Levels of Measurement
What they are;
They are the different ways numbers can be used.
Numbers can be used as tags or labels, where the size of the number is arbitrary. Barcodes and social security numbers are two examples. We could use the name of the merchandise or person equally well, but we choose numbers instead. The fact that the barcode for milk is higher than for razor blades (is it?), or that your SSN is higher than mine ( is it?) tells us nothing. In surveys they often use arbitrary numbers to code variables such as religion, ethnicity, major in college or gender.
In fact, with nominal variables measurement means: classifying cases in (unordered) groups. (A group can have a single member, as it is the case with SSN.) The groups must be
--- all-inclusive: they must cover all cases
--- mutually exclusive: each case must belong to one and only one group
A nominal level variable that can take only two values (yes/no, male/female, Hungarian/non-Hungarian) is called a dichotomy.
Numbers can be also used to order. On most airplanes there are three classes of passengers: first class, business class and economy class (1,2,3). These three categories are
--- all-inclusive and
--- mutually exclusive,
but they also have an order to them. We know that first class is better than economy and that business is in between. Just how much better first class is compared to business, and business compared to economy varies from airline to airline, and even from flight to flight. The groups for ordinal variables are
--- ordered but the distance between two adjacent categories may vary.
Numbers can also be used to express quantities. The amount you pay for your plane ticket, the number of miles you fly, or the degree Fahrenheit at your destination, are all quantities. Here you know that the difference between a ticket price of $100 and $101 is the same as the difference between $99 and $100: a single dollar. There is a ‘yardstick’: in this case, money. Therefore the numbers are not just
--- all-inclusive (all prices can be expressed with a dollar figure) and
--- mutually exclusive (no single price can take two different numbers) and
--- ordered (if you paid $100 and I paid $200 I paid more), but
--- there is a constant distance between any two adjacent categories (a dollar difference is a dollar difference).
There is a further distinction between interval and ratio level variables. Interval variables have an arbitrary 0 point and as a result, you cannot calculate ratios. E.g.: the 0 point on the Fahrenheit scale is arbitrary (it is about –18 degree Celsius). As a result 40 Fahrenheit is not half as warm as 80. The same values on the Celsius scale would be ( +4.5 +26.5 suggesting that the same two levels of heat has a ratio not 1:2 but almost 1:6. Ratio level variables, such as ticket price, miles you fly do have a non-arbitrary 0 point. A free ticket (costing $0) or a grounded plane (traveling no distance) would have an absolute 0 score.