How To Calculate Expected Frequency From Observed Frequency

Calculating expected frequency from observed frequency is a fundamental concept in statistics, essential for various applications ranging from genetics and market research to social sciences. Understanding this calculation allows us to determine if the observed data significantly deviates from what we would expect under a specific hypothesis or model. This article will explore the methods to calculate expected frequency, providing clarity and practical examples to ensure you grasp the concept thoroughly.

Understanding Observed and Expected Frequencies

To delve into the calculation, it’s crucial to differentiate between observed and expected frequencies.

• Observed Frequency: This refers to the actual counts or data collected from an experiment or observation. For example, if you roll a die 60 times and observe the number '1' appearing 12 times, the observed frequency for '1' is 12.
• Expected Frequency: This is the frequency we anticipate seeing if a particular hypothesis is true. In the case of the die, if it's fair, we'd expect each number (1 through 6) to appear approximately 10 times in 60 rolls. Thus, the expected frequency for each number is 10.

The comparison between these frequencies helps in performing statistical tests, such as the Chi-square test, to assess the validity of a hypothesis.

Basic Calculation of Expected Frequency

The basic formula for calculating expected frequency is:

Expected Frequency = (Probability of the event) x (Total number of trials)

This formula is used when you have a specific probability associated with an event and you want to know how many times you'd expect that event to occur in a given number of trials.

Example 1: Fair Coin Toss

Let’s consider a simple scenario: tossing a fair coin 100 times.

• Event: Getting heads
• Probability of the event: 0.5 (since a fair coin has an equal chance of landing on heads or tails)
• Total number of trials: 100

Using the formula:

Expected Frequency of Heads = (0.5) x (100) = 50

Therefore, if you toss a fair coin 100 times, you would expect to see heads approximately 50 times.

Example 2: Rolling a Fair Die

Now, let’s consider rolling a fair six-sided die 60 times.

• Event: Rolling a '3'
• Probability of the event: 1/6 (since there is one '3' out of six possible outcomes)
• Total number of trials: 60

Using the formula:

Expected Frequency of Rolling a '3' = (1/6) x (60) = 10

Thus, in 60 rolls of a fair die, you would expect to roll a '3' approximately 10 times.

Calculating Expected Frequency in Contingency Tables

Contingency tables, also known as cross-tabulation tables, are used to analyze the relationship between two or more categorical variables. Calculating expected frequencies in contingency tables is crucial for performing Chi-square tests of independence.

Constructing a Contingency Table

A contingency table displays the frequency distribution of variables. The rows represent one variable, and the columns represent another. Here’s a basic example:

Formula for Expected Frequency in Contingency Tables

The formula to calculate the expected frequency for each cell in a contingency table is:

Expected Frequency = (Row Total x Column Total) / Grand Total

Where:

• Row Total is the total number of observations in the row.
• Column Total is the total number of observations in the column.
• Grand Total is the total number of observations in the entire table.

Example: Calculating Expected Frequencies

Using the contingency table above, let's calculate the expected frequencies for each cell:

1. Expected Frequency for Group X, Category A:

   • Row Total (Group X) = 100
   • Column Total (Category A) = 75
   • Grand Total = 200
   • Expected Frequency = (100 x 75) / 200 = 37.5

2. Expected Frequency for Group X, Category B:

   • Row Total (Group X) = 100
   • Column Total (Category B) = 125
   • Grand Total = 200
   • Expected Frequency = (100 x 125) / 200 = 62.5

3. Expected Frequency for Group Y, Category A:

   • Row Total (Group Y) = 100
   • Column Total (Category A) = 75
   • Grand Total = 200
   • Expected Frequency = (100 x 75) / 200 = 37.5

4. Expected Frequency for Group Y, Category B:

   • Row Total (Group Y) = 100
   • Column Total (Category B) = 125
   • Grand Total = 200
   • Expected Frequency = (100 x 125) / 200 = 62.5

Here’s the updated contingency table with expected frequencies:

Note: Values in parentheses are the expected frequencies.

Chi-Square Test and Expected Frequencies

The Chi-square test is a statistical test used to determine if there is a significant association between two categorical variables. It compares the observed frequencies with the expected frequencies to assess whether the differences are due to chance or a real relationship.

Formula for Chi-Square Statistic

The Chi-square statistic is calculated using the formula:

χ² = Σ [(Observed Frequency - Expected Frequency)² / Expected Frequency]

Where:

• χ² is the Chi-square statistic
• Σ means “sum of”
• Observed Frequency is the actual frequency observed in the data
• Expected Frequency is the frequency we would expect under the null hypothesis

Steps to Perform a Chi-Square Test

1. State the Hypotheses:
   • Null Hypothesis (H₀): There is no association between the variables.
   • Alternative Hypothesis (H₁): There is an association between the variables.
2. Construct a Contingency Table:
   • Organize the observed frequencies into a table.
3. Calculate Expected Frequencies:
   • Use the formula (Row Total x Column Total) / Grand Total to find the expected frequency for each cell.
4. Calculate the Chi-Square Statistic:
   • Use the formula χ² = Σ [(Observed Frequency - Expected Frequency)² / Expected Frequency] to calculate the Chi-square statistic.
5. Determine the Degrees of Freedom:
   • Degrees of Freedom (df) = (Number of Rows - 1) x (Number of Columns - 1)
6. Find the Critical Value:
   • Use a Chi-square distribution table or calculator to find the critical value for the chosen significance level (e.g., α = 0.05) and degrees of freedom.
7. Compare the Chi-Square Statistic to the Critical Value:
   • If the Chi-square statistic is greater than the critical value, reject the null hypothesis.
   • If the Chi-square statistic is less than or equal to the critical value, fail to reject the null hypothesis.
8. Draw a Conclusion:
   • Based on the comparison, conclude whether there is a significant association between the variables.

