How Do You Calculate Expected Frequency

In the realm of statistics, understanding the expected frequency is crucial for analyzing categorical data and determining whether observed results deviate significantly from what is expected by chance. This concept forms the backbone of various statistical tests, particularly the chi-square test, which is widely used to assess the independence of variables or the goodness-of-fit of a theoretical distribution.

The Essence of Expected Frequency

Expected frequency represents the number of times an event or category is anticipated to occur in a sample, assuming a specific hypothesis or model is true. It's a theoretical value derived from the probabilities associated with each category and the total sample size. By comparing the observed frequencies (actual counts) with the expected frequencies, we can quantify the discrepancy between what we see in the data and what we expect to see under a certain assumption.

Why Calculate Expected Frequency?

The calculation of expected frequency is pivotal for several reasons:

• Hypothesis Testing: It provides a benchmark against which to compare observed data, enabling us to test hypotheses about the relationships between variables or the distribution of data.
• Statistical Inference: It allows us to draw conclusions about a population based on a sample, by assessing whether the observed data are consistent with the expected pattern.
• Decision Making: It informs decision-making in various fields, from marketing and healthcare to social sciences and engineering, by providing insights into the likelihood of different outcomes.

Key Components

Before diving into the calculation methods, let's identify the key components involved:

• Observed Frequency (O): The actual number of times an event or category occurs in the sample data.
• Expected Frequency (E): The theoretical number of times an event or category is expected to occur, based on a specific hypothesis.
• Total Sample Size (N): The total number of observations in the sample.
• Probabilities (P): The probabilities associated with each category or event, derived from the hypothesis being tested.

Methods to Calculate Expected Frequency

The specific method for calculating expected frequency depends on the nature of the data and the hypothesis being tested. Here, we explore the most common scenarios and their corresponding calculation techniques.

1. Equal Probability

In the simplest case, we assume that each category has an equal probability of occurring. This scenario is common when we have no prior knowledge or reason to believe that one category is more likely than another.

Formula:

E = N / k

Where:

• E = Expected frequency for each category
• N = Total sample size
• k = Number of categories

Example:

Suppose we roll a fair six-sided die 60 times and want to know the expected frequency of each number (1 to 6).

• N = 60 (total number of rolls)
• k = 6 (number of sides on the die)

E = 60 / 6 = 10

Therefore, we expect each number to appear approximately 10 times.

2. Unequal Probabilities

When we have prior knowledge or a specific hypothesis that suggests unequal probabilities for different categories, we need to incorporate these probabilities into the calculation.

Formula:

E = N * P

Where:

• E = Expected frequency for a specific category
• N = Total sample size
• P = Probability of that category occurring

Example:

Consider a bag of marbles with the following distribution:

• Red: 50%
• Blue: 30%
• Green: 20%

If we draw 100 marbles from the bag (with replacement), the expected frequencies for each color would be:

• Red: E = 100 * 0.50 = 50
• Blue: E = 100 * 0.30 = 30
• Green: E = 100 * 0.20 = 20

3. Contingency Tables and the Chi-Square Test

Contingency tables are used to analyze the relationship between two or more categorical variables. The chi-square test is often employed to determine whether there is a statistically significant association between these variables. The calculation of expected frequencies is a crucial step in this process.

Scenario:

We have two categorical variables:

• Variable A: Two levels (A1, A2)
• Variable B: Two levels (B1, B2)

The data is organized in a 2x2 contingency table:

	B1	B2	Total
A1	O11	O12	R1
A2	O21	O22	R2
Total	C1	C2	N

Where:

• Oij = Observed frequency in cell (i, j)
• Ri = Row total for row i
• Ci = Column total for column j
• N = Total sample size

Formula:

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

Eij = (Ri * Ci) / N

Where:

• Eij = Expected frequency for cell (i, j)
• Ri = Row total for row i
• Ci = Column total for column j
• N = Total sample size

Example:

Let's say we want to investigate whether there is an association between smoking (Variable A: Smoker, Non-smoker) and lung cancer (Variable B: Yes, No). We collect data from 200 individuals and create the following contingency table:

	Lung Cancer (Yes)	Lung Cancer (No)	Total
Smoker	40	60	100
Non-smoker	10	90	100
Total	50	150	200

To calculate the expected frequencies:

• E11 (Smoker, Lung Cancer): (100 * 50) / 200 = 25
• E12 (Smoker, No Lung Cancer): (100 * 150) / 200 = 75
• E21 (Non-smoker, Lung Cancer): (100 * 50) / 200 = 25
• E22 (Non-smoker, No Lung Cancer): (100 * 150) / 200 = 75

The expected frequency table would be:

	Lung Cancer (Yes)	Lung Cancer (No)
Smoker	25	75
Non-smoker	25	75

These expected frequencies are then used in the chi-square test to determine whether the observed association between smoking and lung cancer is statistically significant.

