Summarising Qualitative Data in Statistics

Data Presentation and Summarization Process

Data presentation is an essential step that must be performed before further statistical analysis can be carried out.
Summarizing and displaying data enables researchers, managers, and decision-makers to observe important features of the data set.
Effective data presentation provides insight into the appropriate type of statistical model and analysis that should be utilized.
Data must be summarized and presented in meaningful, readable, and understandable ways to ensure it is useful for decision-making and problem-solving.
Common models of data presentation include:
- Tabular Presentation: Data is arranged in a table with rows and columns. This format is particularly handy when handling many different units of measure.
- Graphical Presentation: This interacts with the audience's visual system and can present immense amounts of data very quickly.

Tabular Presentation of Qualitative Data

Qualitative data can be organized into three primary tabular formats:
- Frequency Distribution.
- Relative Frequency Distribution.
- Contingency Tables.

1. Frequency Distribution

Definition: A tabular model of data summary that shows categories or variables together with the corresponding frequency (the number of items) in each of several non-overlapping classes.
Structure: It is a list or table containing the values of a variable (or a set of ranges) and the corresponding frequency with which each value or range occurs.
Objectives:
- To summarize raw data.
- To condense data into a more useful form.
- To allow for quick visual interpretation.
- To provide insights that cannot be obtained by looking only at original raw data.
Example 1: Students' Nationality (Nominal Data)
- Malaysian: $245$
- Indian: $98$
- American: $12$
- Irish: $115$
- Total ( $n$ ): $470$
Example 2: Degree of Agreement (Ordinal Data)
- Survey claim: Local graduate Architects perform better than Overseas graduate Architects.
- Rank 1: Strongly agree — Frequency: $20$
- Rank 2: Agree somewhat — Frequency: $30$
- Rank 3: Not sure — Frequency: $20$
- Rank 4: Disagree somewhat — Frequency: $15$
- Rank 5: Strongly disagree — Frequency: $15$
- Total: $100$
Example 3: Staff Employment Categories (Nominal Data)
- Research at IIUM regarding local car ownership among staff.
- Academic staff: $20$
- Professional & management staff: $30$
- Support staff: $20$
- Total: $70$

2. Relative Frequency Distribution

Definition: A tabular summary showing the relative frequency for each class. It represents the fraction or proportion of the total number of data items belonging to a specific class.
Formula:
- $\text{Relative Frequency} = \frac{\text{Frequency of Class}}{\text{Total Frequency}}$
- $\text{Relative Frequency Percentage} = \frac{\text{Frequency of Class}}{\text{Total Frequency}} \times 100$
Categorical Examples:
- Nationality (from $n=470$ ):
  - Malaysian: $\frac{245}{470} \times 100 = 52.0\%$
  - Indian: $\frac{98}{470} \times 100 = 21.0\%$
  - American: $\frac{12}{470} \times 100 = 2.5\%$
  - Irish: $\frac{115}{470} \times 100 = 24.5\%$
  - Total: $100\%$
- Employment Category (from $n=70$ ):
  - Academic staff: $\frac{20}{70} = 0.29$
  - Professional & management staff: $\frac{30}{70} = 0.42$
  - Support staff: $\frac{20}{70} = 0.29$
  - Total: $1.0$

3. Contingency Tables

Definition: A table showing responses of subjects to one variable as a function of another variable. It is a way of summarizing relationships between two or more variables.
Characteristics:
- The population or sample is "cross-classified" or subjected to a two-way classification.
- The table contains frequencies classified by values of two or more qualitative variables simultaneously.
- It includes a "Marginal total" for each row and column, and a "Grand total" for the entire sample.
Independence vs. Contingency:
- Contingency: If the proportions of individuals in different columns vary between rows, the variables are contingent (often tested using the Chi-square test).
- Independence: If there is no contingency, the two variables are independent, meaning there is no relationship between them.
Example (Gender vs. Nationality):
- Malaysian: Male ( $145$ ), Female ( $100$ ), Total: $245$
- Indian: Male ( $53$ ), Female ( $45$ ), Total: $98$
- American: Male ( $8$ ), Female ( $4$ ), Total: $12$
- Irish: Male ( $85$ ), Female ( $30$ ), Total: $115$
- Total: Male ( $291$ ), Female ( $179$ ), Grand Total: $470$

Graphical Presentation of Qualitative Data

Qualitative data can be graphically represented through:
- Bar Graphs.
- Clustered Bar Graphs.
- Pie Charts.

