📂Data Science

Scales in Statistics: Nominal, Ordinal, Interval, Ratio

Overview

Generally, creating data in the real world refers to the action of observing a phenomenon or experiment and recording it, which is called measuring.

Definition ¹

There are four known scales for measuring data:

1. Nominal: The purpose is classification itself.
2. Ordinal: Order is assigned.
3. Interval: It is determined within a certain range.
4. Ratio: It is expressed in proportion to a certain unit.

Explanation

Nominal and ordinal scales are used for qualitative variables, while interval and ratio scales are used for quantitative variables.

This definition is not arbitrarily listed; the higher the number, the more detailed the scale becomes, and logically, it can be seen that the earlier scale includes the later number in a hierarchical relationship. All ratio scales are nominal, but not all nominal scales are ratio. $$ 명목 \supset 순서 \supset 구간 \supset 비율 $$

For a smooth and detailed explanation, let’s consider a recurring example. After the first midterm of this year ended, we gathered 100 freshmen from the statistics department in a room called ’laboratory’ and collected, in other words, measured their information. Among them, the information of two people, Adam and Eve, was as follows.

• Adam: Weight 92kg, Midterm 30 points, IQ Rank 12th, Male
• Eve: Weight 46kg, Midterm 90 points, IQ Rank 48th, Female

Here, weight is a ratio scale, midterm score is an interval scale, IQ rank is an ordinal scale, and gender is a nominal scale.

Ratio

Ratio scales include absolute temperature, etc., and from its definition, “proportionality” appears, allowing for scalar multiplication. For example, Adam’s weight of 92kg can be expressed as $$ 92 = 2 \cdot 46 $$ suggesting that Adam’s weight is twice that of Eve’s.

It may seem like a lackluster statement, but surprisingly, this is not obvious. For instance, if you consider Celsius temperature instead of absolute temperature, Celsius 60 degrees does not mean twice as hot as Celsius 30 degrees; it’s simply 30 degrees hotter.

Interval

Interval scales, such as Celsius temperature, allow for addition or subtraction, but the numbers do not necessarily need to represent proportionality. Of course, all ratio scales can be represented as the product of some unit, but the inverse is not true. For example, Adam received 30 points on the midterm, and Eve got 90 points, but this does not mean “the difference in their abilities is three times.” To clarify, it’s not that the score itself isn’t three times; rather, it doesn’t lead to a meaningful proportional relationship. As a more extreme example, imagine Adam’s score as $x$:

• $x = 1 \sim 2$: Just a difference of $1$ points in one question, but with $1$ points, the difference is 90 times, and with $2$ points, the difference is 45 times, which seems too extreme.
• $x = 0$: It can’t be expressed as a multiple at all. Multiplication cannot escape from the curse of $0$, and conversely, it can be understood that a scale going beyond the interval, to the ratio scale, is free from such $0$.
• $x = -90$: Some professors even deduct points for incorrect, unnecessary answers, making scores negative. Adam’s grade being $-1$ times that of Eve’s is hard to comprehend.

Once again, it’s not that scalar multiplication itself isn’t possible; rather, it doesn’t make sense to do so.

Order

Ordinal scales include almost everything with a hierarchy. The most overlooked part by non-majors is that, although an order exists, only comparisons of magnitude can be made, and arithmetic operations are meaningless. For instance, Adam’s IQ is 12th, and Eve’s IQ is 48th, but you can’t say the difference between them and a 25th placed student is the same. More drastically, the difference between 1st and 2nd place is not the same as between 49th and 50th place.

These can only be distinguished by ranking alone, “how much difference” is not provided, and adding or subtracting ranks is presumed meaningless. Calculating the difference in rank numbers is possible but lacks consistent meaning; if meaningful, it would already be an interval scale.

Nominal

Nominal scale now only needs to be distinguishable. Practically, only qualitative variables are considered, but indeed, thinking about its definition, it includes all the scales discussed so far. For example, Adam is distinguished as a male and Eve as a female, but in fact, there was not a single feature shared between the two people. In this sense, height, midterm scores, IQ rank, and gender can be immediately recognized as nominal scales.

But should we force it? Even if height differs by a very small $\varepsilon > 0$, should it be forcedly distinguished? Recalling the explanations of other scales, rather than ‘is it possible or not’ itself, the focus was on ‘what does it mean’, suggesting that nominal scales are essentially only qualitative variables, although they might be represented as numbers.

Be Alert

Don’t take scales lightly and arrogantly just because they appear easy. As detailed as in this post or not, ordinary people tend to believe they have a good grasp of the concept of scales. However, surprisingly, arbitrary interpretations and foolish conclusions are rampant, all because ’this should be easy,’ leading to an unquestioning trust in their thinking and judgment.

You who are reading this text are special. At least in terms of handling data, you are definitively different from most non-majors. If you do not stay alert, it must be assumed that no one knows these obvious things. Let’s always stay alert.