MEASUREMENT

Best reference: Stevens, S. S. (1946). On the theory of scales of measurement. Science, 103, 677-680.

Stevens is the person credited with originating the scales of measurement referred to by researchers, psychometricians, and statisticians. These levels should become familiar to you as they are part of the assumptions to be met by various statistical procedures. When inappropriate analysis procedures are used, the results are meaningless. Unfortunately, there are far too many of these violations in research studies, and your ability to identify them will enable you to assess the degree of confidence you can have in reported results.

Measurement is defined as the assignment of numbers to the result of some instrument (test, attitude scale, physical scale) or more broadly, to objects or events, according to specific rules. Measurement exists in a variety of forms and in four levels. The following must be made explicit: (1) the specific rule for the assignment of numbers, (2) the mathematical properties of the scale, i.e., the level of measurement, and (3) the statistical operations appropriate for that level.

The level of operation is dependent on the character of the basic operations performed. These operations are limited by the nature of the characteristic being measured and how we decide to measure it. The four levels of measurement are: (1) nominal, (2) ordinal, (3) interval, and (4) ratio.

Nominal

There are characteristics that we measure that can only be put in groups or categories that do NOT indicate amount of. For example, eye color is measured most frequently by color, not by degree or amount of a particular hue. We generally have categories of blue, brown, hazel; and this usually is all we need. When we are measuring characteristics in this manner, we are setting all categories as equal in value and are simply differentiating --- not making a value judgment.

Examples of characteristics that are usually measured at the nominal level:

Political affiliation (democrat, independent, republican, no party affiliation)

Religious affiliation (Catholic, Protestant, other)

Gender (female, male)

These are always measured in discrete groups, each subject or object is in one group or another - they can not straddle groups or belong to more than one group. A person is either male or female.

For characteristics measured at the nominal level, only the descriptive statistics of mode, number of cases, contingency correlation are appropriate.

Ordinal

There are characteristics that we measure indicating which subjects/participants have more of an attribute than the other subjects/participants. When we are able to rank subjects/participants or objects, we have progressed from the nominal level to the ordinal level.

Examples of characteristics that are usually measured at the ordinal level:

Academic rank at a university (instructor, assistant, associate, full)

Student's academic class (freshman, sophomore, junior, senior, graduate)

Science Fair contest (first, second, third, honorable mention)

These are always measured in discrete groups, each subject or object is in one group or another - they can not straddle groups or belong to more than one group. A person has won either first, second, third place or honorable mention.

For characteristics measured at the ordinal level, median and percentiles are appropriate in addition to statistics appropriate for the nominal level.

Interval

There are characteristics that we measure that we can not only indicate amounts of in terms of more or less, but also how much more or how much less. When the measurement is of equal intervals, we have progressed from ordinal to interval.

Examples of characteristics that are usually measured at the interval level:

Temperature measured on Centigrade or Fahrenheit scales

Achievement tests

Attitude scales

These may be measured in discrete groups or along a continuous scale. Temperature is considered a continuous scale. Achievement tests may be considered as continuous if they are reported as percent correct, but may be considered discrete if reported as number correct without partial credit.

For characteristics measured at the interval level, most commonly used statistics are appropriate (one exception would be the coefficient of variation).

Ratio

There are characteristics that we measure that we can indicate amounts of in terms of more or less and how much more or how much less in terms of ratios (i.e., twice as much or three times as much). When a characteristic can logically be assumed to have a true zero point and measured with enough precision to allow for comparisons in terms of twice as much of an attribute, it is considered to be ratio level measurement.

Examples of characteristics that are usually measured at the nominal level:

Weight

Number of children

Temperature measured on the Absolute Scale

These may be measured in discrete groups or along a continuous scale. Temperature is considered a continuous scale. Number of children is considered discrete (forget what you heard about the size of the average family being 4.14 - this is a good example of how statistics are inappropriately applied and how it leads to confusion in interpretation). For characteristics measured at the ratio level, all known statistics are appropriate.