Interval-Level Measurement

The numerical observations or scores obtained from measuring some definable attribute of a set of objects are at the interval level of measurement if the following three conditions exist:

• 1.
  The order of the numbers corresponds to the rank order of the objects with respect to the attribute being measured.
• 2.
  The difference between any two consecutive numbers on the measurement scale is the same regardless of which pair of adjacent numbers is considered.
• 3.
  The zero point of the number scale used represents an arbitrary origin and does not indicate complete absence of the property being measured.

Scales that satisfy all three conditions are also described as equal-interval scales or equal-unit scores. The diagram in Figure 1 graphically displays a segment excerpted from an equal-interval scale. The letters a, b, c, d, and e in the diagram represent an ordered set of five consecutive numbers such as the examples in each row of Table 1.

Figure 1 Schematic Diagram of a Scale

The first condition specifies that the numbers satisfy the requirements of an ordinal scale including the property of transitivity. This means that the following inequalities hold:

$$e>d>c>b>a.$$

The second condition specifies that all intervals throughout the range of the scale must be the same size. Hence, the following algebraic relationships must be true:

$$\left(b–a\right) = \left(c–b\right) = \left(d–c\right) = \left(e–d\right).$$

The third condition specifies that the numeral zero does not indicate that an object assigned to that value on the scale completely lacks the measured characteristic. Ratio-level scales have equal intervals and a true zero that indicates a complete absence of the target attribute. In contrast, interval-level scales have equal increments but the zero does not designate a null value.

When an equal-interval scale exists, differences in the numbers convey meaning about the magnitude of the differences between objects with respect to the trait or attribute being measured. Hence, users may be justified in making inferences about how much more or less of the trait certain individuals have. Equal-interval scales are especially useful for measuring within-person growth or change.

Because interval-level measurements do not have a meaningful zero, ratio comparisons of the numbers assigned to individual objects are not meaningful. For example, a person who received a score of 60 on an equal-interval scale does not have twice as much of the characteristic measured as an individual with a score of 30. Similarly, an individual who receives a score of 2 does not have half as much of the targeted trait as a person with a score of 4.

However, ratio comparisons of differences in interval scale values are meaningful. For example, in Figure 1, the difference (e − a) divided by the difference (d − b) produces a ratio of 2:1 regardless of which of the four sets of sample numbers are used. This ratio remains the same if a constant (k) is added to each number in the numerator and to each number in the denominator of a ratio of differences. It does not matter what value of k is used as long as the same value is added to each term in the numerator and to each term in the denominator.

...