Expected frequencies

Definition

Expected frequencies are the theoretical frequencies that we anticipate observing in a statistical test, based on a specific hypothesis and the sample size. They serve as a baseline for comparison in hypothesis testing, especially when evaluating categorical data. The calculation of expected frequencies is essential for conducting chi-square tests, allowing researchers to determine whether there are significant differences between observed and expected outcomes.

5 Must Know Facts For Your Next Test

1. Expected frequencies are calculated using the formula: $$E = \frac{(row\ total)(column\ total)}{grand\ total}$$, where E represents expected frequency.
2. In chi-square tests, the expected frequency must be at least 5 for valid results; if many expected frequencies are less than this, data may need to be combined.
3. The sum of all expected frequencies should equal the sum of all observed frequencies, ensuring that they represent the same dataset.
4. Expected frequencies help assess whether observed deviations from a distribution are due to random chance or suggest a significant pattern.
5. If the observed frequency significantly differs from the expected frequency, it can lead to rejecting the null hypothesis.

Review Questions

• How do you calculate expected frequencies for a chi-square test and why is this calculation important?
  • To calculate expected frequencies for a chi-square test, you use the formula: $$E = \frac{(row\ total)(column\ total)}{grand\ total}$$. This calculation is crucial because it provides a baseline against which we can compare observed frequencies. By establishing these expected values, we can determine if there are significant discrepancies that may indicate a relationship or difference between categorical variables. The comparison between observed and expected frequencies ultimately helps in making conclusions about the data.
• Discuss the implications of having expected frequencies less than 5 in chi-square tests and what adjustments may be necessary.
  • Having expected frequencies less than 5 can jeopardize the validity of a chi-square test's results. When many cells have low expected counts, it may violate the assumptions of the test, leading to unreliable conclusions. In such cases, researchers often combine categories or collect more data to ensure that all expected frequencies meet the minimum threshold. This adjustment helps maintain the integrity of the test and allows for a more accurate assessment of relationships between variables.
• Evaluate how expected frequencies contribute to hypothesis testing in chi-square analysis and their role in interpreting results.
  • Expected frequencies are foundational in hypothesis testing within chi-square analysis because they provide a reference point for assessing the fit of observed data to a specified model. By comparing observed frequencies to these expected values, researchers can identify if differences are statistically significant or likely due to chance. This evaluation plays a crucial role in interpreting results; significant deviations from expected frequencies suggest that the null hypothesis may not hold true. Thus, understanding expected frequencies is vital for drawing meaningful conclusions from categorical data.

Related terms

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

Null hypothesis: A statement that assumes no effect or no difference in the population, which researchers aim to test against using statistical methods.

Degrees of freedom: A value that describes the number of independent values or quantities which can be assigned to a statistical distribution; it influences the critical values in hypothesis testing.