Expected Frequencies

Definition

Expected frequencies refer to the theoretical frequency of occurrence of an event or outcome, based on a statistical model or hypothesis. In the context of Chi-square tests, expected frequencies are calculated to determine how well the observed data fits the expected distribution under the null hypothesis. They are crucial in assessing whether any observed differences among categorical variables are statistically significant.

5 Must Know Facts For Your Next Test

1. Expected frequencies are calculated by multiplying the total number of observations by the proportion expected under the null hypothesis for each category.
2. In Chi-square tests, the sum of all expected frequencies should equal the total number of observations in the data set.
3. When expected frequencies are too low (usually less than 5), it may violate assumptions of the Chi-square test, requiring alternative methods.
4. Expected frequencies play a key role in determining the Chi-square statistic, which is calculated using the formula: $$ ext{Chi-square} = \sum \frac{(O - E)^2}{E}$$, where O is observed frequency and E is expected frequency.
5. The comparison of observed and expected frequencies helps identify whether there is a significant deviation from what would be expected if the null hypothesis were true.

Review Questions

• How are expected frequencies calculated in a Chi-square test, and why are they important?
  • Expected frequencies are calculated by taking the total number of observations and multiplying it by the proportion that is expected under the null hypothesis for each category. They are important because they provide a benchmark against which observed frequencies can be compared. This comparison helps to determine whether any differences are statistically significant or could have occurred by chance.
• What challenges arise when expected frequencies are too low, and how can researchers address these issues in their analysis?
  • When expected frequencies are too low, typically below 5, it can invalidate the assumptions of the Chi-square test, leading to unreliable results. Researchers can address these issues by combining categories to increase expected frequencies or by using alternative statistical methods such as Fisher's Exact Test that do not have this limitation. This ensures that their analysis remains valid and meaningful.
• Evaluate the role of expected frequencies in determining whether observed differences between categorical variables indicate a real association or are simply due to random variation.
  • Expected frequencies play a critical role in evaluating whether observed differences between categorical variables reflect a genuine association or are simply due to random chance. By comparing observed data against calculated expected frequencies under the null hypothesis, researchers can assess the significance of their findings through Chi-square statistics. If significant deviations exist, this suggests that the variables may be associated rather than just coincidental, providing valuable insights into patterns within the data.

Related terms

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

Null Hypothesis:

A statement asserting that there is no effect or difference, which serves as the basis for statistical testing.

Degrees of Freedom: A parameter used in statistical tests that represents the number of independent values in a calculation, impacting the critical value needed for significance.