How To Compute Expected Frequency

How to Compute Expected Frequency: A Comprehensive Guide

Understanding expected frequency is crucial in various statistical analyses, particularly in hypothesis testing using the chi-squared test. This comprehensive guide will walk you through the process of computing expected frequencies, explaining the underlying concepts and providing practical examples. We'll cover different scenarios, address common misconceptions, and equip you with the knowledge to confidently tackle expected frequency calculations in your own work. By the end, you'll not only know how to calculate expected frequencies but also why they are essential for statistical inference.

Introduction to Expected Frequency

In statistics, expected frequency represents the theoretical frequency of an event assuming a specific hypothesis is true. It's the number of times you would expect to observe a particular outcome if your null hypothesis (the hypothesis of no effect or no difference) were correct. This contrasts with the observed frequency, which is the actual number of times an outcome occurred in your sample data. The difference between observed and expected frequencies is crucial for determining whether your data supports or refutes your null hypothesis. We often use the chi-squared test to statistically compare these frequencies.

The calculation of expected frequency depends on the type of statistical test and the specific research question. This guide focuses on calculating expected frequencies primarily within the context of the chi-squared test for independence and goodness-of-fit.

Calculating Expected Frequency: Chi-Squared Test of Independence

The chi-squared test of independence assesses whether two categorical variables are associated. To perform this test, you need to calculate the expected frequencies for each cell in your contingency table. A contingency table displays the frequencies of observations for each combination of categories across two or more variables.

Steps to Calculate Expected Frequency for a Chi-Squared Test of Independence:

1. Create a Contingency Table: Organize your data into a contingency table. This table shows the observed frequencies for each combination of categories.

2. Calculate Row and Column Totals: Sum the observed frequencies for each row and each column.

3. Calculate the Grand Total: Sum all the observed frequencies in the table. This represents the total number of observations in your dataset.

4. Calculate Expected Frequency for Each Cell: For each cell in the contingency table, the expected frequency is calculated using the following formula:

   Expected Frequency (E) = (Row Total * Column Total) / Grand Total

   This formula essentially estimates the frequency you'd expect in each cell if the two variables were independent.

Example:

Let's say we're investigating the relationship between gender and preference for coffee or tea. We collect data from 100 people:

Let's calculate the expected frequency for the cell representing "Male" and "Coffee":

• Row Total (Male) = 50
• Column Total (Coffee) = 55
• Grand Total = 100

Expected Frequency (Male, Coffee) = (50 * 55) / 100 = 27.5

We repeat this calculation for each cell:

Now we have both the observed and expected frequencies, which we can use to perform the chi-squared test.

Calculating Expected Frequency: Chi-Squared Goodness-of-Fit Test

The chi-squared goodness-of-fit test assesses whether a sample distribution matches a hypothesized distribution. Here, the expected frequencies are derived from the hypothesized distribution.

Steps to Calculate Expected Frequency for a Chi-Squared Goodness-of-Fit Test:

1. Define the Hypothesized Distribution: This could be a uniform distribution (equal probabilities for all categories), a normal distribution, or any other distribution specified by your research question.

2. Determine Expected Probabilities: Based on your hypothesized distribution, determine the probability of observing each category.

3. Calculate Expected Frequencies: Multiply each expected probability by the total number of observations in your sample.

Example:

Suppose we roll a six-sided die 60 times. Under the null hypothesis that the die is fair, we expect each face to appear with equal probability (1/6).

• Total observations = 60
• Expected probability for each face = 1/6

Expected frequency for each face = (1/6) * 60 = 10

Therefore, we expect each face (1, 2, 3, 4, 5, 6) to appear 10 times. We would then compare these expected frequencies to the observed frequencies from our 60 rolls to see if the die is indeed fair.

Important Considerations and Potential Pitfalls

• Sample Size: The chi-squared test is generally considered reliable when the expected frequency in each cell is at least 5. If expected frequencies are less than 5, consider combining categories or collecting more data. Fisher's exact test might be a more appropriate alternative in such cases.

• Independence of Observations: The chi-squared test assumes that the observations are independent. If your data violates this assumption, the results of the test may be unreliable.

• Interpretation: A significant chi-squared test indicates that there is a statistically significant difference between the observed and expected frequencies. This does not necessarily imply a strong association or a large effect size; it simply suggests that the observed differences are unlikely to have occurred by chance alone. Always consider the context of your research and the magnitude of the differences.

Frequently Asked Questions (FAQ)

Q: What happens if the expected frequency is not a whole number?

A: It's perfectly acceptable for expected frequencies to be decimals. The formula calculates the theoretical frequency, and it doesn't have to be a whole number.

Q: Can I use expected frequency with other statistical tests besides the chi-squared test?

A: While the concept of expected frequency is most prominently used in chi-squared tests, the underlying principle of comparing observed and expected values applies to other statistical methods, such as binomial and Poisson tests, though the calculation of expected frequencies will differ.

Q: What if my expected frequencies are all very close to my observed frequencies?

A: This would result in a non-significant chi-squared test, suggesting that your data is consistent with the null hypothesis (no significant difference or association).

Q: What are some common reasons for large discrepancies between observed and expected frequencies?

A: Discrepancies might stem from a flawed null hypothesis, sampling error, a real effect or association between the variables, or violations of the assumptions underlying the statistical test.

Conclusion

Calculating expected frequencies is a fundamental step in several statistical analyses, most notably the chi-squared test. Understanding how to compute expected frequencies, interpreting the results, and being aware of potential pitfalls are essential for conducting robust and meaningful statistical analyses. Remember to always check your assumptions, consider your sample size, and interpret the results within the broader context of your research question. By mastering these techniques, you enhance your ability to draw accurate and informative conclusions from your data. This guide has provided you with a solid foundation for working confidently with expected frequencies in your statistical endeavors. Remember to practice with different examples to solidify your understanding.