Levels of Measurement

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Measurement Hierarchy

Level	Properties	Observations Reflect	Examples	Type of Data
Ratio	True zero Equal intervals Order classification	Measurable differences in total amount	weight income reaction time family size	quantitative
Interval	Equal Intervals Order classification	Measurable differences in amount	Fahrenheit temperature IQ Score* grade point average* verbal aptitude score*	quantitative
Ordinal	Order Classification	Differences in degree	attitude toward abortion academic letter grade movie ratings	quantitative
Nominal	Classification	Differences in kind	sex / gender ethnic background political affiliation major in college	qualitative

*Approximates interval measurement

Nominal Measurement involves sorting observations into different classes or categories

Reflects only differences in kind, not differences in degree or amount
Nominal scale is the most rudimentary form of measurement.

Ordinal Measurement involves classification reflecting differences in degree (i.e. more or less).

If replies to the question "Should all abortions be legal?" are classified as either strongly agree, agree, neutral, disagree, or strongly disagree, measurement is ordinal. Strongly agree represents more agreement than agree; by the same token, agree represents more agreement than neutral, and so forth. No claim can be made about the amount of difference between adjacent categories. Although strongly agree represents more agreement than agree, we don’t know how much more.

Interval Measurement involves classification reflecting equal intervals.

Fahrenheit scale possesses equal intervals; it’s appropriate to interpret the difference between 40° F and 60° F as representing exactly the same amount of heat as does the difference between 60° F and 80° F.

Ratio Measurement permits interpretation of one observation as exceeding another not only by a certain

amount, but also by a certain ratio, such as "twice as much".

The distinctive property of ratio measurement is a true zero. If people are weighed on a bathroom scale, measurement is ratio. Since the bathroom scale possesses a true zero, numerical readings reflect the total amount of a person’s weight, and it’s appropriate to describe one person’s weight as a certain ratio of another’s. It can be claimed that the weight of a 300-pound person is twice that of a 150-pound person.

In absence of true zero, numbers fail to reflect the total amount being measured. For example, a reading of 0° on the Fahrenheit scale does not reflect the complete absence of heat. In fact, true zero equals –459.4° F on the scale. It would be inappropriate to claim that 80° F is twice as hot as 40° F. An appropriate claim could be salvaged by adding 459.4° F to each of these numbers: 80° becomes 539.4° and 40° becomes 499.4° . Clearly, 539.4° is not twice as hot as 499.4° .

Mathematical Preliminaries

Summation Notation

Expression 1: also

- The summation operation is signaled by (Capital Greek S, called "sigma")

- notation below the sigma tells us to start w/person #1

- notation above the sigma tells us to add through to person #5

- X_i to right of sigma is general term stands for individual X scores

Expression 2: also

- This means that each X score should be first squared and then summed.

Expression 3: also

- This is not the same as Expression 2.

- A general rule that we follow throughout this book is to perform any mathematical operations within parentheses before performing the operations outside the parentheses.

- In Expression 3, the parentheses signal that the summation operation should be executed first (that is, the X scores should be summed) and then this sum should be squared.

Expression 4: also

- This means that for each pair of scores, each X score first should be multiplied by its corresponding Y score, and then these products should be summed.

Expression 5: also

- Where c represents a constant. Suppose that c=2. Then this expression indicates that we should subtract 2 from each X score, square each difference, and, last, sum these squared differences.

Expression 6: also

- Where both c and k represent constants. This means for each individual, multiply the difference between X and c by the difference between Y and k, and then sum the resulting products.