# Frequencies and chi-square distances for Simple Correspondence Analysis

Find definitions and interpretation for every statistic that is provided for frequencies and chi-square distances for simple correspondence analysis.

## Contingency Table

The contingency table tallies observations according to multiple categorical variables. The rows and columns in the table correspond to the categorical variables. The table includes marginal totals for each level of the variables.

The contingency table for simple correspondence analysis is a two-way table that tallies observations for two variables. You can also categorize observations for three or four variables by using the Combine sub-dialog box to cross the variables and create the rows and/or columns of a two-way table.

### Interpretation

Use the contingency table to view the observed frequency for each cell defined by a row category and column category. Use the column and row totals to see the total frequency for each category.

## Expected Frequencies

The expected frequency is the count of observations that is 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.

## Observed – Expected Frequencies

The observed − expected frequency is the difference between the count of actual observations in the cell and the count of observations in the cell that you expect if the variables are independent.

### Interpretation

Use the difference between the observed and expected frequencies to look for evidence of possible associations in the data. If two variables are associated, then the distribution of observations for one variable differs depending on the category of the second variable. As a result, the magnitude of the difference between the observed frequency and the expected frequency is relatively large. If the two variables are independent, then the distribution of observations for one variable is similar for all categories of the second variable. As a result, the magnitude of the difference between the observed frequency and the expected frequency is relatively small.

## Chi-Square Distances

Minitab displays each cell's contribution to the chi-square statistic as the chi-square distance. The chi-square distance for each cell 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 total chi-square is the sum of the values for all cells.

### Interpretation

You can compare the chi-square distances for each cell to assess which cells contribute most to the total chi-square. If the observed and expected cell frequencies differ greatly, the chi-squared value for the cell is larger. Therefore, a larger chi-square distance in a cell suggests a stronger association between the row and column categories than is expected by chance.

## Relative Inertias

Cell inertia is the chi-squared value in the cell divided by the total frequency for the contingency table. The sum of all the cell inertias is the total inertia, or simply the inertia. The relative inertia for a cell is the cell inertia divided by the total inertia. The relative inertia for a row is the sum of the cell inertias for the row divided by the total inertia. The relative inertia for a column is the sum of the cell inertias for the column divided by the total inertia.

### Interpretation

Use relative inertia to assess the strength of the associations between categories and contributions to variation in the data. Higher values generally indicate a stronger association and a greater proportion of the total variability from expected values in the data.

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