Expected frequencies

Definition

Expected frequencies are the theoretical frequencies that one would expect to observe in a statistical analysis, based on a specific hypothesis. They are calculated under the assumption that the null hypothesis is true and are essential for performing chi-square tests, which assess how well observed data fits an expected distribution or to determine independence between categorical variables.

5 Must Know Facts For Your Next Test

1. Expected frequencies are computed based on the proportion of each category in the population when using chi-square tests.
2. In goodness-of-fit tests, expected frequencies help determine how well a sample distribution matches a theoretical distribution.
3. For tests of independence, expected frequencies are calculated by multiplying the row total by the column total and dividing by the grand total.
4. Expected frequencies should ideally be 5 or more to ensure that the chi-square approximation is valid.
5. When observed frequencies deviate significantly from expected frequencies, it indicates that the null hypothesis may be rejected.

Review Questions

• How are expected frequencies calculated in the context of chi-square tests?
  • Expected frequencies are calculated differently depending on whether you are conducting a goodness-of-fit test or a test of independence. In a goodness-of-fit test, they are determined based on the total sample size multiplied by the proportion expected for each category under the null hypothesis. In a test of independence, expected frequencies for each cell in a contingency table are computed by taking the product of the corresponding row and column totals divided by the grand total.
• Discuss the importance of expected frequencies in evaluating the results of chi-square tests.
  • Expected frequencies are crucial for evaluating chi-square test results because they provide a baseline against which observed frequencies can be compared. A significant difference between observed and expected frequencies suggests that the observed data does not fit well with what was expected under the null hypothesis. This can lead to rejecting the null hypothesis and concluding that there is an effect or association present in the data. Therefore, accurate calculation of expected frequencies is essential for valid statistical inference.
• Evaluate how changing expected frequencies affects the conclusions drawn from chi-square tests.
  • Changing expected frequencies can significantly impact the outcomes and interpretations of chi-square tests. If expected frequencies are too low, it may violate assumptions required for valid results, leading to unreliable conclusions about relationships or distributions. Conversely, if they are calculated correctly but then altered inappropriately during testing, it could result in misleading significance levels and erroneous rejections or acceptances of null hypotheses. Understanding how these adjustments affect overall analysis is key to ensuring robust and accurate statistical interpretations.

Related terms

Null Hypothesis: A statement that there is no effect or no difference, which serves as a starting point for statistical testing.

Chi-Square Statistic: A measure used in chi-square tests that calculates the difference between observed and expected frequencies to determine if a significant relationship exists.

Degrees of Freedom: A concept used in statistical tests that refers to the number of values in a calculation that are free to vary, impacting the shape of the chi-square distribution.