Chi-square test is a primordial technique employed by statisticians to evaluate the hypothesis concerned with an association between two variables. This article will take you through an understanding of the Chi-square test especially when used with ordinal data.

In this article, we will learn the general concept of the test, the assumption placed on it and how and in what manner all these results may be understood and analyzed.

What is Chi-Square Test?

Chi-square test can be described as the statistical method that is used in testing the variation between the categories’ explanations of the collected data. It is mainly applied to examine hypotheses relating to the correlation between two variables. It is used comprehensively in numerous areas of studies including marketing, political science, and biology to determine if the distribution of categorical variables deviates from what is expected by chance.

Chi-Square Test Formula

The formula for the Chi-square statistic (χ²) is:

[Tex]\chi^2 = \sum \frac{(O_i – E_i)^2}{E_i}[/Tex]

where:

• O_i is Observed Frequency
• E_i is Expected frequency

Each component represents the observed and expected frequencies for each category. It is a mathematical formula used when arriving at the test statistic which is the sum of the squared difference between the frequency that occurred, and that which was expected, all divided by the expected frequency.

How to Calculate the Chi-Square Statistic

The steps to calculate Chi-Square Statistics are as follows:

Step 1: Define Hypotheses: Squash your null hypothesis (H₀) and alternative hypotheses (H₁).

Step 2: Create a Contingency Table: Tabulate observed frequencies.

Step 3: Calculate Expected Frequencies: For each cell, use [Tex]E_i = \frac{(\text{row total} \times \text{column total})}{\text{grand total}}[/Tex]

Step 4: Determine χ²: Using the above-mentioned formula ascertain the corresponding value of Chi-square for the categorical data.

Step 5: Determine Degrees of Freedom: The following should be applied as ((r–1)X(c–1)) where r is the number of rows & c is the number of columns.

Step 6: Compare with Critical Value: Here it is necessary to check the obtained Chi-square statistic against the Chi-square critical value from distribution table.

Expected vs. Observed Frequencies

Referring to the Chi-square test, observed frequencies ‘O’ represent true data values gathered using a study or an experiment. On the other hand, expected frequencies ‘E’ are believed frequencies which would exist if the two variables under investigation are independent of each other.

Assumptions for Chi-Square Test

Various assumption for Chi-square test includes:

• Independence: Observations must be independent of each other.
• Expected Frequency: Each expected frequency should be at least 5.
• Categorical Data: Data must be in categorical form.

Meeting these assumptions ensures the validity of the Chi-square test results.

What is Ordinal Data?

Ordinal data are quantitative data that can be ranked in order but the intervals between them are not fixed. The main difference between nominal data and ordinal data is that the former is characterized by ordering while the intervals between terms are irregular.

Examples include:

• Customer satisfaction ratings (e.g., Excellent, Good, Fair, Poor)
• Education levels (e.g., High School, Bachelor’s, Master’s).

Using Chi-Square Test with Ordinal Data

To use chi-square test with ordinal data follow the steps added below:

Step 1: Organize Data: Create a contingency table with ordinal categories.

Step 2: Compute Expected Frequencies: As before, calculate for each cell.

Step 3: Calculate Chi-Square Statistic: Use the same formula.

Step 4: Evaluate Results: Check the Chi-square value against the critical value.

Challenges with ordinal data include ensuring the data’s order is considered without assuming equal intervals between categories.

How to Interpret Chi-Square Test Results

The p-value shows the likelihood of these associations existing by mere coincidence. If the p-value is less than the specifically set significance level for instance 0.05 then the null hypothesis is rejected, implying a relationship between the two variables.

Understanding Statistical Significance

The term statistical significance refers to the results which cannot be attributed to random occurrence. Communicate these results clearly, emphasizing their practical implications and limitations.

Example of Chi-Square with Ordinal Data

Let’s consider a hypothetical dataset of customer satisfaction and service quality ratings. The observed frequencies are as follows: