected counts

The observed count is the actual number of observations in a sample that belong to a category.

The expected count is the frequency that would be expected in a cell, on average, if the variables are independent. Minitab calculates the expected counts as the product of the row and column totals, divided by the total number of observations.

Interpretation

You can compare the observed values and the expected values for each cell in the output table. In these results, the observed cell count is the first number in each cell, and the expected count is the second number in each cell.

If two variables are associated, then the distribution of observations for one variable will differ depending on the category of the second variable. If two variables are independent, then the distribution of observations for one variable will be similar for all categories of the second variable. In this example, from column 1, row 2 of the table, the observed count is 76, and the expected count is 60.78. The observed count seems to be much larger than would be expected if the variables were independent.

Tabulated Statistics: Machine ID, Worksheet columns

Rows: Machine ID Columns: Worksheet columns 1st shift 2nd shift 3rd shift All 1 48 47 48 143 56.08 46.97 39.96 2 76 47 32 155 60.78 50.91 43.31 3 36 40 34 110 43.14 36.13 30.74 All 160 134 114 408 Cell Contents Count Expected count

All row and column counts

Minitab displays the marginal counts for rows and columns.
Row counts
The sum of counts across each table row.
Column counts
The sum of counts down each table column.
Total
The sum of counts for all cells. The sum of all row counts is equal to the sum of all column counts.

Interpretation

Use the marginal counts to understand how the counts are distributed between the categories.

In these results, the total for row 1 is 143, the total for row 2 is 155, and the total for row 3 is 110. The sum of all the rows is 408. The total for column 1 is 160, the total for column 2 is 134, and the total for column 3 is 114. The sum of all the columns is 408.

Tabulated Statistics: Machine ID, Worksheet columns

Rows: Machine ID Columns: Worksheet columns 1st shift 2nd shift 3rd shift All 1 48 47 48 143 56.08 46.97 39.96 2 76 47 32 155 60.78 50.91 43.31 3 36 40 34 110 43.14 36.13 30.74 All 160 134 114 408 Cell Contents Count Expected count

Contribution to Chi-square

Minitab displays each cell's contribution to the chi-square statistic, which quantifies how much of the total chi-square statistic is attributable to each cell's divergence.

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

Interpretation

In these results, the sum of the chi-square from each cell is the Pearson chi-square statistic which is 11.788. The largest contributions are from Machine 2, on the 1st and 3rd shift. The smallest contributions are from the 2nd shift, on Machines 1 and 2.

Chi-Square Test for Association: Machine ID, Worksheet columns

Rows: Machine ID Columns: Worksheet columns 1st shift 2nd shift 3rd shift All 1 48 47 48 143 56.08 46.97 39.96 1.1637 0.0000 1.6195 2 76 47 32 155 60.78 50.91 43.31 3.8088 0.2998 2.9530 3 36 40 34 110 43.14 36.13 30.74 1.1809 0.4151 0.3468 All 160 134 114 408 Cell Contents Count Expected count Contribution to Chi-square
Chi-Square Test Chi-Square DF P-Value Pearson 11.788 4 0.019 Likelihood Ratio 11.816 4 0.019

Pearson Chi-Square and Likelihood Ratio Chi-Square

Minitab performs a Pearson chi-square test and a likelihood-ratio chi-square test. Each chi-square test can be used to determine whether or not the variables are associated (dependent).
Pearson chi-square test

The Pearson chi-square statistic (χ2) involves the squared difference between the observed and the expected frequencies.

Likelihood-ratio chi-square test

The likelihood-ratio chi-square statistic (G2) is based on the ratio of the observed to the expected frequencies.

Interpretation

Use the chi-square statistics to test whether the variables are associated.

In these results, both the chi-square statistics are very similar. Use the p-values to evaluate the significance of the chi-square statistics.
Chi-Square Test Chi-Square DF P-Value Pearson 11.788 4 0.019 Likelihood Ratio 11.816 4 0.019

When the expected counts are small, your results may be misleading. For more information, see the Data considerations for Chi-Square Test for Association

DF

The degrees of freedom (DF) is the number of independent pieces of information on a statistic. The degrees of freedom for a table is (number of rows – 1), multiplied by (number of columns – 1).

Interpretation

Minitab uses the degrees of freedom to determine the p-value associated with the test statistic.

In these results, the degrees of freedom (DF) is 4.
Chi-Square Test Chi-Square DF P-Value Pearson 11.788 4 0.019 Likelihood Ratio 11.816 4 0.019

P-value

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 no association between two categorical variables exist.

Minitab uses the chi-square statistic to determine the p-value.

Note

Minitab does not display the p-value when any expected count is less than 1 because the results can be invalid.

Interpretation

To determine whether the variables are independent, 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 concluding that an association between the variables exists when there is no actual association.
P-value ≤ α: The variables have a statistically significant association (Reject H0)
If the p-value is less than or equal to the significance level, you reject the null hypothesis and conclude that there is a statistically significant association between the variables.
P-value > α: Cannot conclude that the variables are associated (Fail to reject H0)
If the p-value is larger than the significance level, you fail to reject the null hypothesis because there is not enough evidence to conclude that the variables are associated.
In these results, the p-value = 0.019. Because the p-value is less than α, you reject the null hypothesis. You can conclude that the variables are associated.
Chi-Square Test Chi-Square DF P-Value Pearson 11.788 4 0.019 Likelihood Ratio 11.816 4 0.019

Raw residuals

The raw residuals are the differences between observed counts and expected counts.
Observed count
The observed count is the actual number of observations in a sample that belong to a category.
Expected count

The expected count is the frequency that would be expected in a cell, on average, if the variables are independent. Minitab calculates the expected counts as the product of the row and column totals, divided by the total number of observations.

Interpretation

You can compare the observed values and the expected values in the output table.

In these results, the cell count is the first number in each cell, the expected count is the second number in each cell, and the raw residual is the third number in each cell. Machine 2, 2nd shift has the largest raw residual, which means that the greatest difference between expected and actual defects is found on Machine 2 during the 2nd shift.
Rows: Machine ID Columns: Worksheet columns 1st shift 2nd shift 3rd shift All 1 48 47 48 143 56.08 46.97 39.96 -8.078 0.034 8.044 2 76 47 32 155 60.78 50.91 43.31 15.216 -3.907 -11.309 3 36 40 34 110 43.14 36.13 30.74 -7.137 3.873 3.265 All 160 134 114 408 Cell Contents Count Expected count Residual
A better way to compare observed counts and expected counts is with the standardized residuals.

Standardized residuals

The standardized residuals are the raw residuals (or the difference between the observed counts and expected counts), divided by the square root of the expected counts.

Interpretation

You can compare the standardized residuals in the output table to see which category of variables have the largest difference between the expected counts and the actual counts relative to sample size, and seem to be dependent. For example, you can assess the standardized residuals in the output table to see the association between machine and shift for producing defects.

In these results, the cell count is the first number in each cell, the expected count is the second number in each cell, and the standardized residual is the third number in each cell. The positive standardized residuals indicate that there were more defective handles than expected. The negative standardized residuals indicate that there were less defective handles than expected.
Rows: Machine ID Columns: Worksheet columns 1st shift 2nd shift 3rd shift All 1 48 47 48 143 56.08 46.97 39.96 -1.0788 0.0050 1.2726 2 76 47 32 155 60.78 50.91 43.31 1.9516 -0.5476 -1.7184 3 36 40 34 110 43.14 36.13 30.74 -1.0867 0.6443 0.5889 All 160 134 114 408 Cell Contents Count Expected count Standardized residual

Adjusted residuals

The adjusted residuals are the raw residuals (or the difference between the observed counts and expected counts) divided by an estimate of the standard error. Use adjusted residuals to account for the variation due to the sample size.

Interpretation

You can compare the adjusted residuals in the output table to see which categories have the largest difference between the expected counts and the actual counts relative to sample size. For example, you can see which machine or shift has the largest difference between the expected number of defectives and the actual number of defectives.

In these results, the cell count is the first number in each cell, the expected count is the second number in each cell, and the adjusted residual is the third number in each cell. The positive adjusted residuals indicate that there were more defective handles than expected, adjusted for sample size. The negative adjusted residuals indicate that there were less defective handles than expected, adjusted for sample size.
Rows: Machine ID Columns: Worksheet columns 1st shift 2nd shift 3rd shift All 1 48 47 48 143 56.08 46.97 39.96 -1.7169 0.0076 1.8602 2 76 47 32 155 60.78 50.91 43.31 3.1788 -0.8485 -2.5707 3 36 40 34 110 43.14 36.13 30.74 -1.6309 0.9199 0.8117 All 160 134 114 408 Cell Contents Count Expected count Adjusted residual
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