1. Bar Graph

Definition: A graphical device depicting summarized qualitative data using bars or rectangles of fixed width drawn above each class label.
Key Features:
- The height of the bar represents the frequency or percentage for that category.
- Vertical Bar Graph: The Vertical axis ( $Y$ ) represents frequency or relative frequency; the Horizontal axis ( $X$ ) contains class labels.
- Horizontal Bar Graph: The Vertical axis ( $Y$ ) contains class labels; the Horizontal axis ( $X$ ) represents frequency or relative frequency.
Construction Steps:
- 1. Draw vertical and horizontal axes and label them with categories or frequency scales.
- 2. Construct a rectangle over each category with a height equal to the frequency/percentage.
- 3. Ensure the bases are of equal width and are separated to emphasize that each class is a separate category.

2. Clustered Bar Graph

Definition: Also known as a grouped bar chart or multi-series bar chart. It is an extension of the bar chart used for plotting numerical values across levels of two categorical variables instead of one.
Construction Mechanics:
- Bars are grouped by position for a single categorical variable (the primary variable).
- The secondary category level within each group is differentiated by color.
- These can be constructed directly from a contingency table.
Example (Cafeteria Cleanliness - Café A, B, C):
- The primary variable is the Cafeteria's Name.
- The secondary variable is the Cleanliness Rating (Poor, Satisfactory, Good).
- Each café has a cluster of three colored bars representing the frequencies of those ratings.

3. Pie Chart

Definition: A graphical device for presenting relative frequency distributions. It consists of a circle divided into sectors (pie slices).
Key Features:
- The size of each slice represents the frequency or percentage of that category.
- The angle (degree) of each sector is proportional to the relative frequency.
Angle Calculation:
- A full circle contains $360$ degrees.
- Calculation: $\text{Angle in Degrees} = \text{Relative Frequency Percentage} \times 360$
- Example: If a class has a relative frequency of $42\%$ , the angle is $0.42 \times 360 = 151.2^{\circ}$ .
Example: Investment Portfolio
- Stocks: Amount $46.5$ ( $42\%$ ), Angle $151.2^{\circ}$
- Bonds: Amount $32.0$ ( $29\%$ ), Angle $104.4^{\circ}$
- CD: Amount $15.5$ ( $14\%$ ), Angle $50.4^{\circ}$
- Savings: Amount $16.0$ ( $15\%$ ), Angle $54.0^{\circ}$
- Total: Amount $110$ , Percentage $100\%$ , Degrees $360^{\circ}$
Example: Breakdown of Assessments
- Assignment: $10\%$ , Angle $36^{\circ}$
- Project 1: $17\%$ , Angle $61^{\circ}$
- Project 2: $18\%$ , Angle $65^{\circ}$
- Quiz 1: $5\%$ , Angle $18^{\circ}$
- Quiz 2: $10\%$ , Angle $36^{\circ}$
- EOSE (End of Semester Examination): $40\%$ , Angle $144^{\circ}$
- Total: $100\%$ , Degrees $360^{\circ}$

Questions & Discussion

In-Class Exercise Task:
- 1. Conduct a survey within the section regarding the first choice of degree programme students wish to pursue in KAED, IIUM Gombak.
- 2. Form a frequency table and relative frequency distribution for all students based on the collected data.
- 3. Construct a contingency table incorporating "gender" as a variable.
- 4. Construct the following diagrams based on the data: Bar graph, Clustered bar graph, and Pie chart.