Calculating Expected Frequencies Using Probability

Introduction

Key Concepts

Understanding Expected Frequencies

Expected frequency refers to the theoretical frequency of an event occurring in a large number of trials, assuming that all outcomes are equally likely. It is a crucial component in the chi-square test for goodness of fit and independence, allowing statisticians to compare observed data with expected data under a specific hypothesis.

Formula for Expected Frequency

The expected frequency (\(E\)) of a particular event can be calculated using the formula:

where:

• n is the total number of trials or observations.
• p is the probability of the event occurring in a single trial.

For instance, if there are 100 trials (\(n = 100\)) and the probability of success in each trial is 0.3 (\(p = 0.3\)), the expected frequency of success is:

Calculating Expected Frequencies in Categorical Data

In categorical data, expected frequencies are used to determine if there is a significant difference between observed and expected outcomes. The general steps to calculate expected frequencies in such cases are:

1. Determine the total number of observations (\(n\)).
2. Identify the categories and their corresponding probabilities based on the hypothesis or theoretical distribution.
3. Apply the formula \(E = n \times p\) for each category to find the expected frequencies.

Consider a simple example where a die is rolled 60 times, and the hypothesis is that the die is fair. The probability of each face (1 through 6) occurring is \(p = \frac{1}{6}\). Thus, the expected frequency for each face is:

Expected Frequencies in Two-Way Tables

When dealing with two categorical variables, expected frequencies are calculated for each cell in a contingency table to assess independence. The formula for the expected frequency in a cell corresponding to row \(i\) and column \(j\) is:

Let's consider an example where we're examining the relationship between gender (Male, Female) and preference for a new product (Like, Dislike). Suppose the contingency table is as follows:

To calculate the expected frequency for males who like the product (\(E_{Male, Like}\)):

Chi-Square Test and Expected Frequencies

The chi-square test is a statistical test used to determine whether there is a significant difference between observed and expected frequencies. The test statistic is calculated as:

where:

• \(\chi^2\) is the chi-square statistic.
• O represents the observed frequency.
• E denotes the expected frequency.

If the calculated chi-square statistic exceeds the critical value from the chi-square distribution table at a specified significance level, the null hypothesis is rejected, indicating a significant difference between observed and expected frequencies.

Example: Calculating Expected Frequencies

Let's take a practical example. Suppose we have a deck of 52 cards, and we draw 10 cards. We want to calculate the expected frequency of drawing an Ace.

• Total cards (\(n\)) = 10
• Number of Aces in the deck = 4
• Probability of drawing an Ace (\(p\)) = \(\frac{4}{52} = \frac{1}{13}\)

Using the formula for expected frequency:

Thus, the expected frequency of drawing an Ace in 10 trials is approximately 0.769.

Law of Large Numbers and Expected Frequencies

The Law of Large Numbers states that as the number of trials increases, the relative frequency of an event gets closer to the expected probability. In the context of expected frequencies, this means that observed frequencies tend to converge to expected frequencies as the sample size becomes large.

For example, flipping a fair coin (where \(p = 0.5\)) 100 times is likely to produce a number of heads close to 50. However, with only 10 flips, significant deviations from 5 are more probable.

Applications of Expected Frequencies

Expected frequencies are employed in various fields including:

• Market Research: To determine the expected consumer behavior under certain conditions.
• Medicine: In clinical trials to compare observed patient outcomes against expected results.
• Social Sciences: To assess the association between different demographic variables.
• Genetics: To predict the expected distribution of genetic traits.

Limitations of Expected Frequencies

While expected frequencies are useful, they have certain limitations:

• Sample Size Dependency: The accuracy of expected frequencies improves with larger sample sizes.
• Assumption of Independence: Expected frequencies often assume that categories are independent, which may not always hold true.
• Sensitivity to Distribution: If the underlying probability distribution is incorrect, expected frequencies will be misleading.

Calculation Using Software Tools

While expected frequencies can be calculated manually, software tools like Microsoft Excel, R, and Python can handle complex calculations efficiently. For example, in Excel, the expected frequency can be calculated using the formula:

=Total_Observations * (Probability)

In more advanced statistical software, built-in functions and packages can perform chi-square tests and other analyses that utilize expected frequencies, providing more comprehensive insights with greater ease.