Example: Chi-Square Test

Let’s use the contingency table from the previous example:

1. Calculate the Chi-Square Statistic:

   χ² = [(30 - 37.5)² / 37.5] + [(70 - 62.5)² / 62.5] + [(45 - 37.5)² / 37.5] + [(55 - 62.5)² / 62.5]

   χ² = [7.5² / 37.5] + [7.5² / 62.5] + [7.5² / 37.5] + [7.5² / 62.5]

   χ² = [56.25 / 37.5] + [56.25 / 62.5] + [56.25 / 37.5] + [56.25 / 62.5]

   χ² = 1.5 + 0.9 + 1.5 + 0.9 = 4.8

2. Determine the Degrees of Freedom:

   df = (2 - 1) x (2 - 1) = 1 x 1 = 1

3. Find the Critical Value:

   For α = 0.05 and df = 1, the critical value from a Chi-square distribution table is 3.841.

4. Compare the Chi-Square Statistic to the Critical Value:

   Since 4.8 > 3.841, we reject the null hypothesis.

5. Draw a Conclusion:

   There is a significant association between the group and the category.

Advanced Scenarios and Considerations

Unequal Probabilities

In some cases, the probabilities of events may not be equal. For example, in genetics, certain traits might be more likely to occur due to dominant genes.

Example: Genetic Traits

Suppose we are studying the inheritance of flower color in a plant species. The gene for flower color has two alleles: R (red) and r (white). According to Mendelian genetics, the expected ratio for the offspring of a heterozygous cross (Rr x Rr) is:

• RR (Red): 25%
• Rr (Red): 50%
• rr (White): 25%

In a sample of 200 offspring, we observe the following frequencies:

• Red Flowers: 140
• White Flowers: 60

To calculate the expected frequencies:

• Expected Frequency of RR (Red) = 0.25 x 200 = 50
• Expected Frequency of Rr (Red) = 0.50 x 200 = 100
• Expected Frequency of rr (White) = 0.25 x 200 = 50

We can then perform a Chi-square test to determine if the observed frequencies deviate significantly from the expected Mendelian ratios.

Small Sample Sizes

When dealing with small sample sizes, the Chi-square test may not be appropriate because the expected frequencies in some cells might be too low (usually less than 5). In such cases, alternative tests like Fisher's exact test are more suitable.

Example: Small Sample in Medical Study

Suppose a medical study investigates the effectiveness of a new drug. The results are summarized in the following contingency table:

Calculating the expected frequencies:

• Expected Frequency (Drug, Improved) = (10 x 7) / 20 = 3.5
• Expected Frequency (Drug, Not Improved) = (10 x 13) / 20 = 6.5
• Expected Frequency (Placebo, Improved) = (10 x 7) / 20 = 3.5
• Expected Frequency (Placebo, Not Improved) = (10 x 13) / 20 = 6.5

In this case, some expected frequencies are less than 5, which might make the Chi-square test unreliable. Fisher's exact test would be a better choice to analyze this data.

Yates's Correction for Continuity

Yates's correction is a method used to adjust the Chi-square statistic when dealing with 2x2 contingency tables, especially when sample sizes are small. It reduces the overestimation of the Chi-square value by subtracting 0.5 from the absolute difference between observed and expected frequencies before squaring.

Corrected Chi-Square Formula:

χ² = Σ [(|Observed Frequency - Expected Frequency| - 0.5)² / Expected Frequency]

Real-World Applications

1. Market Research:

   Companies use observed and expected frequencies to analyze customer preferences. For instance, a company might observe the number of customers who prefer different product features and compare these observations to expected preferences based on market surveys.

2. Genetics:

   In genetics, these calculations are crucial for determining if observed genetic ratios align with expected Mendelian ratios. This helps in understanding inheritance patterns and identifying genetic mutations.

3. Social Sciences:

   Researchers use contingency tables and Chi-square tests to analyze relationships between social variables, such as education level and income, or political affiliation and voting behavior.

4. Healthcare:

   In clinical trials, observed and expected frequencies are used to assess the effectiveness of new treatments. Researchers compare the number of patients who improve with the treatment to the number expected to improve based on chance or placebo effects.

Common Pitfalls to Avoid

1. Incorrectly Calculating Expected Frequencies:

   Ensure you are using the correct formulas for calculating expected frequencies, especially in contingency tables. Double-check your calculations to avoid errors.

2. Applying Chi-Square Test to Non-Categorical Data:

   The Chi-square test is designed for categorical data. Avoid using it with continuous variables, as it can lead to incorrect conclusions.

3. Ignoring Small Sample Size Issues:

   Be aware of the limitations of the Chi-square test with small sample sizes. If expected frequencies are too low, consider using alternative tests like Fisher's exact test.

4. Misinterpreting Results:

   Rejecting the null hypothesis only indicates that there is a significant association between the variables, but it does not prove causation. Further research may be needed to establish causal relationships.

Conclusion

Calculating expected frequency from observed frequency is a foundational skill in statistics, enabling us to evaluate hypotheses and understand the relationships between variables. Whether you're analyzing coin tosses, genetic traits, or social phenomena, mastering these calculations provides valuable insights. By understanding the formulas, considering the context, and avoiding common pitfalls, you can confidently use expected frequencies in various statistical analyses. This comprehensive guide has equipped you with the knowledge and tools to accurately calculate expected frequencies and apply them in real-world scenarios, enhancing your analytical capabilities and decision-making processes.