We saw that measurement is the representation of relations between people on a certain property, by using corresponding relations between numbers. Consider body length. For this property we can distinguish different body lengths, we can order them, compare differences in body length and even compare ratios. We can use the numerical relations to represent all these relations. For body length, all four possible relations. Inequality, order, differences and ratios can be meaningfully interpreted. Unfortunately, this is not the case for psychological and social properties. For most properties in social sciences, we can only determine some of these relations. The term measurement level is used to indicate what type of relation can be meaningfully interpreted. If the only relation you can determine is that of inequality, distinguishing between values, then we call this a nominal variable. An instrument that can only differentiate between values is said to have a nominal measurement level. Examples are nationality, sex or pet preference. A German has a different nationality than a Brit. Women are of a different sex than men. Dog people have a different preference for pets than cat people or hamster people. One value doesn't represent a greater degree of the property than any other value. They're just different. A German doesn't have more nationality than a Brit. Women don't have more sex, well let's say more gender, than men. And being a dog person doesn't mean you have a stronger animal preference than a cat or hamster person. There's no order to these properties. Ordinal variables allow for differentiation. And ordering of values. Suppose I want to measure math ability and use the number of correct answers on a math test with ten questions. The higher someone’s math ability, the more answers they get right. We can order people's test scores to reflect their order in math ability. But differences or ratios of scores don't reflect differences or ratios of math ability. We have no way of showing that the difference between a score of four and five is the same in terms of different math ability as the difference between a score of seven and eight. Sure, the difference in right answers is the same, but how can we show this corresponds to an equal difference in math ability? We can't. And the same goes for ratios. Someone with a score of ten, doesn't have twice the mental math ability of someone with a score of five. It actually remains to be seen if the test scores are measured at the ordinal level. That is, if they accurately reflect the order in math ability. What if someone with a score of one spent all their time on the hardest question and got it right? Or someone else focused on the easier questions and got a score of three? Only if the questions are equally difficult can we use the test scores to accurately reflect the ordering of students on math ability. In that case the math test is said to measure at the ordinal level. For interval variables, it's possible to distinguish and order values, but also to interpret differences between values. Temperature is a good example. Suppose I'm heating up four pans filled with water on a stove, and I measure temperature with a thermometer in degrees Fahrenheit. A pan of water reading 90 degrees Fahrenheit is hotter than one that reads 80. We can verify this by sticking our hand in. And the same goes for two pans reading 40 and 50 degrees. We can also verify that when we heat up the 80 degree water to 90 degrees the expansion of the liquid like the quicksilver in a thermometer, is the same as the expansion when we heat up water at 40 degrees to 50. So the difference between 80 and 90 and 40 and 50 is the same. We can't say however, that water at 80 degrees Fahrenheit is twice as hot as water at 40 degrees. This is because the zero point for temperature is arbitrarily defined. The value zero doesn't correspond to the absence of temperature. It corresponds to the temperature required to freeze brine or salt water. The Celsius scale defines zero as the temperature at which fresh water freezes. If we consider the same temperatures as before but now in degrees Celsius, we see that 32.2 minus 22.6 is 5.6. Just like 10 minus 4.4 is 5.6, but 26.6 is nowhere near twice 4.4. This is because the scales use different zero points. Unlike interval variables, ratio variables have a non-arbitrary zero point. That's the same for any scale you might choose. Of course, length is an obvious example. The absence of length, zero length is the same whether you measure in inches or in centimeters. Variables measured at the interval or ratio level unfortunately are very rare in the social sciences. One final remark, the structure of a property doesn't have to be fully captured by a measurement instrument. Take age, a ratio property. I can measure age by asking respondents to indicate their age in years, thereby preserving the ratio level. I could also ask them whether they are under 20, 20 to 39, 40 to 59, or 60 or older, assigning the scores 1, 2, 3, and 4. By creating age categories we no longer know exactly how old someone is. We can say that people in a higher category are older, but not by how much. By categorizing the variable we've lost the ratio and interval information.
