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Expected counts in Chi-Square tests are calculated by multiplying the total number of observations by the proportion of each category in the sample. This is done for each category in the contingency table. The expected count represents the number of observations that would be expected in each cell of the table if there was no association between the variables being tested. These expected counts are then compared to the observed counts in the table to determine the degree of association between the variables and ultimately, the significance of the Chi-Square test.
In statistics, there are two common types of Chi-Square tests in which you will have to calculate expected counts:
1. – Used to determine whether or not a categorical variable follows a hypothesized distribution.
2. – Used to determine whether or not there is a significant association between two categorical variables.
The following examples show how to calculate expected counts for each of these tests.
Example 1: Expected Counts for Chi-Square Goodness of Fit Test
Suppose a store owner claims that an equal number of customers come into his shop each weekday.
To test this hypothesis, he records the number of customers that come into the shop on a given week and finds the following:
To find the expected count of customers each day, we can use the following formula:
Expected count = Expected percentage * Total count
Recall that the shop owner expects an equal amount of customers to come into the shop each day. Thus, the expected percentage of customers that come in on a given day is 20% of the total customers for the week.
This means we can calculate the expected frequency of customers each day as:
Expected count = 20% * 250 total customers = 50
Once we have the expected counts, we can proceed to calculate the Chi-Square test statistic and the corresponding p-value to determine if the shop owner’s claim is likely to be true.
Note: explains how to perform this exact Chi-Square Goodness of Fit test in Excel.
Example 2: Expected Counts for Chi-Square Test of Independence
We take a simple random sample of 500 voters and survey them on their political party preference. The following table shows the results of the survey:
To calculate the expected count of each cell in the table, we can use the following formula:
Expected count = (row sum * column sum) / table sum
For example, the expected value for Male Republicans is: (230*250) / 500 = 115.
We can repeat this formula to obtain the expected value for each cell in the table:
Once we have the expected counts, we can proceed to calculate the Chi-Square test statistic and the corresponding p-value to determine if there is a statistically significant association between gender and political party preference.
Note: explains how to perform this exact Chi-Square Test of Independence in Excel.
Additional Resources
The following resources provide additional information about Chi-Square tests: