Expected frequency

Definition

Expected frequency refers to the anticipated number of occurrences of an event in a statistical experiment, based on the total number of observations and the probability of that event. It is a crucial concept in analyzing data, particularly in relation to contingency tables and bar charts, where it helps to determine how closely the observed frequencies align with what would be expected if there were no association between variables.

5 Must Know Facts For Your Next Test

1. Expected frequency is calculated by multiplying the total number of observations by the probability of an event occurring.
2. In a contingency table, expected frequencies are used to compare with observed frequencies to assess whether there is a significant association between two categorical variables.
3. For any cell in a contingency table, the expected frequency can be calculated using the formula: $$E_{ij} = \frac{(Row \ total) \times (Column \ total)}{Grand \ total}$$.
4. When conducting a Chi-square test, each expected frequency must be at least 5 to ensure the validity of the test results.
5. If the observed frequencies significantly deviate from the expected frequencies, it suggests that there may be an association between the variables being studied.

Review Questions

• How is expected frequency calculated and why is it important when analyzing data in contingency tables?
  • Expected frequency is calculated by multiplying the total number of observations by the probability of an event. It is important in analyzing data because it provides a baseline for comparison against observed frequencies. By using expected frequencies in contingency tables, researchers can assess whether there is an association between categorical variables, which can lead to significant insights about relationships in the data.
• Discuss how expected frequencies contribute to the results of a Chi-square test and what assumptions must be met.
  • Expected frequencies are central to the Chi-square test as they serve as the theoretical benchmark for comparison against observed frequencies. For accurate results, it's crucial that each expected frequency is at least 5; this ensures that the Chi-square approximation holds true. If this condition isn't met, the results may not be reliable, leading to incorrect conclusions about associations between variables.
• Evaluate the implications of finding that observed frequencies significantly differ from expected frequencies in a contingency table analysis.
  • Finding that observed frequencies significantly differ from expected frequencies implies that there may be an association or relationship between the variables being analyzed. This deviation suggests that one variable may influence or depend on another, prompting further investigation into causal relationships. Such findings can have important implications for decision-making and understanding underlying patterns in data, shaping future research directions and strategies.

Related terms

Observed frequency: The actual count of occurrences of an event or category in a dataset.

Chi-square test: A statistical test used to determine whether there is a significant association between categorical variables by comparing observed and expected frequencies.

Marginal distribution: The distribution of values for one variable in a contingency table, found by summing the counts across rows or columns.