Expected contingency frequency
If we have a contingency table, we can use it to find the expected frequency of each cell in the table, which gives what we would expect if the two variables were independent of each other.
Calculating the expected frequency
Knowing the row total, column total, and grand total (the total of all the frequencies in the table), we can calculate the expected frequency for each cell using this formula:
\text{Expected frequency} = \frac{\text{Row total} \times \text{Column total}}{\text{Grand total}}
Sometimes, we refer to the expected frequency as
Example
Suppose we have the following contingency table showing the number of people who like different types of music, split by their age group:
| Age Group | Pop | Instrumental |
|---|---|---|
| Under 18 | 30 | 10 |
| 18-35 | 50 | 20 |
| 35+ | 20 | 30 |
Let’s first add the row and column totals:
| Age Group | Pop | Instrumental | Total |
|---|---|---|---|
| Under 18 | 30 | 10 | 40 |
| 18-35 | 50 | 20 | 70 |
| 35+ | 20 | 30 | 50 |
| Total | 100 | 60 | 160 |
Now, let’s calculate the expected frequency for the cell corresponding to “Under 18” and “Pop”:
- Row total for under 18:
40 - Column total for pop:
100 - Grand total:
160 - Expected frequency:
\frac{40 \times 100}{160} = 25
We can do that for the entire table, to get:
| Age Group | Pop | Instrumental | Total |
|---|---|---|---|
| Under 18 | 25 | 15 | 40 |
| 18-35 | 43.75 | 26.25 | 70 |
| 35+ | 31.25 | 18.75 | 50 |
| Total | 100 | 60 | 160 |
flashcards
| Question | Answer |
|---|---|
| What is the formula for calculating the expected frequency of a cell in a contingency table? | |
| What is the symbol sometimes used to refer to the expected frequency? | |
| In a contingency table, what would the expected frequency represent if the two variables were independent? | The value we would expect in each cell if the variables were independent of each other. |
| In the example table, what is the expected frequency for “Under 18” and “Pop”? | |
| In the example, what is the grand total for the contingency table? | 160 |
| What is the row total for the “18-35” age group in the example? | 70 |
| What is the column total for “Instrumental” in the example table? | 60 |
| What is the expected frequency for the “35+” and “Instrumental” cell in the example? | |
| In the example table, what is the total for the “Pop” column? | 100 |
| What is the expected frequency for the “Under 18” and “Instrumental” cell? |