Data Types and Summary Measures

1. The Landscape of Data: A Classification

At the highest level, data variables can be split into two main groups: categorical (qualitative) and numerical (quantitative).

• Categorical variables represent groupings or labels. Examples include gender, country of origin, or satisfaction level (e.g., "satisfied," "neutral," "dissatisfied").
• Numerical variables represent measurable quantities. Examples include height, temperature, or the number of clicks on a webpage.

This course will focus heavily on analyzing categorical data, but understanding the complete picture is vital. The flowchart below gives you a map of the main data types we'll discuss.

To get a formal introduction to these concepts, please watch the first part of the following video. It clearly defines the four main levels of measurement: nominal, ordinal, interval, and ratio.

As you saw in the video, we can summarize the hierarchy of data as follows:

1. Nominal: Purely categorical. Values are distinct labels with no inherent order.
   • Examples: Eye color (blue, brown, green), political affiliation, type of car.
2. Ordinal: Categorical with a meaningful order, but the intervals between the ranks are not necessarily equal or measurable.
   • Examples: Education level (High School, Bachelor's, Master's, PhD), customer satisfaction (Very Unsatisfied, Neutral, Very Satisfied). The gap between "High School" and "Bachelor's" isn't the same as between "Master's" and "PhD".
3. Interval: Numerical data that is ordered and has equal intervals between values, but lacks a "true zero." A value of zero does not mean the absence of the quantity.
   • Examples: Temperature in Celsius or Fahrenheit (0°C doesn't mean no heat), IQ scores. You can't say 20°C is "twice as hot" as 10°C.
4. Ratio: The most informative level. It is ordered, has equal intervals, and a "true zero," which represents the complete absence of the attribute.
   • Examples: Height, weight, age, income in dollars. A weight of 0 kg means no weight. You can say someone who is 2 meters tall is twice as tall as someone who is 1 meter tall.

For the purposes of this course, we will primarily group these into three buckets: Nominal, Ordinal, and Numerical (which combines Interval and Ratio).

2. A Closer Look at Categorical Data

Let's now focus on nominal and ordinal data, as they are the stars of categorical data analysis. The infographic below provides a side-by-side comparison.

To deepen your functional understanding, let's explore these types with more examples. The following reading and video provide concise definitions and practical illustrations. A special type of nominal data, binary data, consists of only two categories (e.g., Yes/No, Pass/Fail). It's extremely common in data science.

Now, watch the following video. It offers a slightly different perspective and a helpful applied example at the end, which reinforces the concepts.

Functional Understanding Check

A crucial insight for any data scientist is that the way you collect data determines its type.

Consider the "Exploratory Data Analysis" video (LINK) you watched earlier. It gave an excellent example where a survey asked about the number of books in a home, but provided these response options: '0-10', '11-50', '51-100', '101+'.

Question: What data type is this variable as collected by the survey, and why? How would the data type change if the survey simply asked respondents to enter the exact number of books they own?

3. How to Summarize Categorical Data

Now that you can identify nominal and ordinal data, how do you describe a dataset composed of them? The choice of summary measure is dictated by the data's properties.

We use:

• Measures of Central Tendency: To find a "typical" value (e.g., mode, median).
• Measures of Distribution/Variability: To describe how the data is spread across categories (e.g., frequencies, range).
• Visualizations: To see the distribution at a glance (e.g., bar charts, pie charts).

The following table summarizes which measures are appropriate for each categorical data type.

Why these measures?

• Mode: The most frequent category. It works for both nominal and ordinal data because it doesn't require any sense of order.
• Frequencies/Proportions: Simple counts of how many observations fall into each category. This is the most fundamental way to summarize any categorical variable.
• Median: The middle value after the data has been ordered. This is why it's suitable for ordinal data but meaningless for nominal data (you can't find the "middle" eye color).
• Mean (Average): You should generally avoid calculating the mean for categorical data. What is the average of "blue," "green," and "brown"? The question makes no sense. While some researchers compute means for ordinal data (like 1-5 satisfaction scores), this assumes the intervals are equal, which is often not true. This is a common but questionable practice.
• Range & IQR: These measures of spread rely on ordering the data from minimum to maximum. Therefore, they are applicable to ordinal data but not nominal data.

To solidify your understanding of these summary measures, please review the following resource.

Conclusion

In this lesson, we established the critical first step of any data analysis workflow: classifying variables.

Key Takeaways:

• Data can be broadly classified as categorical (labels) or numerical (quantities).
• Categorical data is further divided into:
  • Nominal data: Unordered categories (e.g., Male, Female).
  • Ordinal data: Ordered categories where the distance between them is not uniform (e.g., Low, Medium, High).
• The type of data determines the appropriate summary measures:
  • For nominal data, use frequencies, proportions, and the mode. Visualize with bar or pie charts.
  • For ordinal data, you can also use the median, range, and inter-quartile range (IQR) because the data has a meaningful order.

Understanding these distinctions is not just academic; it has profound practical implications for choosing the right statistical tests and models, which is the entire focus of this course.

Preview of the Next Lesson:

Now that you can describe a single categorical variable, we will move on to exploring the relationship between two categorical variables. We will learn how to construct and interpret a contingency table, a fundamental tool for categorical data analysis.