Expected Frequencies

Definition

Expected frequencies refer to the theoretical frequency distribution of events in a statistical experiment, assuming that the null hypothesis is true. They are calculated based on the total number of observations and the proportions predicted by the model being tested. Expected frequencies play a crucial role in various statistical tests, particularly those involving categorical data, where they help assess how well observed data fits with what is expected under certain hypotheses.

5 Must Know Facts For Your Next Test

1. Expected frequencies are essential for calculating the Chi-Square statistic in tests such as goodness-of-fit and independence tests.
2. They are computed by multiplying the total sample size by the probability of each category under the null hypothesis.
3. In contingency tables, expected frequencies are used to determine if there is a significant association between two categorical variables.
4. When expected frequencies are low (usually below 5), results may be unreliable, and alternative statistical methods may need to be considered.
5. Expected frequencies help researchers identify discrepancies between observed data and what is predicted, guiding conclusions about hypotheses.

Review Questions

• How are expected frequencies calculated and what role do they play in hypothesis testing?
  • Expected frequencies are calculated by multiplying the total number of observations by the probability of each category based on the null hypothesis. They serve as a benchmark to compare observed frequencies during hypothesis testing. By comparing these two sets of frequencies, researchers can determine if there is a statistically significant difference, which helps in deciding whether to reject or fail to reject the null hypothesis.
• Discuss how expected frequencies relate to Chi-Square tests, particularly in analyzing categorical data.
  • Expected frequencies are fundamental to Chi-Square tests, including goodness-of-fit and tests of independence. In these tests, researchers compare the observed frequencies of data to the expected frequencies under the null hypothesis. A significant difference suggests that the observed distribution is unlikely under the assumed model, indicating that there may be a relationship between variables or that the observed data does not fit the expected distribution.
• Evaluate how changes in expected frequencies impact conclusions drawn from statistical tests involving contingency tables.
  • Changes in expected frequencies can significantly impact conclusions drawn from statistical tests using contingency tables. If the expected frequencies increase due to a larger sample size or different probabilities, this could lead to a more accurate assessment of relationships between variables. Conversely, if expected frequencies drop below acceptable thresholds (e.g., less than 5), it may invalidate test results, leading researchers to reconsider their analysis methods or even their hypotheses, emphasizing the importance of appropriate sample sizes and theoretical distributions.

Related terms

Observed Frequencies: The actual counts or frequencies of occurrences in each category based on collected data.

Null Hypothesis: A statement that there is no effect or no difference, which is tested against an alternative hypothesis in statistical analysis.

Chi-Square Statistic: A measure used to assess how the observed frequencies differ from the expected frequencies, calculated as the sum of squared differences between observed and expected values divided by the expected values.