Expected Frequency

Definition

Expected frequency refers to the anticipated count of occurrences in each category of a categorical variable, calculated based on the assumption of independence or a specific distribution. It plays a crucial role in statistical tests, particularly when determining how well observed data fits a given model. By comparing observed frequencies to expected frequencies, researchers can assess whether any significant discrepancies exist, which helps in making conclusions about the underlying data relationships.

5 Must Know Facts For Your Next Test

1. Expected frequencies are calculated under the null hypothesis, which typically assumes that there is no association between the variables being studied.
2. In a chi-square test for independence, expected frequencies are derived from the row and column totals of a contingency table.
3. To ensure validity, each expected frequency should ideally be 5 or greater; otherwise, it may lead to inaccurate results in the chi-square test.
4. Expected frequency calculations are crucial in goodness-of-fit tests, where they help determine how well the observed data conforms to a specific theoretical distribution.
5. The difference between observed and expected frequencies can indicate whether a relationship exists between the variables or if any variation is due to random chance.

Review Questions

• How do you calculate expected frequencies in a chi-square test for independence?
  • Expected frequencies in a chi-square test for independence are calculated using the formula: $$E = \frac{(Row \ Total) \times (Column \ Total)}{Grand \ Total}$$. This means for each cell in the contingency table, you multiply the total count of its row by the total count of its column and then divide by the overall total. This calculation allows us to assess how many counts we would expect if there were no association between the variables.
• Discuss why it's important for expected frequencies to be at least 5 in chi-square tests.
  • Having expected frequencies of at least 5 is important because it ensures that the chi-square approximation to the sampling distribution is valid. When expected frequencies are too low, the test may not accurately reflect the true underlying distribution, leading to unreliable results. Low expected counts can cause the chi-square statistic to be skewed, making it harder to determine if any observed differences are statistically significant or just due to random variation.
• Evaluate how discrepancies between observed and expected frequencies can inform researchers about variable relationships.
  • Discrepancies between observed and expected frequencies provide critical insights into potential relationships between variables. If the observed frequency significantly deviates from what is expected under the null hypothesis, it suggests that an association may exist that warrants further investigation. By analyzing these differences, researchers can form hypotheses about underlying mechanisms driving these relationships and ultimately contribute to a deeper understanding of their data.

Related terms

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

Observed Frequency: The actual count of occurrences in each category of a variable, measured from the collected data.

Goodness-of-Fit: A statistical test that determines how well a theoretical distribution fits the observed data by comparing observed and expected frequencies.