Calculating expected frequencies for chi-squared tests is pretty easy once you understand the steps. Let’s break it down for two types of tests: goodness-of-fit tests and tests for independence in contingency tables.

For Goodness-of-Fit Tests:

1. Find the Total Observations: First, count how many total observations, or items, you have. Let's call this number $N$.

2. Know the Expected Ratio: You need to know what you expect for each category. For example, if you are testing a die, you would expect each number (1 through 6) to show up about 1 out of 6 times.

3. Calculate Expected Frequencies: To find the expected frequency for each category, multiply the total number of observations ($N$) by the expected proportion for that category.

   For example, the formula is: $E_i = N \times p_i$ Here, $E_i$ is the expected frequency for category $i$, and $p_i$ is the expected proportion.

For Chi-Squared Tests for Independence:

1. Make a Contingency Table: Start by putting your data into a two-way table where you can see two different groups.

2. Add Up Rows and Columns: Calculate the total for each row ($R_j$) and each column ($C_i$).

3. Calculate Expected Frequencies: For each box in the table, find the expected frequency using this formula: $E_{ij} = \frac{R_j \times C_i}{N}$ Here, $E_{ij}$ is the expected frequency for the box located in row $i$ and column $j$.

By following these simple steps, you can find the expected frequencies that are very important for your chi-squared analysis!