How To Find The Expected Frequency

Let's dive into the concept of expected frequency, a cornerstone of statistical analysis, particularly when dealing with categorical data. Understanding how to calculate expected frequencies is crucial for conducting chi-square tests and making informed decisions based on observed data.

Understanding Expected Frequency

Expected frequency represents the theoretical frequency of a particular outcome in a sample, assuming a certain hypothesis is true. It is the frequency we expect to see if there's no association between the variables being studied. In simpler terms, it's a benchmark we use to compare against the observed frequencies, which are the actual frequencies we collect from our data. The difference between expected and observed frequencies helps us determine if any observed patterns are statistically significant or simply due to random chance.

Think of it like this: you flip a fair coin 100 times. You expect to see heads approximately 50 times and tails approximately 50 times. These are your expected frequencies. However, you might actually observe 55 heads and 45 tails. These are your observed frequencies. The question then becomes: is the difference between 50/50 (expected) and 55/45 (observed) large enough to suggest the coin is not fair? Calculating expected frequencies is the first step in answering questions like this.

The Importance of Expected Frequency

Why bother calculating expected frequencies? They serve a vital role in statistical hypothesis testing, specifically with the chi-square test. The chi-square test assesses the independence of two categorical variables or the goodness-of-fit of a theoretical distribution to observed data. Here’s why expected frequencies are so important:

• Providing a Baseline: Expected frequencies provide a theoretical baseline against which to compare observed frequencies. Without this baseline, it's impossible to determine if the observed frequencies deviate significantly from what would be expected by chance alone.
• Calculating the Chi-Square Statistic: The chi-square statistic, the core of the chi-square test, is calculated directly from the differences between observed and expected frequencies. The larger the difference between these frequencies, the larger the chi-square statistic, and the stronger the evidence against the null hypothesis (which usually assumes independence or no difference).
• Validating Statistical Tests: The chi-square test relies on certain assumptions, one of which is that the expected frequencies are sufficiently large (usually, a minimum expected frequency of 5 in each cell of the contingency table). If expected frequencies are too small, the chi-square approximation becomes unreliable, and alternative tests (like Fisher's exact test) may be more appropriate.

Calculating Expected Frequency: The Formulas and Examples

There are two main scenarios where you'll need to calculate expected frequencies:

1. Chi-Square Test for Independence: This test examines whether two categorical variables are independent of each other. For example, is there a relationship between smoking status and lung cancer?
2. Chi-Square Goodness-of-Fit Test: This test examines whether an observed frequency distribution matches an expected distribution. For example, does the distribution of M&M colors in a bag match the manufacturer's claimed distribution?

Let's examine the formulas and examples for each of these scenarios:

1. Chi-Square Test for Independence

In a chi-square test for independence, you're working with a contingency table. A contingency table is a table that summarizes the frequency distribution of two or more categorical variables.

Formula:

The expected frequency for each cell in the contingency table is calculated as follows:

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

Where:

• Row Total is the sum of all frequencies in the row containing the cell.
• Column Total is the sum of all frequencies in the column containing the cell.
• Grand Total is the total number of observations in the entire table.

Example:

Let's investigate whether there's a relationship between exercise habits and weight status. We collect data from 200 individuals and categorize them into two groups: "Regular Exercise" and "No Regular Exercise." We also categorize their weight status as "Healthy Weight" and "Overweight." The observed frequencies are shown in the contingency table below:

Now, let's calculate the expected frequencies for each cell:

• Cell 1: Regular Exercise, Healthy Weight

  • E = (Row Total * Column Total) / Grand Total
  • E = (80 * 90) / 200
  • E = 36

• Cell 2: Regular Exercise, Overweight

  • E = (80 * 110) / 200
  • E = 44

• Cell 3: No Regular Exercise, Healthy Weight

  • E = (120 * 90) / 200
  • E = 54

• Cell 4: No Regular Exercise, Overweight

  • E = (120 * 110) / 200
  • E = 66

Here's the contingency table with both observed (O) and expected (E) frequencies:

Interpretation:

The expected frequency of 36 for the "Regular Exercise, Healthy Weight" cell means that if exercise habits and weight status were independent, we would expect to see 36 individuals in this category. The observed frequency is 60, suggesting that more people than expected who exercise regularly have a healthy weight. The chi-square test will quantify whether these differences are statistically significant.

2. Chi-Square Goodness-of-Fit Test

In a chi-square goodness-of-fit test, you're comparing an observed distribution to a theoretically expected distribution. This theoretical distribution can be based on prior knowledge, a mathematical model, or a hypothesis.

