Definition and formula for expected frequencies in statistics

📚 Definition of Expected Frequencies

In statistics, the expected frequency refers to the anticipated number of occurrences for each category or outcome in a sample. It's what we'd predict based on a theoretical model or prior knowledge, assuming the null hypothesis is true. Essentially, it represents the 'ideal' distribution we'd expect to see.

📜 History and Background

The concept of expected frequencies emerged alongside the development of hypothesis testing, particularly the chi-square test. Karl Pearson's work in the early 20th century laid the foundation for comparing observed data with expected outcomes, allowing statisticians to assess the goodness-of-fit of various models. This historical context shows how expected frequencies became crucial for statistical inference.

🔑 Key Principles

• ⚖️ Theoretical Basis: Expected frequencies are derived from a specific theoretical distribution (e.g., uniform, normal) or a hypothesis about the population.
• 🔢 Calculation: They are calculated based on the sample size and the probabilities associated with each category or outcome.
• 📊 Comparison: The core purpose is to compare expected frequencies with the observed frequencies to determine if there is a statistically significant difference.
• 🎯 Null Hypothesis: Expected frequencies are calculated assuming the null hypothesis is true.

🧮 Formula for Calculating Expected Frequencies

The formula for calculating the expected frequency ($E$) for a category is given by:

$E = N * p$

Where:

• 💯 $N$ represents the total number of observations in the sample.
• 🧪 $p$ represents the probability of that specific category occurring.

🌍 Real-world Examples

Example 1: Fair Dice Roll

Suppose you roll a fair six-sided die 60 times. What are the expected frequencies for each number (1 through 6)?

Since the die is fair, the probability of rolling any specific number is $1/6$.

Therefore, the expected frequency for each number is: $E = 60 * (1/6) = 10$

This means you'd expect to roll each number approximately 10 times.

Example 2: Coin Toss

You flip a fair coin 100 times. What are the expected frequencies for heads and tails?

The probability of heads is $0.5$, and the probability of tails is also $0.5$.

Expected frequency for heads: $E_{heads} = 100 * 0.5 = 50$

Expected frequency for tails: $E_{tails} = 100 * 0.5 = 50$

You would expect approximately 50 heads and 50 tails.

Example 3: Chi-Square Test for Categorical Data

Let's say we want to test if the distribution of M&M colors in a bag matches the distribution claimed by the manufacturer. The manufacturer claims the following percentages: Brown (30%), Yellow (20%), Red (20%), Blue (10%), Orange (10%), Green (10%). We have a bag with 500 M&Ms.

These expected frequencies are what we would compare to the *observed* frequencies (the actual count of each color in your bag) using a chi-square test.

🏁 Conclusion

Understanding expected frequencies is fundamental to many statistical tests and analyses. By comparing what we observe with what we expect, we can draw meaningful conclusions about the underlying processes and make informed decisions based on data.