Even More About Measurement 3

5. What Do The Numbers Mean? Case studies don’t use numbers. And N=1 studies limit the use of numbers to counting. In contrast, most other approaches to research use numbers to measure and describe groups of people. But what meaning do the numbers have?

Obviously, variables do not always use numbers in the same way. You might want to find the average age of a group of people but it’s unlikely, for example, that you will want to calculate the average ID number. You know intuitively that averaging ID numbers, room numbers, or Social Security numbers isn’t very useful. Such numbers aren’t used for their numerical value but simply as names.

A number which substitutes for a name makes no mathematical assumptions. A marathon runner with a high number of his back doesn’t necessarily run faster than one with a small number. The numbers are only used to be able to tell the difference between contestants. Such numbers are at the lowest level of assumption, and are said to be at a nominal level of measurement.

There are four levels of assumptions which can be made about numbers. At the nominal scale, we assume that the numbers we obtain can be used to simply distinguish between entities. The numbers on the jerseys of football players, for example, help us to distinguish between players. It makes no sense to add these numbers together, or find their average; each number is used as a name (nom).

The exuberance of this runner is not coming in 8^th place. Although I’d be thrilled with that performance, his smile is for winning the race. The number 8 is just a nominal marker. It has no mathematical value.

In contrast, the second level of measurement, ordinal, makes two assumptions about its numbers. An ordinal scale distinguishes between members plus places them in order. Ranking children from tallest to shortest is an ordinal measurement. Winners of a race can be placed in order of 1, 2, and 3 (first, second, and third) but it would be silly to find the average of these numbers. An ordinal scale is like a footrace in a snowstorm: it can tell who came in first but it can’t tell how far apart the runners are.

An interval scale includes both of the previous assumptions plus the assumption that the distances between numbers (intervals) are equal. The distance between a score of 8 and a score of 9 on a spelling test is the same distance apart as 3 and 4. Using an interval scale, we could tell the difference between players, find out who came in first, and determine by how much our spelling star won.

Notice, that an interval scale assumes equal intervals. In the case of a test, equal intervals means that each item is equally difficult. When the steps are not equal, the scale is ordinal. Consequently, a lot of teacher-made tests look as though they are based on an interval scale but are in fact making ordinal measurements.

The final level of measurement is ratio. A ratio scale includes the previous three assumptions and adds an absolute zero. Because of their absolute zeros, ratio scales have a unique characteristic: they can be used to make ratio comparisons. We can say that a task took twice as long (a ratio of 2 to 1), or that an object weighs a third as much (a ratio of 1 to 3). Our judgments can be described in relation to each other. We can’t do that with nominal, ordinal or interval scales.

A 0 on a spelling test doesn’t mean that the person cannot spell anything at all, only that those selected words couldn’t be spelled. The zero is not absolute. Similarly, a 0 on a Fahrenheit thermometer doesn’t indicate a total lack of heat (if it did we couldn’t have minus degrees). In contrast, time, distance, and weight are all ratio scales. A 0 on these scales indicates the total absence of that factor.

There are two problems with ratio scales. First, ratio scales are very rare. We often use interval scales (e.g., intelligence scales, reading tests, personality inventories) or ordinal scales (e.g., rating scales), but do not often use ratio scales.

The second problem is that measurement levels often are ignored. It is common for executives, teachers and others to treat ordinal and interval data as if they were on a ratio scale. Rating scales (1 to 5, 1 to 7, 1 to 10) are ordinal in nature. This is important to understand because some people make the mistake of saying that Group A did twice as well as Group B in the last survey.

When our measurements do not meet the assumptions of a ratio scale, we cannot say that a person with an IQ of 140 is twice as smart as a person with an IQ of 70. Nor can we say that a person who scores 0 on our extroversion scale is not extraverted. These are interval scales.

The way we measure determines the strength of the conclusions we draw. If we label horses as “jumpers” and “non-jumpers,” we have not made any assumptions about which is better, only that they are different. This is a nominal scale. Similarly, if we differentiate between managers and engineers at a nominal level, we make no assumptions concerning which status is best.

At an ordinal level, we could rate horses on their jumping ability or personnel on their sales ability. We could use a scale of 1 to 5, for example. Notice, that we could use a two-level scale: thumbs up, thumbs down. The only difference between a two-level ordinal scale and the nominal scale mentioned above is in the assumptions. If we assume that jumpers are better to have than non-jumpers or that sellers are better than non-sellers, the underlying scale is not nominal but ordinal.

Prejudice is a good example of this. Distinguishing between Asians and Whites, north and south, or tall and short is a nominal description. Yet, if underlying our distinctions there is an assumption of one being better than another, we have moved to a different level of measurement.

It should be clear that the number of spots on an ordinal scale is arbitrary. We are still at an ordinal level when rating on a 10-point scale, a 50-point scale or a 87-point scale. It is the underlying assumptions which determine an item’s level of measurement.

To move up to an interval scale in our horse testing, we could set up a course with obstructions for the horses to jump. Again, the number of hurtles included in the course is arbitrary and does not affect the level of its measurement. And, it should also be clear that a score of zero on our hypothetical course does not mean that a given horse cannot jump at all. We may have made all of the jumps too high for any horse to successfully clear.

If we measured how fast each horse ran the course, or how high each one jumped, the measurements would be on a ratio scale. Then, and only then, could we say that one horse jumped twice as high or ran half as fast.

In social science much of our data is ordinal. When we build a test, we usually don’t make each item of equal difficulty (one of the assumptions for an interval scale). Consequently, our measurements are more like rating scales than precision scientific instruments. Although some of our rating systems are quite complex, the data does not allow us to make fine distinctions between people. We can say one person is more generous, skilled or intelligent than another, but not by how much.

Just as horse-jumping courses usually are composed of items with varying difficulty, items of sales ability differ in difficulty. We do it to save time. With a few items of increasing difficulty, we can distinguish between poor performance and great performance. Without thinking about it, though, we have shifted the underlying level of measurement to an ordinal scale.

This shift is not necessarily bad. It allows us to make gross distinctions with only a few items. But researchers should know which level of measurement they are using. Without such knowledge, they are relying on assumptions which might not be true. We should not fool ourselves into thinking that we are measuring with more precision than is actually present.

Clearly, every level of measurement can be useful. Our tests of increasing difficulty are valuable. We don’t have to measure everything as ratio data. We can use nominal, ordinal, interval and ratio data. All are useful. Levels of measurement are themselves nominal: one level is not better than another.

 

NOW YOU CHOOSE:
    Day 1: Measurement
    A Bit More About Measurement
    Even More About Measurement 1
    Even More About Measurement 2
    Even More About Measurement 3
    Basic Facts About Measurement
    Vocabulary
    Quiz 1
    Summary