Expected Frequencies

Definition

Expected frequencies refer to the anticipated or predicted frequencies of observations in each category or cell of a contingency table, assuming a particular hypothesis or model is true. They are a crucial component in statistical tests that evaluate the goodness-of-fit of observed data to an expected distribution, as well as tests for homogeneity across different populations.

5 Must Know Facts For Your Next Test

1. Expected frequencies are calculated based on the assumption that the null hypothesis is true, which means the observed data follows a specific distribution or pattern.
2. In the Goodness-of-Fit test, expected frequencies are used to compare the observed frequencies in each category to the expected frequencies under the null hypothesis.
3. The Test for Homogeneity examines whether the distribution of a categorical variable is the same across different populations by comparing the expected frequencies in each cell of a contingency table.
4. The difference between observed and expected frequencies is a key component in calculating the test statistic for both the Goodness-of-Fit test and the Test for Homogeneity.
5. Expected frequencies are essential in determining whether the observed data provides sufficient evidence to reject the null hypothesis and conclude that the data does not fit the expected distribution or pattern.

Review Questions

• Explain the role of expected frequencies in the Goodness-of-Fit test.
  • In the Goodness-of-Fit test, expected frequencies represent the anticipated frequencies of observations in each category or cell, assuming the null hypothesis (that the data follows a specific distribution) is true. The test compares the observed frequencies to the expected frequencies to determine if the observed data provides sufficient evidence to reject the null hypothesis and conclude that the data does not fit the expected distribution.
• Describe how expected frequencies are used in the Test for Homogeneity.
  • The Test for Homogeneity examines whether the distribution of a categorical variable is the same across different populations or groups. Expected frequencies are calculated for each cell of the contingency table, based on the assumption that the distributions are homogeneous across the populations. The test then compares the observed frequencies to the expected frequencies to determine if there is sufficient evidence to reject the null hypothesis of homogeneity.
• Analyze the importance of accurately calculating expected frequencies in these statistical tests.
  • Accurately calculating expected frequencies is crucial for the validity and interpretation of both the Goodness-of-Fit test and the Test for Homogeneity. If the expected frequencies are not properly determined, the test statistics and p-values may be inaccurate, leading to incorrect conclusions about the fit of the data to the expected distribution or the homogeneity of the populations. Careful consideration of the assumptions and calculations underlying the expected frequencies is essential for drawing reliable statistical inferences from these tests.

Related terms

Contingency Table: A tabular arrangement of data where the frequencies or counts of observations are displayed in rows and columns, typically used to analyze the relationship between two categorical variables.

Goodness-of-Fit Test: A statistical test that evaluates whether the observed frequencies of a categorical variable match the expected frequencies under a hypothesized distribution or model.

Test for Homogeneity: A statistical test that determines whether the distribution of a categorical variable is the same across different populations or groups.