Expected Frequency

Definition

Expected frequency refers to the anticipated or predicted number of observations in each category or cell of a contingency table, assuming the null hypothesis is true. It is a crucial concept in various statistical tests, including the goodness-of-fit test, test of independence, and chi-square goodness-of-fit analysis.

5 Must Know Facts For Your Next Test

1. The expected frequency is calculated based on the assumption that the null hypothesis is true, and it represents the number of observations that would be expected in each cell if there were no relationship between the variables.
2. In a goodness-of-fit test, the expected frequencies are used to determine whether the observed frequencies in a sample differ significantly from the expected frequencies under a hypothesized distribution.
3. For a test of independence, the expected frequencies are used to determine whether the variables in a contingency table are independent or if there is a significant relationship between them.
4. The chi-square goodness-of-fit test compares the observed frequencies in each cell of a contingency table to the expected frequencies, and the resulting chi-square statistic is used to determine the p-value and assess the statistical significance.
5. Accurate calculation of expected frequencies is crucial for the proper interpretation of the results in these statistical tests, as it directly affects the conclusions drawn about the relationships or differences between variables.

Review Questions

• Explain the role of expected frequency in the context of a contingency table.
  • In a contingency table, the expected frequency represents the number of observations that would be expected in each cell if the null hypothesis of independence between the variables were true. The expected frequencies are calculated based on the row and column totals of the table, and they are used to determine whether the observed frequencies differ significantly from the expected frequencies. This comparison is the basis for statistical tests like the chi-square test of independence, which assess the strength of the relationship between the variables in the contingency table.
• Describe how expected frequency is used in a goodness-of-fit test.
  • In a goodness-of-fit test, the expected frequencies are used to determine whether the observed frequencies in a sample differ significantly from the expected frequencies under a hypothesized distribution. The test compares the observed frequencies to the expected frequencies, and the resulting chi-square statistic is used to assess the statistical significance of the difference. If the observed frequencies deviate significantly from the expected frequencies, the null hypothesis (that the data follows the hypothesized distribution) is rejected, indicating a poor fit between the data and the hypothesized distribution.
• Analyze the importance of accurately calculating expected frequencies in the context of a test of independence.
  • In a test of independence, the expected frequencies are crucial for determining whether the variables in a contingency table are independent or if there is a significant relationship between them. The test compares the observed frequencies in each cell to the expected frequencies, which are calculated under the assumption of independence. If the observed frequencies deviate significantly from the expected frequencies, the null hypothesis of independence is rejected, indicating a significant relationship between the variables. Accurate calculation of expected frequencies is essential for the proper interpretation of the test results and the conclusions drawn about the relationship between the variables.

Related terms

Contingency Table: A table that displays the frequency distribution of two or more categorical variables, allowing for the examination of the relationship between them.

Null Hypothesis: A statement that there is no significant difference or relationship between the variables being tested, which is assumed to be true until proven otherwise.

Chi-Square Statistic: A test statistic used to determine whether there is a significant difference between the observed and expected frequencies in one or more categories.