How To Calculate Expected Frequency From Observed Frequency

How to Calculate Expected Frequency from Observed Frequency: A Comprehensive Guide

Understanding how to calculate expected frequency from observed frequency is crucial in various statistical analyses, particularly in hypothesis testing, like the chi-squared test. This guide will walk you through the process, explaining the underlying concepts and providing practical examples to solidify your understanding. We'll cover different scenarios and address common challenges encountered during the calculation.

What is Observed Frequency?

Observed frequency refers to the number of times an event actually occurs in a sample. It's the raw data you collect during your experiment or observation. For example, if you're observing the number of heads and tails in 100 coin tosses, the observed frequency of heads might be 53 and tails 47. These are the actual counts from your experiment.

What is Expected Frequency?

Expected frequency, on the other hand, is the number of times you expect an event to occur based on a theoretical model or hypothesis. It's the frequency you'd predict if your hypothesis were true. In the coin toss example, if the coin is fair, your expected frequency for both heads and tails would be 50 (100 tosses * 0.5 probability of heads/tails). This is the predicted count based on your assumption of a fair coin.

The Importance of Expected Frequency

Calculating expected frequency is fundamental to several statistical tests, especially:

• Chi-square test: This test compares observed frequencies with expected frequencies to determine if there's a significant difference between them. A significant difference suggests that your observed data doesn't align with your expected values, potentially leading to the rejection of your null hypothesis.

• Goodness-of-fit tests: These tests assess how well a sample distribution fits a theoretical distribution. Expected frequencies are calculated based on the theoretical distribution, allowing for a comparison with the observed frequencies from your sample data.

• Contingency table analysis: When analyzing categorical data using contingency tables, expected frequencies are crucial in determining if there's an association between variables. The calculation of expected frequencies in this context involves marginal probabilities.

Calculating Expected Frequency: Different Scenarios

The method for calculating expected frequency depends on the context of your study. Here are several common scenarios:

Scenario 1: Simple Probability

This scenario is the most straightforward. You have a known probability for an event, and you want to calculate the expected frequency given a certain number of trials.

Formula:

Expected Frequency (E) = Probability (P) * Number of Trials (N)

Example:

Suppose you roll a six-sided die 60 times. What is the expected frequency of rolling a "3"?

• Probability (P) of rolling a 3 = 1/6
• Number of Trials (N) = 60
• Expected Frequency (E) = (1/6) * 60 = 10

Therefore, you'd expect to roll a "3" 10 times.

Scenario 2: Chi-Square Test with Contingency Tables

Contingency tables are used to analyze the relationship between two or more categorical variables. Calculating expected frequencies in this case involves a slightly more complex process.

Formula:

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

Example:

Let's say we're investigating the relationship between gender (male/female) and preference for coffee (like/dislike). Our observed data is:

To calculate the expected frequency for males who like coffee:

• Row Total (Males) = 50
• Column Total (Like Coffee) = 55
• Grand Total = 100
• Expected Frequency = (50 * 55) / 100 = 27.5

We repeat this calculation for each cell in the contingency table to obtain the full set of expected frequencies. Note that the expected frequencies may not be whole numbers.

Scenario 3: Goodness-of-Fit Test

In goodness-of-fit tests, you're comparing your observed distribution to a theoretical distribution (e.g., normal distribution, Poisson distribution).

The calculation of expected frequencies here depends heavily on the specific theoretical distribution. For instance, if you're testing if your data follows a normal distribution, you would use the parameters of the normal distribution (mean and standard deviation) to calculate the probability of each category and then multiply by the total number of observations. This process often requires statistical software or tables for accurate calculations.

For instance, if you hypothesize your data follows a Poisson distribution with a mean (λ) of 3, you would use the Poisson probability mass function to calculate the probability for each possible count (0, 1, 2, etc.) and then multiply those probabilities by the total number of observations to get your expected frequencies.

Checking Your Calculations

After calculating the expected frequencies, it's crucial to perform a few checks:

• Summation Check: The sum of the expected frequencies should always equal the sum of the observed frequencies (the grand total). If they don't, there's an error in your calculations.

• Reasonableness Check: The expected frequencies should be reasonably close to the observed frequencies, especially with a large sample size. If there's a substantial difference, it may warrant a closer examination of your data or your hypotheses. Consider whether there may be sampling bias, outliers, or whether your underlying assumptions are correct.

• Expected Frequency Rule: As a general rule of thumb, for the chi-squared test to be valid, each expected frequency should be at least 5. If an expected frequency is less than 5, you might need to combine categories or use a different statistical test. This is because the chi-squared approximation becomes less accurate with small expected frequencies.

Software for Calculating Expected Frequencies

While manual calculation is valuable for understanding the process, statistical software packages like SPSS, R, SAS, and Python (with libraries like SciPy and Statsmodels) significantly simplify the calculation of expected frequencies, particularly in more complex scenarios involving large datasets or intricate statistical tests. These tools not only calculate expected frequencies but also perform the entire hypothesis testing process, including calculating the chi-square statistic and p-value.

Conclusion

Calculating expected frequencies is a fundamental step in several important statistical analyses. Understanding the different methods for calculating expected frequencies, depending on your scenario, allows you to correctly interpret your results and make informed conclusions. Always remember to check your calculations for accuracy and ensure that the assumptions for your chosen statistical test are met. Mastering this skill is essential for anyone working with statistical data and conducting hypothesis tests. Remember to always focus on the underlying logic of the calculations and interpret the results in the context of your research question.