The Data: The 4 Scales of Measurement

The 4 scales of measurement. I understand the instinctive reaction: to skip an article that tastes like an unexciting introduction to a topic considered trivial.
However, I ask readers for an effort that I think is worth making. The concepts presented in this article are basic and precisely for this reason they have fundamental value and importance.
Assimilating these concepts means building a solid foundation for the topics that will follow.

Put concisely, but resolutely: we take nothing for granted, because nothing is taken for granted.

Quantitative and Qualitative Data

Let’s start with some fundamental concepts that will always be with us.

Data can be classified into 2 main types:

• Quantitative
• Qualitative (or Categorical)

Important
In statistics, the entire group we are studying is called the population.
The individuals (which can be living beings or things) in the population are called units.
The characteristics of the units we are studying are called variables.

These variables can be quantitative or qualitative (also called categorical).

The 4 Levels of Measurement

Data can be measured at different levels, depending on the type of variable and the level of detail recorded.
American psychologist Stanley Smith Stevens proposed a classification of 4 levels of measurement (or scales of measurement) in 1946, which is still widely used today.

So we are talking about:

• Nominal Measurement
• Ordinal Measurement
• Interval Measurement
• Ratio Measurement

The difference between these 4 types of measurement scales is based on some salient characteristics:

• The order
• The distance between observations
• The presence and inclusion of a zero with a real meaning

Nominal Scale

A nominal measurement is one in which the values of the variables are names. In this case we have:

• The order of observations does not matter
• The distance is not maintained
• There is no true zero

Let’s use examples from the world of web traffic data analysis, as it’s “daily bread” for those involved in SEO.

Think of the country of origin of visits to a website. Simplifying greatly, I consider visits coming from 4 countries:

Italy
France
UK
USA

We can count the visits from each of these countries:

Country        Visits
Italy        3305
France       1850
UK            1938
USA           2214

We are clearly dealing with a nominal measurement.
This is because:

• The order doesn’t matter (the table is readable even if I change the position of the various countries).
• The distance between categories is not relevant. (It would be if we were treating the data in terms of ratios).
• Zero is not needed (indicating the complete absence of views, and therefore that country would not appear in the report…).

For this type of measurement, the suitable chart type is the bar chart, or the histogram.

Ordinal Measurement

An ordinal measurement involves collecting information in which the order is important.

In terms of salient characteristics:

• The order of observations matters.
• Ordinal measurement does not preserve distance. The distance between two consecutive values has no meaning. (For example, the distance between the first and second observation can be in the order of thousands of units, while that between the fifth and sixth may be only a few units…).
• There is no meaningful zero.

Returning to our example of website visits by country:

Country        Position
Italy        1
USA           2
UK            3
France       4

We have established an order. The distance between the values of the various countries is unknown. Zero does not exist.

The appropriate chart type for ordinal measurements is the histogram or the bar chart.

Interval Scale

In interval scales, the distance between two values has specific meaning.
A typical example is a questionnaire where the answers are coded on a scale ranging, for example, from:

1 = I like it very little
to
10 = I like it very much

The characteristics of interval measurements are:

• The order of responses/observations is relevant.
• The distance is relevant.
• There is no zero with a real meaning. (Although the data could be scaled so that 0 could be counted).

This type of measurement is very common in surveys.

Appropriate chart types for representation are bar charts, line charts, and scatterplots.

The most appropriate statistics for interval measurements are the mean, median, variance, standard deviation, skewness, and kurtosis.

Ratio Scale

Now we come to the most common type of measurement in web data analysis: the ratio.

A ratio measurement expresses the relationship between the magnitude of a continuous quantity and a unit magnitude of the same kind.

A variable measured in this way includes not only the concept of order and interval, but also the idea of “nothing,” or absolute zero. Therefore:

• The order of responses/observations matters.
• The ratio expresses an interpretable distance.
• There is a true zero.

Staying within the field of web metrics, a typical example is the ratio between the number of visits and goals (conversions).

Appropriate charts are: histograms, bar or line charts, and scatterplots.

Appropriate statistics are: median, mean, variance, standard deviation, skewness, and kurtosis.

Complexity of Measurement Types

Stevens’ categorization of measurement scales shows us an increase in the complexity of measurement types. Schematically:

Or in a table:

	Nominal	Ordinal	Interval	Ratio
Order	No	Yes	Yes	Yes
Interpretable Distance	No	No	Yes	Yes
True Zero	No	No	No	Yes

Some types of measurement levels can be transformed into others. The transformation can take place from the most complex to the least complex, never vice versa. And in the transformation, of course, we lose information.

Working with Data Using the Correct Tools

We have seen how nominal or ordinal data are qualitative data. Therefore, we cannot perform normal arithmetic operations on them or directly use statistical indices such as the mean, standard deviation, skewness, or kurtosis. However, we can use a series of non-parametric tools, such as contingency tables or the chi-squared test of independence.

For quantitative data, we obviously have the possibility to operate with the tools of basic arithmetic (we can add, subtract, multiply, divide), as well as take advantage of the possibility of calculating the mean, variance, standard deviation, kurtosis, and skewness. We also have parametric analysis tools at our disposal, such as correlation indices, regression calculations, and ANOVA.

For the distinction between parametric and non-parametric analysis tools, I refer you to this article.