Formula:

The calculation of expected frequencies depends on the nature of the theoretical distribution. Here are a few common scenarios:

• Equal Proportions: If you expect each category to have an equal proportion, then:

  Expected Frequency (E) = Total Number of Observations / Number of Categories
  

• Specific Proportions: If you expect each category to have a specific proportion based on a theory or prior knowledge, then:

  Expected Frequency (E) = Total Number of Observations * Expected Proportion
  

Example 1: Equal Proportions

A researcher wants to determine if a six-sided die is fair. They roll the die 600 times and record the number of times each face appears. The observed frequencies are:

Since a fair die should have an equal probability of landing on each face, the expected frequency for each face is:

• E = Total Number of Observations / Number of Categories
• E = 600 / 6
• E = 100

Therefore, the expected frequency for each face is 100.

Example 2: Specific Proportions

A candy company claims that its bags of mixed candies contain the following proportions: 30% red, 20% blue, 20% green, 15% yellow, and 15% orange. A consumer buys a bag of 200 candies and counts the number of each color. The observed frequencies are:

To calculate the expected frequencies, we multiply the total number of candies by the expected proportion for each color:

• Red: E = 200 * 0.30 = 60
• Blue: E = 200 * 0.20 = 40
• Green: E = 200 * 0.20 = 40
• Yellow: E = 200 * 0.15 = 30
• Orange: E = 200 * 0.15 = 30

Here's a table summarizing the observed and expected frequencies:

Interpretation:

The expected frequency of 60 for red candies means that if the company's claim is accurate, we would expect to see 60 red candies in a bag of 200. The observed frequency is 50, suggesting there might be fewer red candies than claimed. Again, the chi-square test will determine if this difference is statistically significant.

Common Mistakes and Pitfalls

Calculating expected frequencies is generally straightforward, but here are some common mistakes to avoid:

• Incorrectly Calculating Totals: Ensure that you are calculating row totals, column totals, and grand totals accurately. A mistake in these basic calculations will propagate through the entire process.
• Applying the Wrong Formula: Use the correct formula depending on whether you are performing a test for independence or a goodness-of-fit test. Mixing up the formulas will lead to incorrect results.
• Forgetting the Minimum Expected Frequency Rule: Be mindful of the rule that expected frequencies should generally be at least 5. If some expected frequencies are too low, consider combining categories (if meaningful) or using a different statistical test.
• Misinterpreting Expected Frequencies: Remember that expected frequencies are theoretical values based on a specific hypothesis. They don't represent what actually happened, but rather what should have happened if the hypothesis were true.

Beyond the Basics: Advanced Considerations

While the formulas presented above cover the most common scenarios, there are some advanced considerations to keep in mind:

• Yates' Correction for Continuity: When performing a chi-square test with a 2x2 contingency table (two rows and two columns), Yates' correction for continuity is sometimes applied to adjust the chi-square statistic. This correction reduces the chi-square value slightly, making the test more conservative (less likely to find a significant result).
• Alternatives to Chi-Square: When the assumptions of the chi-square test are violated (e.g., small expected frequencies), alternative tests like Fisher's exact test or the G-test may be more appropriate.
• Software and Statistical Packages: Statistical software packages like SPSS, R, and Python (with libraries like SciPy) can automate the calculation of expected frequencies and the execution of chi-square tests. These tools can save time and reduce the risk of calculation errors.

FAQ

• What happens if my expected frequencies are too low?

  If you have expected frequencies less than 5, the chi-square approximation may not be accurate. Consider combining categories (if it makes sense theoretically) to increase the expected frequencies. Alternatively, use a more appropriate test, such as Fisher's exact test.

• Can I use the chi-square test with continuous data?

  No, the chi-square test is designed for categorical data. If you have continuous data, you'll need to categorize it into discrete bins before applying the chi-square test.

• How do I interpret the results of a chi-square test after calculating expected frequencies?

  After calculating the chi-square statistic (which uses the observed and expected frequencies), you compare it to a critical value from the chi-square distribution or calculate a p-value. A small p-value (typically less than 0.05) indicates that the observed differences between the observed and expected frequencies are statistically significant, leading you to reject the null hypothesis.

• Is there a difference between expected count and expected frequency?

  No, the terms "expected count" and "expected frequency" are often used interchangeably. They both refer to the theoretical frequency expected under a specific hypothesis.

Conclusion

Calculating expected frequencies is a fundamental step in conducting chi-square tests, which are powerful tools for analyzing categorical data. By understanding the formulas, avoiding common mistakes, and considering advanced considerations, you can confidently apply the chi-square test to answer a wide range of research questions. Remember, expected frequencies provide the essential baseline needed to determine if observed patterns are statistically significant or simply due to chance. Mastering this concept will significantly enhance your ability to analyze data and draw meaningful conclusions.