4. Goodness-of-Fit Test

The goodness-of-fit test assesses how well a sample distribution matches a theoretical distribution. In this case, the expected frequencies are derived from the theoretical distribution being tested.

Scenario:

We want to test whether the observed distribution of a variable follows a specific distribution (e.g., normal distribution, Poisson distribution).

Steps:

1. Define the theoretical distribution: Specify the parameters of the theoretical distribution (e.g., mean and standard deviation for a normal distribution).
2. Calculate probabilities: Determine the probability of each category or interval based on the theoretical distribution.
3. Calculate expected frequencies: Multiply the probabilities by the total sample size to obtain the expected frequencies for each category.

Formula:

E = N * P

Where:

• E = Expected frequency for a specific category
• N = Total sample size
• P = Probability of that category occurring based on the theoretical distribution

Example:

Suppose we want to test whether the number of customers arriving at a store per hour follows a Poisson distribution with a mean of 5. We observe the number of customers arriving for 100 hours and obtain the following data:

Number of Customers	Observed Frequency
0	2
1	8
2	15
3	20
4	22
5	18
6	9
7	4
8 or more	2

To calculate the expected frequencies, we need to determine the probabilities of each number of customers arriving based on the Poisson distribution with a mean of 5. Using the Poisson probability formula:

P(x) = (e^-λ * λ^x) / x!

Where:

• P(x) = Probability of observing x events
• λ = Mean number of events (5 in this case)
• e = Euler's number (approximately 2.71828)
• x! = Factorial of x

We can calculate the probabilities for each number of customers and then multiply by the total sample size (100) to obtain the expected frequencies.

Note: The calculation of Poisson probabilities can be done using statistical software or calculators.

5. Regression Models

In regression analysis, expected frequencies can be derived from the predicted values generated by the regression model. These expected frequencies can then be compared to the observed frequencies to assess the goodness-of-fit of the model.

Scenario:

We have a regression model that predicts the frequency of a certain event based on one or more predictor variables.

Steps:

1. Fit the regression model: Estimate the parameters of the regression model using the observed data.
2. Generate predicted values: Use the regression model to predict the frequency of the event for each observation in the sample. These predicted values represent the expected frequencies.

Formula:

The specific formula for calculating the expected frequency depends on the type of regression model being used. For example, in linear regression:

E = a + bX

Where:

• E = Expected frequency
• a = Intercept of the regression line
• b = Slope of the regression line
• X = Value of the predictor variable

In logistic regression, the expected frequency is calculated using the logistic function:

E = N * (1 / (1 + e^-(a + bX)))

Where:

• E = Expected frequency
• N = Total sample size
• a = Intercept of the logistic regression model
• b = Coefficient of the predictor variable
• X = Value of the predictor variable
• e = Euler's number (approximately 2.71828)

Example:

Suppose we have a linear regression model that predicts the number of ice cream cones sold per day based on the temperature. The regression equation is:

Ice Cream Cones = 10 + 2 * Temperature

If the temperature on a particular day is 25 degrees Celsius, the expected number of ice cream cones sold would be:

E = 10 + 2 * 25 = 60

This expected frequency can then be compared to the actual number of ice cream cones sold on that day to assess the accuracy of the regression model.

Practical Considerations and Common Pitfalls

While calculating expected frequencies might seem straightforward, there are several practical considerations and potential pitfalls to be aware of:

• Sample Size: The chi-square test and other statistical tests based on expected frequencies are sensitive to sample size. Small sample sizes can lead to inaccurate results. As a general rule, each expected frequency should be at least 5. If this condition is not met, consider combining categories or using alternative statistical tests.
• Independence: The chi-square test assumes that the observations are independent. If the data are not independent, the test results may be invalid.
• Theoretical Distribution: When using the goodness-of-fit test, it's crucial to choose an appropriate theoretical distribution. The choice of distribution should be based on prior knowledge or theoretical considerations.
• Software and Tools: Statistical software packages (e.g., R, SPSS, SAS) provide functions for calculating expected frequencies and performing chi-square tests. These tools can simplify the process and reduce the risk of errors.
• Interpretation: It's important to interpret the results of statistical tests based on expected frequencies in the context of the research question and the limitations of the data. Statistical significance does not necessarily imply practical significance.

Conclusion

The calculation of expected frequency is a fundamental skill in statistics, essential for analyzing categorical data, testing hypotheses, and making informed decisions. By understanding the different methods for calculating expected frequencies and being aware of the potential pitfalls, researchers and practitioners can effectively utilize this concept to gain valuable insights from their data. From simple scenarios with equal probabilities to more complex situations involving contingency tables and regression models, the principles of expected frequency provide a powerful framework for statistical inference and data analysis.