Expected Frequency Formula
Expected Frequency Formula - Related Videos
01:08
Determination of Expected Frequency
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2023
Suppose one wants to test independence between the two variables of a contingency table. The values in the table constitute the observed frequencies of the dataset. But how does one determine the expected frequency of the dataset? One of the important assumptions is that the two variables are independent, which means the variables do not influence each other. For independent variables, the statistical probability of any event involving both variables is calculated by multiplying the individual...
01:19
Expected Frequencies in Goodness-of-Fit Tests
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2023
A goodness-of-fit test is conducted to determine whether the observed frequency values are statistically similar to the frequencies expected for the dataset. Suppose the expected frequencies for a dataset are equal such as when predicting the frequency of any number appearing when casting a die. In that case, the expected frequency is the ratio of the total number of observations (n) to the number of categories (k). Hence, the expected frequency of any number appearing when casting a die will...
01:15
Expected Value
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2023
The expected value is known as the "long-term" average or mean. This means that over the long term of experimenting over and over, you would expect this average. The expected average is represented by the symbol μ. It is calculated as follows:In the equation, x is an event, and P(x) is the probability of the event occurring.The expected value has practical applications in decision theory.This text is adapted from Openstax, Introductory Statistics, Section 4.2 Mean or Expected Value and...
01:19
Expected Income, Expected Utility, and Risk Aversion II
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2025
John is evaluating a job offer from a company where his income will be uncertain. If the company performs well, John will earn an annual income of $81,000; otherwise, he will earn $49,000. It is assumed that either outcome has an equal chance, assigning a probability of 0.5 to each. This results in an expected income of $65,000. His decision-making is affected by the diminishing marginal utility of income. John evaluates his options based on their utility. Expected utility accounts for risk...
02:52
Chemical Formulas
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2020
A chemical formula presents information about the proportions of atoms constituting a particular chemical compound or molecule, mainly using symbols of elements and numbers. At times other symbols, such as dashes, parentheses, brackets, commas, plus, and minus signs, are also used. A chemical formula can be one of three types – molecular, empirical, and structural. Molecular Formula The molecular formula represents a molecule or a chemical compound using chemical symbols (to indicate the...
FAQs
The theoretical probability determines the proportion of observations assigned to a category under the chosen model. Because the calculation multiplies that probability by the total number of observations, changing the model’s probability changes the expected count even when the dataset stays the same. This makes the result a model-based benchmark rather than a description of the observed data.
In a contingency table, row totals and column totals determine how the grand total is distributed across categories. Multiplying the relevant row total by the relevant column total, then dividing by the grand total, produces the table cell’s expected count. The result represents the count implied by those marginal totals under the comparison model.
A difference shows that the data do not match the specified probability model perfectly, but the difference alone does not establish a meaningful departure. Chi-square analysis evaluates whether the overall pattern of discrepancies is consistent with chance or instead suggests an association between categorical variables. Expected frequencies therefore provide the reference point for interpreting observed counts.
The same comparison principle serves two related statistical purposes. In goodness-of-fit analysis, observed category counts are compared with counts predicted by a theoretical probability model. In a contingency table, cell counts are compared with values derived from row and column totals to assess association between categorical variables. The analysis changes according to the structure of the data and the question.
First identify the categories or table cells being analyzed, the total number of observations, and the theoretical probability for each category when using a single-category model. For a contingency table, collect each row total, column total, and the grand total. These inputs establish the appropriate baseline before observed and expected counts are compared.
A basic workflow calculates the expected count for every category or contingency-table cell, places those values alongside the observed counts, and then evaluates the discrepancies through a chi-square test. The resulting comparison addresses whether the departures from the model can reasonably be attributed to chance or whether they indicate a categorical association.