An expected frequency is a theoretical frequency that we expect to occur in an experiment.
This type of frequency occurs most often in two types of Chi-Square tests:
This tutorial explains how to calculate expected frequencies for each of these two tests.
Expected Frequency in a Chi-Square Goodness of Fit Test
A Chi-Square goodness of fit test is used to determine whether or not a categorical variable follows a hypothesized distribution. With this type of test, we compare the observed frequencies of a categorical variable with the expected frequencies.
For example, suppose a shop owner claims that an equal number of customers come into his shop each weekday. To test this hypothesis, an independent researcher records the number of customers that come into the shop on a given week and finds the following:
To calculate the expected frequency of customers each day, we can use the following formula:
Expected frequency = Expected percentage * Total count
For this particular example, 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 frequency = 20% * 250 total customers = 50
Expected Frequency in a Chi-Square Goodness Test of Independence
A Chi-Square Test of Independence is used to determine whether or not there is a significant association between two categorical variables. With this type of test, we also compare a set of observed frequencies with a set of expected frequencies.
For example, suppose we want to know whether or not gender is associated with political party preference. 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 frequency of each cell in the table, we can use the following formula:
Expected frequency = (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:
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