Expected Frequencies

Definition

Expected frequencies refer to the anticipated count of observations in each category or cell of a contingency table, based on the assumption that there is no association between the variables being studied. They are calculated under the null hypothesis, which states that any observed differences in frequencies are due to random chance. Understanding expected frequencies is crucial for conducting chi-square tests, as they serve as a baseline for comparison against observed frequencies.

5 Must Know Facts For Your Next Test

1. Expected frequencies are calculated by multiplying the row total by the column total and then dividing by the overall total for each cell in the contingency table.
2. In chi-square tests, if the observed frequencies significantly differ from the expected frequencies, it indicates that there may be an association between the variables.
3. For valid chi-square tests, each expected frequency should ideally be 5 or more to ensure the approximation to the chi-square distribution is appropriate.
4. If any expected frequency is less than 5, it's recommended to combine categories or use an alternative statistical method.
5. The formula for calculating expected frequencies in a two-way table is: $$E = \frac{(Row\ Total) \times (Column\ Total)}{Grand\ Total}$$

Review Questions

• How do expected frequencies relate to observed frequencies in the context of a chi-square test?
  • Expected frequencies are crucial in a chi-square test because they provide a benchmark against which observed frequencies can be compared. The test assesses whether the differences between observed and expected frequencies are significant, indicating a potential relationship between the categorical variables. A larger difference suggests that the variables may be associated rather than independent.
• Discuss how one would determine if the expected frequencies are appropriate for conducting a chi-square test.
  • To determine if expected frequencies are appropriate for a chi-square test, one must check that each expected frequency is at least 5. If some expected frequencies fall below this threshold, adjustments should be made, such as combining categories or applying different statistical techniques. This ensures that the chi-square approximation holds true and the test results remain valid.
• Evaluate the implications of using expected frequencies in hypothesis testing and what it means for interpreting results.
  • Using expected frequencies in hypothesis testing allows researchers to quantify and understand patterns in categorical data. If observed frequencies significantly diverge from expected frequencies, it challenges the null hypothesis and suggests a potential relationship between variables. Proper interpretation requires careful consideration of how well expected frequencies meet assumptions, as failing to do so could lead to misleading conclusions about variable associations and the overall validity of study results.

Related terms

Chi-square Test: A statistical test used to determine whether there is a significant association between categorical variables by comparing observed and expected frequencies.

Null Hypothesis: A statement suggesting that there is no effect or association between variables, often tested using statistical methods like the chi-square test.

Contingency Table:

A tabular representation that displays the frequency distribution of variables and is used to calculate expected frequencies in chi-square tests.