Advanced Concepts

Mathematical Derivation of Expected Frequency Formula

The expected frequency formula \(E = n \times p\) stems from the principles of probability theory, specifically the expectation value in discrete probability distributions.

In a binomial distribution, the expected value (mean) is given by:

where:

• n is the number of trials.
• p is the probability of success on a single trial.

This expectation value represents the long-term average or central tendency of the distribution, which aligns with the concept of expected frequency in the context of repeated trials.

Multinomial Distributions and Expected Frequencies

In cases where there are more than two outcomes for each trial, the multinomial distribution generalizes the binomial theorem. The expected frequencies for each category in a multinomial distribution are calculated as:

where:

• n is the total number of trials.
• p_i is the probability of the ith outcome.

This concept is particularly useful in scenarios involving multiple categories, such as rolling a die or categorizing survey responses.

Expectation in Continuous Probability Distributions

While expected frequencies are typically discussed in the context of discrete probability distributions, the concept extends to continuous distributions through the expected value or mean. The expected value in a continuous distribution is calculated by:

where:

• f(x) is the probability density function of the continuous random variable \(X\).

This integral represents the theoretical mean or average value of \(X\) over an infinite number of observations.

Advanced Chi-Square Tests Utilizing Expected Frequencies

The chi-square test can be extended beyond the goodness-of-fit test to include tests of independence and homogeneity. In these advanced applications, expected frequencies are calculated under more complex hypotheses:

• Test of Independence: Determines whether two categorical variables are independent of each other.
• Test of Homogeneity: Assesses whether different populations have the same distribution of categorical variables.

In these tests, the expected frequencies are computed based on the marginal totals of the contingency tables under the null hypothesis of independence.

Maximum Likelihood Estimation (MLE) and Expected Frequencies

Maximum Likelihood Estimation is a method used in statistics to estimate the parameters of a probability distribution by maximizing a likelihood function. In the context of expected frequencies, MLE can be used to derive expected frequencies that best fit the observed data under the model.

For example, in a multinomial setting, the MLE for the probability \(p_i\) of outcome \(i\) is:

where \(O_i\) is the observed frequency of outcome \(i\), and \(n\) is the total number of trials. The expected frequency is then:

This shows that under MLE, the expected frequencies align with observed frequencies, which is a foundational concept in statistical modeling.

Bayesian Approaches to Expected Frequencies

Bayesian statistics incorporates prior beliefs with evidence from data to update the probability estimates. In the context of expected frequencies, a Bayesian approach might involve adjusting expected frequencies based on prior information or beliefs about the distribution of outcomes.

For instance, if prior knowledge suggests a certain bias in a process, the expected frequencies can be adjusted accordingly before analyzing the observed data, leading to more informed and nuanced conclusions.

Interdisciplinary Connections: Expected Frequencies in Economics and Engineering

Expected frequencies play a vital role beyond pure mathematics, particularly in fields like economics and engineering:

• Economics: In risk assessment and decision-making, expected frequencies help in evaluating the probability of various economic outcomes, such as market fluctuations or investment returns.
• Engineering: In quality control and reliability engineering, expected frequencies are used to assess the likelihood of component failures or defects in manufacturing processes.

These interdisciplinary applications highlight the versatility and importance of expected frequencies in real-world problem-solving.

Case Study: Expected Frequencies in Genetics

Consider a case study in genetics where the expected frequencies of different genotypes are calculated under the assumption of Hardy-Weinberg equilibrium. Suppose we are examining a population where the frequency of allele A is \(p\) and allele a is \(q = 1 - p\).

The expected genotype frequencies are:

• AA: \(p^2\)
• Aa: \(2pq\)
• aa: \(q^2\)

If the population has 1000 individuals and \(p = 0.6\), the expected frequencies are:

• AA: \(1000 \times (0.6)^2 = 360\)
• Aa: \(1000 \times 2(0.6)(0.4) = 480\)
• aa: \(1000 \times (0.4)^2 = 160\)

These expected frequencies can then be compared with observed data to determine if the population is in Hardy-Weinberg equilibrium or if other evolutionary forces are at play.

Comparison Table

Summary and Key Takeaways