Expected Frequency

Definition

The expected frequency is the anticipated or predicted frequency of an outcome in a statistical analysis, particularly in the context of contingency tables, goodness-of-fit tests, and tests of independence. It represents the expected number of observations in a particular cell or category under the null hypothesis.

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 is used to compare with the observed frequencies in a statistical test.
2. In a contingency table, the expected frequency for each cell is calculated by multiplying the row total and column total for that cell, and then dividing by the grand total.
3. The goodness-of-fit test and the test of independence both use the expected frequency to determine if the observed data fits the expected distribution or if there is a significant relationship between the variables.
4. The comparison of the chi-square tests, such as the goodness-of-fit test and the test of independence, involves comparing the test statistic, which is based on the difference between the observed and expected frequencies.
5. In the chi-square goodness-of-fit lab, the expected frequency is used to calculate the test statistic and determine if the observed data fits the expected distribution.

Review Questions

• Explain how the expected frequency is calculated in a contingency table and why it is an important component of the analysis.
  • In a contingency table, the expected frequency for each cell is calculated by multiplying the row total and column total for that cell, and then dividing by the grand total. This expected frequency represents the anticipated number of observations in each cell under the assumption that the null hypothesis is true, which is the hypothesis of no significant difference or relationship between the variables. The expected frequency is a crucial component of the analysis because it is used to compare with the observed frequencies and determine if there is a statistically significant difference, as in the case of a chi-square test of independence or goodness-of-fit test.
• Describe the role of the expected frequency in the goodness-of-fit test and the test of independence, and how it is used to compare the observed and expected data.
  • The expected frequency is a central component of both the goodness-of-fit test and the test of independence. In the goodness-of-fit test, the expected frequency represents the anticipated frequency of observations in each category under the null hypothesis that the data follows a specific distribution. The test statistic is calculated based on the differences between the observed and expected frequencies, and this statistic is then used to determine if the observed data significantly deviates from the expected distribution. Similarly, in the test of independence, the expected frequency represents the anticipated frequency of observations in each cell of the contingency table under the null hypothesis of no association between the variables. The test statistic is again calculated based on the differences between the observed and expected frequencies, and this is used to assess the statistical significance of the relationship between the variables.
• Analyze the importance of the expected frequency in the comparison of the chi-square tests, such as the goodness-of-fit test and the test of independence, and explain how it contributes to the interpretation of the results.
  • The expected frequency is a crucial component in the comparison of the chi-square tests, such as the goodness-of-fit test and the test of independence. In both tests, the expected frequency is used to calculate the test statistic, which is then compared to a critical value to determine the statistical significance of the results. The difference between the observed and expected frequencies is what drives the test statistic, and a larger difference indicates a greater deviation from the null hypothesis. By comparing the test statistic to the critical value, the researcher can determine the probability of observing the given data under the null hypothesis, and make a decision about whether to reject or fail to reject the null hypothesis. The expected frequency, therefore, plays a central role in the interpretation of the results, as it provides the benchmark against which the observed data is evaluated.

Related terms

Contingency Table: A table that displays the frequencies or counts of observations that fall into different categories defined by two or more variables.

Null Hypothesis: The statistical hypothesis that there is no significant difference or relationship between the variables being tested.

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