How To Compute Expected Frequency

Understanding and Computing Expected Frequency: A Comprehensive Guide

Expected frequency, a cornerstone of statistical analysis, represents the average number of times an event is predicted to occur based on probability. Understanding how to compute expected frequency is crucial in various statistical tests, including the chi-square test, which assesses the difference between observed and expected frequencies to determine if a relationship exists between variables. This comprehensive guide will walk you through the concept, calculation methods, and practical applications of expected frequency, ensuring a thorough understanding even for beginners.

What is Expected Frequency?

In simple terms, expected frequency is the anticipated number of occurrences of a particular outcome in a sample, given the known or assumed probabilities of that outcome. It's a theoretical value, calculated based on probability distributions and sample size, and serves as a benchmark against which observed frequencies (actual counts from data) are compared. The discrepancy between observed and expected frequencies often forms the basis for determining statistical significance. For instance, if you're studying coin flips, the expected frequency of heads is 50% of the total number of flips, assuming a fair coin. This contrasts with the observed frequency, which is the actual number of heads you counted in your experiment.

Methods for Computing Expected Frequency

The calculation of expected frequency varies depending on the context. Let's explore the most common scenarios:

1. Expected Frequency in Contingency Tables:

Contingency tables (also known as cross-tabulations) are used to analyze the relationship between two or more categorical variables. In this context, expected frequencies are calculated for each cell in the table. The formula is:

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

Let's illustrate with an example. Suppose we are investigating the relationship between gender and preference for coffee (tea vs. coffee). We collect the following data:

To calculate the expected frequency for males who prefer coffee:

• Row Total (Male) = 100
• Column Total (Coffee) = 110
• Grand Total = 200

E = (100 * 110) / 200 = 55

Therefore, the expected frequency of males preferring coffee is 55. We repeat this calculation for each cell in the table. Significant deviations between observed and expected frequencies suggest a relationship between gender and coffee preference.

2. Expected Frequency in Binomial Distributions:

A binomial distribution models the probability of a certain number of successes in a fixed number of independent Bernoulli trials (trials with only two possible outcomes, e.g., success/failure, heads/tails). The expected frequency for a specific number of successes (k) is:

E(k) = n * p^k * (1-p)^(n-k)

Where:

• n = number of trials
• p = probability of success in a single trial
• k = number of successes

For example, if we flip a fair coin 10 times (n=10), and p = 0.5 (probability of heads), the expected frequency of getting exactly 3 heads (k=3) is:

E(3) = 10 * 0.5³ * (1-0.5)^(10-3) ≈ 1.2

Note that this calculates the expected frequency of exactly 3 heads. To find the expected frequency for getting at least 3 heads, you'd sum the expected frequencies for k=3, k=4,..., k=10.

3. Expected Frequency in Poisson Distributions:

The Poisson distribution models the probability of a certain number of events occurring in a fixed interval of time or space when events occur independently and at a constant average rate. The expected frequency for a given number of events (k) is:

E(k) = (λ^k * e^-λ) / k!

Where:

• λ (lambda) = average rate of events
• k = number of events
• e = Euler's number (approximately 2.71828)
• k! = factorial of k

For instance, if the average number of cars passing a certain point on a highway per minute is 5 (λ=5), the expected frequency of observing exactly 3 cars in one minute (k=3) is:

E(3) = (5³ * e^-5) / 3! ≈ 1.4

4. Expected Frequency from Relative Frequencies:

If you have data representing relative frequencies (proportions or percentages) of different outcomes, you can calculate the expected frequency by multiplying the relative frequency by the total number of observations. For example, if you know that 30% of customers prefer a particular product, and you have a sample of 1000 customers, the expected frequency for that product is:

E = 0.30 * 1000 = 300

Chi-Square Test and Expected Frequency

The chi-square (χ²) test is a widely used statistical test that compares observed frequencies with expected frequencies to determine if there's a statistically significant difference. The test statistic is calculated as:

χ² = Σ [(O - E)² / E]

Where:

• O = observed frequency
• E = expected frequency
• Σ = summation across all categories

A large chi-square value indicates a significant difference between observed and expected frequencies, suggesting a relationship between the variables under investigation. The calculated chi-square value is then compared to a critical value from the chi-square distribution, considering the degrees of freedom, to determine the statistical significance (p-value). A small p-value (typically below 0.05) suggests that the difference between observed and expected frequencies is unlikely to be due to chance alone.

Interpreting Expected Frequencies

Expected frequencies provide a crucial benchmark against which to compare observed data. Significant deviations between observed and expected frequencies often indicate a noteworthy phenomenon worthy of further investigation. However, it's important to remember that:

• Expected frequencies are theoretical values: They represent what is expected based on probability, not what will necessarily be observed.
• Sample size matters: With larger sample sizes, the differences between observed and expected frequencies are more likely to be statistically significant, even if the effect size is small.
• Context is key: The interpretation of expected frequencies must always be considered within the broader context of the study, including the research question, methodology, and limitations.

Frequently Asked Questions (FAQ)

Q1: What if the expected frequency for a cell is less than 5?

A1: In some statistical tests, particularly the chi-square test, having expected frequencies less than 5 in one or more cells can affect the accuracy of the results. Various strategies can be employed, such as combining categories to increase expected frequencies or using alternative statistical tests that are less sensitive to low expected frequencies.

Q2: Can expected frequency be a decimal?

A2: Yes, expected frequency can be a decimal. It represents the average expected count, and it is not limited to whole numbers.

Q3: How do I choose the right statistical test when dealing with expected frequencies?

A3: The choice of statistical test depends on the type of data (categorical or numerical) and the research question. The chi-square test is commonly used for comparing observed and expected frequencies in categorical data. For numerical data, different tests like t-tests or ANOVA might be more appropriate.

Q4: What are the limitations of using expected frequencies?

A4: While expected frequencies are a powerful tool, they are based on assumptions about probability distributions. If these assumptions are violated (e.g., the data is not truly random or the underlying probability distribution is not accurately represented), the calculated expected frequencies may not be reliable.

Conclusion

Computing expected frequency is a vital skill in various statistical analyses. Understanding how to calculate expected frequencies based on different probability distributions and interpreting the results in conjunction with observed frequencies is crucial for drawing valid conclusions from data. While the calculations can seem complex at first, with practice and a solid grasp of the underlying principles, you can confidently apply these methods to analyze data and uncover hidden relationships. Remember to always carefully consider the context of your research, the assumptions of your chosen statistical test, and the potential limitations of your analysis. The accurate and insightful use of expected frequency contributes significantly to rigorous and reliable statistical inference.