Levels of Measurement
It is easy think of all numeric user research data as being equivalent. For the most part numbers seem to be, well … numbers. In reality, however, we process bits of information in different ways using different sorting and measuring rules. Although we often do not stop to think about it in this way, recording and classifying of data is always done according to a scheme.
There are four levels of data measurement. Ranked from top to bottom in order of complexity and information content these are:
- Ratio data
- Interval data
- Ordinal data
- Nominal data
Each level of measurement is characterized by its properties. Nominal measurement has just one property: CLASSIFICATION. Ordinal measurement has two properties: CLASSIFICATION and ORDER. Interval measurement has three properties: CLASSIFICATION, ORDER and EQUAL INTERVALS. Ratio data has four properties: CLASSIFICATION, ORDER, EQUAL INTERVALS and TRUE ZERO. Because the higher levels of measurement contain more properties and more information, they permit a wider variety of interpretations. For this reason here is an important rule:
Always assign the highest permissible level of measurement to a given set of observations.
It is important to understand the differences that follow because different kinds of data require different kinds of statistical tests in order to evaluate them.
Observations reflect: differences in kind
Examples: gender, ethnic background, political affiliation, handedness, major in college
As you can see nominal measurement is simply concerned with sorting observations into categories. Because the single property of nominal data is classification it tells us nothing about differences in degree or amount. Numbers assigned to categories (as identification codes) have no numeric value (we cannot add, subtract, divide or multiply nominal data) and any ordering of categories is arbitrary. This is the most primitive form of measurement. The presence vs. absence of something is a form of nominal measurement (“do you smoke?” YES, NO). Although it is considered a form of measurement the collection of nominal data is more easily thought of as a sorting method.
Properties: classification, order
Observations reflect: differences in degree
Examples: Likert scale categories, rankings, academic letter grade, stages in development
The distinctive property of ordinal measurement is order. On a typical Likert Scale “strongly agree” represents more agreement than “agree”. However, we do not know how much more. Similarly if Jerry Seinfeld is ranked 1st for funniness, and David Letterman is ranked 15th we have no way of knowing how much funnier Seinfeld is than Letterman. We cannot assume that he is fifteen times funnier. He may be more or less than fifteen times funnier. But we do know that he is more funny than Letterman, and more funny than the comedians ranked 2nd through 14th places as well. We know about order but we have no information about the size of the interval between points.
Properties: classification, order, equal intervals
Observations reflect: measurable differences in amount
Examples: IQ scores, degrees of temperature, magnitude estimation scales*
Essentially, interval data are ordinal, but they have an extra property - the ability to meaningfully add and subtract measurements. In interval-scaled data, the gaps between the numbers are comparable, unlike with ordinal data. Any interval has the same meaning regardless of its location on the scale. "X is five inches longer than y" has meaning regardless of the values of X and Y. However, ratios are meaningless on an interval scale because an interval scale has no true zero. Temperature scales are an example of this, so are decibel scales. Zero degrees Fahrenheit does not mean the total absence of temperature. Zero decibels does not mean there is no sound. Furthermore, if it is 80 degrees outside today and it was only 40 degrees outside yesterday we cannot say that today is twice as hot as yesterday. Similarly a sound level of 80 dB is not twice as loud as a sound level of 40 dB. In short, if the data can be ordered and the arithmetic difference is meaningful, then the data are at least interval data.
* Note: Ratings on a continuous 1-10 or 1-100 scale are a form of magnitude estimation and they approximate the properties of interval data. We can use the statistical tools appropriate for interval data to analyze these data because there is an underlying assumption that values given on such a rating scale are considered to have equal intervals.
Properties: classification, order, equal intervals, true zero
Observations reflect: measurable differences in total amount
Examples: weight, income, family size, number of cows in a field
Ratio data are the highest form of data measurement and the form we are most familiar with. For ratio data both differences and ratios are interpretable. Ratio data have a natural zero. Examples of ratio scale data are number of computers you own, weight, height, a bank balance, number of people watching a movie, goals scored by Brazil in the World Cup, etc. Ratio data look a lot like interval data. However, the zero point has a special meaning in ratio-scaled data: it indicates the absence of whatever property is being measured. Ratio data always have the flavor of counting: when you measure the amount of money that you have, you are counting up coins and bills. When you are measuring your height, you are counting the number of inches off the ground to the top of your head. Both ratio and interval data make use of a wide range of statistical analysis tools.
Shifts to more complex levels of measurement are accompanied by more informative observations that in turn permit a wider variety of interpretations and statistical analyses. Even though the numerical measurement of some non-physical characteristics (e.g., measures of intelligence, measures of satisfaction on a continuous rating scale etc.) may fail to attain the true characteristics of interval or ratio data, they are often treated as approximating at least interval data.
And the rule again: Always assign the highest permissible level of measurement to a given set of observations. Here is an example: a list of annual incomes should be designated as ratio data because $0 signifies the complete absence of income. It would be incorrect to treat annual income as interval data even though a difference of $1000 always signifies the same amount of income (equal intervals); or as ordinal data even though different incomes can always be ranked as more or less (order); or as nominal data even though different incomes always represent different classes (classification).
Philip Hodgson, May 2003