Contingency Tables > Expected Frequency

## What is Expected Frequency?

The expected frequency is a probability count that appears in contingency table calculations including the chi-square test. Expected frequencies also used to calculate standardized residuals, where the expected count is subtracted from the observed count in the numerator.

**Observed Frequencies**are counts made from experimental data. In other words, you actually observe the data happening and take measurements. For example, you roll a die ten times and then count how many times each number is rolled. The count is made*after*the experiment.**Expected Frequencies**are counts calculated using probability theory. For example,*before*you roll a six-sided die, you calculate the probability of any one number being rolled as 1/6.

## How to Calculate Expected Frequency by Hand

Expected frequencies are calculated for *each cell* in a contingency table. So if you have, say, 16 cells, you’ll need to perform the steps 16 times (one for each cell). The formula to calculate expected frequency is:

_{ij}= expected frequency for the

*i*th row/

*j*th columm.

_{i}= total in the ith row

_{j}= total in the jth column

**Tip**: You can think of this equation more simply as *(row total * column total) / grand total.*

**Sample question: **What are the expected cell frequencies for the following table?

Step 1: Find T_{i} = total in the ith row. The first cell is in the first row (i=1), which has a total of 114.

Set this number aside for a moment.

Step 2: Find T_{j} = total in the jth column. The first cell is in the first column (j=1), which has a total of 102.

Set this number aside for a moment.

Step 3: Find N, or the **total number of participants/items** in the experiment. In a contingency table, this is usually done for you and is tallied up in the bottom right-hand corner. For this example, the total number of participants is 173.

Step 4: Insert the three numbers from Step 1, 2 , and 3 into the formula and solve:

**The expected cell frequency for cell 1 is 67.214.**

Step 5: Repeat Steps 1 through 4 for each of the other cells.

The solutions for the remaining cells are:

- Cell 2 (top right) = (114 * 71) / /173 = 48.786

- Cell 3 (bottom left) = (59 * 102) / 173 = 34.786
- Cell 4 (bottom right) = (59 * 71) / 173 = 24.214.

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