Once a construct has been operationalized we're ready to start measuring. Unfortunately, measurement is much less straightforward in the social sciences than in the natural sciences, therefore it's extremely important to know what we mean when we say we're measuring depression or political persuasiveness. We should know what we're capturing with the numbers that result from measurement, but more importantly, we should know what information we're not capturing. So what is measurement? Okay, here we go. Measurement is the representation of relations between objects, persons, or groups, on a certain property, by using relations between numbers. Take the property body length. I could determine qualitative relations on this property between three objects, in this case two persons and a cat, by standing them side by side and observing and comparing how far their heads stick out. The first thing I can tell is that they're all of a different length. And I can represent this inequality relation by using different labels or different numbers to represent the inequalities. Of course it would be weird to use the numbers 2, 10, and 14 like this. Because there's another type of relation that we can immediately see which is not reflected in the assigned numbers. I'm talking about the order relation between A, B, and C. Person B is the tallest, so he should receive the highest number, 14. And C is shortest, and so should receive the lowest number, 2. We use the ordering of the numbers to represent the ordering of people and cat in terms of body length, and we don't need to stop there. We can also determine if the difference in length between person A and person B is the same, or larger, than the difference between person A and C. We hold a piece of cardboard over A's head and cut the cardboard where it reaches the top of B's head. Then we hold the piece of cardboard over C's head and compare to A. Suppose the cardboard reaches exactly to A's head. Then the differences in length are the same. We can represent this relation of equal differences by using numbers that differ by the same amount. For example, the difference between the numbers for B and A, 14 and 10, is 4. So we could change the number assigned to C from a 2 to a 6. The equal differences in numbers between B and A, 14 minus 10 is 4. And A and C, 10 minus 6 is 4. Now, accurately reflect the equal differences in body length between A and B, and A and C. If the difference between A and C had been larger than the difference between A and B, then the difference in the corresponding numbers should've reflected this. There's one more type of relation we can observe for the property body length. We can compare ratios of body length. We can take the piece of cardboard we cut out earlier, cut some extra pieces of exactly the same length, and see how many cardboard units C, A, and B are tall. Okay, now suppose it takes two cardboard units to reach the top of C's head, four to reach the top of B's head, and three to reach the top of A's head. This means B is twice as tall as C. We could reflect this relation by changing the numbers again to 9, 12, and 6. We're still using different numbers for different lengths. The ordering of the numbers corresponds to the ordering of body length. The differences between A and B and A and C are the same, 12 minus 9 is 3 and 9 minus 6 is also 3. And now the number for person B is twice as large as the number for C. So you can see that measurement is the representation of empirical relations between objects or persons on a certain property by using the numerical relations between numbers. We can differentiate lengths, order them, compare differences, and ratios of body length. We determine these empirical relations by looking and using some cardboard. Of course, this method is pretty laborious if we want to assess body length for a great number of people. Assigning numbers in a way that captures these empirical relations, for example by using a tape measure, makes our life a lot easier. Especially if we want to compare or aggregate over many people. And, of course, assigning numbers to represent a property allows us to use statistics to help describe and draw conclusions about the property we're interested in.
