Expected frequencies

Definition

Expected frequencies are the theoretical frequencies of occurrences for each category in a contingency table, calculated under the assumption that there is no association between the variables being studied. They serve as a baseline for comparison against observed frequencies when determining if a significant relationship exists between the variables. Expected frequencies are crucial for conducting chi-square tests, as they help assess the degree of deviation between observed data and what would be expected if the null hypothesis were true.

5 Must Know Facts For Your Next Test

1. Expected frequencies are calculated by multiplying the total number of observations by the proportion of each category under the assumption of independence.
2. For a chi-square test to be valid, expected frequencies should generally be 5 or greater in each cell of the contingency table.
3. The formula for expected frequency in a two-way contingency table is: $$E = \frac{(row\ total) \times (column\ total)}{grand\ total}$$.
4. If expected frequencies are too low, it may invalidate the chi-square test results, requiring either data pooling or using alternative tests.
5. Expected frequencies help assess how much the observed counts deviate from what would be expected under the null hypothesis, providing insight into potential associations between variables.

Review Questions

• How do expected frequencies contribute to the validity of the chi-square test?
  • Expected frequencies provide a crucial comparison point for observed frequencies in a chi-square test. They are calculated based on the assumption that there is no association between variables. If expected frequencies are too low or not met according to the criteria of 5 or more per cell, it can invalidate the results of the test, indicating that another method may need to be used.
• What is the significance of calculating expected frequencies in relation to the null hypothesis?
  • Calculating expected frequencies is significant because it directly relates to testing the null hypothesis. The null hypothesis posits that there is no relationship between the categorical variables being analyzed. By comparing observed and expected frequencies, researchers can determine whether deviations from expectation are statistically significant, thus providing evidence to either reject or fail to reject the null hypothesis.
• Evaluate how variations in expected frequencies might indicate different strengths of relationships between categorical variables.
  • Variations in expected frequencies can highlight different strengths of relationships between categorical variables. For instance, if observed frequencies significantly differ from expected frequencies, this suggests a strong relationship or association exists. Conversely, minimal differences indicate a weak or no relationship. Analyzing these variations helps researchers understand not just whether an association exists but also how pronounced that association may be, influencing decisions on further investigation or application.

Related terms

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

Null Hypothesis: A statement that there is no effect or no relationship between two variables, which serves as the basis for statistical testing.

Contingency Table: A table used to display the frequency distribution of variables, showing the relationship between categorical data.