Scales of Measurement

Data comes in various sizes and shapes and it is important to know about these so that the proper analysis can be used on the data. There are usually 4 scales of measurement that must be considered:

1. Nominal Data
   • classification data, e.g. m/f
   • no ordering, e.g. it makes no sense to state that M > F
   • arbitrary labels, e.g., m/f, 0/1, etc

2. Ordinal Data
   • ordered but differences between values are not important
   • e.g., political parties on left to right spectrum given labels 0, 1, 2
   • e.g., Likert scales, rank on a scale of 1..5 your degree of satisfaction
   • e.g., restaurant ratings

3. Interval Data
   • ordered, constant scale, but no natural zero
   • differences make sense, but ratios do not (e.g., 30°-20°=20°-10°, but 20°/10° is not twice as hot!
   • e.g., temperature (C,F), dates

4. Ratio Data
   • ordered, constant scale, natural zero
   • e.g., height, weight, age, length

Some computer packages (e.g. JMP) use these scales of measurement to make decisions about the type of analyses that should be performed. Also, some packages make no distinction between Interval or Ratio data calling them both continuous. However, this is, technically, not quite correct.

Only certain operations can be performed on certain scales of measurement. The following list summarizes which operations are legitimate for each scale. Note that you can always apply operations from a 'lesser scale' to any particular data, e.g. you may apply nominal, ordinal, or interval operations to an interval scaled datum.

• Nominal Scale. You are only allowed to examine if a nominal scale datum is equal to some particular value or to count the number of occurrences of each value. For example, gender is a nominal scale variable. You can examine if the gender of a person is F or to count the number of males in a sample.

• Ordinal Scale. You are also allowed to examine if an ordinal scale datum is less than or greater than another value. Hence, you can 'rank' ordinal data, but you cannot 'quantify' differences between two ordinal values. For example, political party is an ordinal datum with the NDP to left of Conservative Party, but you can't quantify the difference. Another example, are preference scores, e.g. ratings of eating establishments where 10=good, 1=poor, but the difference between an establishment with a 10 ranking and an 8 ranking can't be quantified.

• Interval Scale. You are also allowed to quantify the difference between two interval scale values but there is no natural zero. For example, temperature scales are interval data with 25C warmer than 20C and a 5C difference has some physical meaning. Note that 0C is arbitrary, so that it does not make sense to say that 20C is twice as hot as 10C.

• Ratio Scale. You are also allowed to take ratios among ratio scaled variables. Physical measurements of height, weight, length are typically ratio variables. It is now meaningful to say that 10 m is twice as long as 5 m. This ratio hold true regardless of which scale the object is being measured in (e.g. meters or yards). This is because there is a natural zero.

Frequently Asked Questions (FAQ)

1. What is a natural zero

   Some scales of measurement have a natural zero and some do not. For example, height, weight etc have a natural 0 at no height or no weight. Consequently, it makes sense to say that 2m is twice as large as 1m. Both of these variables are ratio scale.

   On the other hand, year and temperature (C) do not have a natural zero. The year 0 is arbitrary and it is not sensible to say that the year 2000 is twice as old as the year 1000. Similarly, 0C is arbitary (why pick the freezing point of water?) and it again does not make sense to say that 20C is twice as hot as 10C. Both of these variables are interval scale.