# Indices and Scales

The word "scale" is used in a number of different ways in research contexts. One meaning of scale is a device that measures things. In this case, I'm particularly interested in scales that are constructed by combining the answers to a number of questions on a questionnaire into a single value.

## Tests

An interesting example of a scale (or index) is an exam. The exam is meant to measure your competence in a topic. Instead of asking the student "On a 1 to 100 scale, how much of the material do you know?", we ask the student the answers to a whole series of questions. We then add up the number of right answers to get a single value for each student that represents how much they know.

## Underlying Attitudes

Where scales are particularly handy is when we want to measure underlying attitudes such as "liberalness" or "feminism". In some cases, we could probably just ask respondents "On a 0 to 100 scale, how liberal are you?". But respondents cannot always be trusted to give very good answers, partly because they don't know (or don't want you to know), and partly because they have no way of calibrating their scale with that of other respondents.

An alternative approach is to ask them a bunch of questions that indicate how liberal they are. For example, you might ask:

 Are you in favor of a woman's right to an abortion? Do you support gun control? Should firms hire based on affirmative action principles? etc. Yes Yes Yes No No No

If you had a total of 20 questions like this, the proportion of yes's that a respondent gives can be used as a measure of how liberal they are. A person who answers YES to every question is as liberal as you can get. A person who says YES to none of them is not at all liberal.

The individual questions used in this way are known as items. The collection of questions is called a scale. The term scale also refers to the actual total scores obtained by each respondent (its confusing, I know!).

## Indexes and Scales

All scales are indices, but not all indexes are scales. An index really just means that it is a measurement that is constructed by summing up other, simpler, measurements. A scale is an index that in some sense only measures one thing. For example, a final exam in a given course could be thought of as a scale: it measures competence in a single subject. In contrast, a person's gpa can be thought of as an index: it is a combination of a number of separate, independent competencies. In a larger sense, an index does measure one thing: scholastic achievement. But since scholastic achievement is a function of a number of independently varying competencies (a person can be good at math but not good at history), it is not a scale.

## Validity

How do you know whether the scale really measures liberalness? Maybe what most of the questions really indicate is willingness to have the government set rules limiting behavior (like questions 2 and 3). Ultimately, there is no way to be sure. But we can check it a number of ways.

### Face Validity

This means, do the items seem like they refer to liberalness? A question like, 'Are you a liberal?' has a lot of face validity.

### Correlational Validity

If the scale really measures liberalness, then it should correlate with things that liberalness correlates with. For example, we would expect that people scoring high on the* liberalness scale should be more likely to vote for Clinton than for Dole. Similarly, if young people are known to be more liberal, than we expect the liberalness score to correlate (negatively) with age.

### Inter-Item Reliability

A key feature of a scale is that it measures only one thing. What does this mean? One thing it can mean is that the all the items are highly correlated. This means that if you answer positively on question in the scale, you also answer positively on the other items of the scale. Similarly if a person answers negatively on a question in the scale, they also answer negatively on most of the other items in the scale.