Comparison of Observed and Expected Frequencies

Meaning of Observed Frequencies

Observed frequencies refer to the actual number of occurrences recorded in different categories or cells of a frequency distribution or contingency table. These frequencies are denoted by O and represent the real outcomes obtained from surveys, experiments, or secondary data sources. Observed frequencies form the raw data on which statistical analysis is carried out. They show how attributes or variables actually behave in real-life situations and reflect the true pattern of association present in the data.

Meaning of Expected Frequencies

Expected frequencies are theoretical frequencies calculated under the assumption that there is no association between attributes or that the data follows a specified theoretical distribution. These frequencies are denoted by E and indicate what the frequencies should be if the null hypothesis is true. Expected frequencies are not directly observed but are computed mathematically using row totals, column totals, and the grand total of the contingency table.

Formula for Expected Frequencies

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

This formula ensures that the marginal totals remain unchanged while assuming independence between the attributes.

Purpose of Comparing Observed and Expected Frequencies

The main purpose of comparing observed and expected frequencies is to determine whether the difference between them is due to chance or due to a real and significant relationship between variables. If observed frequencies are close to expected frequencies, it indicates independence. Large deviations suggest the presence of association. This comparison is the foundation of inferential statistical tools like the Chi-square test.

Importance in Statistical Analysis

This comparison converts qualitative observations into measurable evidence. It allows researchers to move beyond mere description and make objective conclusions about relationships between variables. In business and economics, it helps in decision-making related to consumer behavior, market segmentation, and policy formulation.

1. Yule’s Coefficient of Association

Yule’s Coefficient of Association is a measure used to determine the degree and direction of association between two qualitative attributes. It is applicable when data is arranged in a 2 × 2 contingency table, where each attribute has only two alternatives, such as presence or absence.

Structure of a 2 × 2 Table

Formula

Yule’s Coefficient (Q) = (ad − bc) / (ad + bc)

Range and Interpretation

The value of Yule’s coefficient lies between –1 and +1. A value of +1 indicates perfect positive association, meaning both attributes move together. A value of –1 indicates perfect negative association, meaning the presence of one implies absence of the other. A value of zero indicates no association.

Merits of Yule’s Coefficient

Yule’s coefficient is simple to calculate and easy to interpret. It is particularly useful in social sciences, management studies, and business research where variables are qualitative in nature. It provides a clear numerical measure of association.

Limitations of Yule’s Coefficient

It is applicable only to 2 × 2 tables and ignores the marginal totals. It does not consider the size of the sample, which may sometimes lead to misleading conclusions.

2. Chi-square (χ²) Test

The Chi-square test is a non-parametric statistical test used to examine whether there is a significant difference between observed and expected frequencies. It helps determine whether attributes are independent or associated, and whether observed data fits a theoretical distribution.

Formula

χ² = Σ [(O − E)² / E]

Where:
O = Observed Frequency
E = Expected Frequency

Nature of the Test

The Chi-square test is based on frequency data and does not require assumptions about the normality of the population. It is widely used in business statistics, economics, sociology, and psychology.

Applications of Chi-square Test

The test is used to study association between attributes, test goodness of fit, and test homogeneity of samples. In business, it is applied in market surveys, consumer preference studies, and quality control.

Decision Rule

If the calculated Chi-square value is greater than the table value at a given level of significance, the null hypothesis is rejected. If it is less, the null hypothesis is accepted.

Assumptions of Chi-square Test

The Chi-square test is based on several important assumptions. The data must be expressed in frequencies and not percentages or ratios. The observations should be independent of each other. The sample should be randomly selected. Expected frequencies should generally not be less than five. The categories must be mutually exclusive and exhaustive. Violation of these assumptions reduces the reliability of the test results.

3. Degrees of Freedom

Degrees of freedom represent the number of independent observations that are free to vary after certain restrictions have been imposed. In Chi-square analysis, degrees of freedom determine the critical value used for hypothesis testing.

Formula for Degrees of Freedom

Degrees of Freedom (df) = (r − 1)(c − 1)

Where:
r = number of rows
c = number of columns

Role in Hypothesis Testing

Degrees of freedom affect the shape of the Chi-square distribution and the critical value against which the calculated value is compared. Higher degrees of freedom result in a flatter distribution.

4. Level of Significance

The level of significance represents the probability of rejecting a true null hypothesis. It indicates the level of risk a researcher is willing to take while making a decision based on sample data.

Common Levels of Significance

The most commonly used levels are 5% (0.05) and 1% (0.01). A 5% level implies a 5% risk of committing a Type I error, while a 1% level indicates stricter testing standards.

Importance in Decision Making

The level of significance provides an objective criterion for accepting or rejecting hypotheses. In business decisions, choosing an appropriate significance level balances risk and reliability.

5. Test of Goodness of Fit

The Chi-square goodness-of-fit test is used to determine whether observed data fits a specified theoretical or expected distribution. It examines how well a theoretical model explains the observed frequencies.

Procedure

First, expected frequencies are calculated based on the theoretical distribution. Then, the Chi-square statistic is computed using observed and expected frequencies. Finally, the calculated value is compared with the table value.

Applications of Goodness of Fit Test

This test is used to verify distributions like binomial, Poisson, and normal distributions. In business, it is used in quality control, demand forecasting, and market research studies.

Conclusion of the Test

If the calculated Chi-square value is less than the table value, the theoretical distribution is considered a good fit. Otherwise, it is rejected.