Scales of Measurement Let: m(Oi ) = our measurement of the amount of some attribute that object i has t(Oi ) = the true amount of that attribute that object i has For a measuring scale to be ordinal, the following two criteria must be met: 1. m(O1 ) m(O2 ) only if t(O1 ) t(O2 ) If two measurements are unequal, the true magnitudes (amounts of the measured attribute) are unequal. 2. m(O1 ) > m(O2 ) only if t(O1 ) > t(O2 ) If measurement 1 is larger than measurement 2, then object 1 has more of the measured attribute than object 2. If the relationship between the Truth and our Measurements is positive monotonic [whenever T increases, M increases], these criteria are met. For a measuring scale to be interval, the above criteria must be met and also a third criterion: 3. The measurement is some positive linear function of the Truth. That is, letting Xi stand for t(Oi ) to simplify the notation: m(Oi ) = a + bXi, b > 0 Thus, we may say that t(O1 ) - t(O2 ) = t(O3 ) - t(O4 ) if m(O1 ) - m(O2 ) = m(O3 ) - m(O4 ), since the latter is (a + bX1) - (a + bX2) = (a + bX3) - (a + bX4), so bX1 - bX2 = bX3 - bX4. Thus, X1 - X2 = X3 - X4. In other words, a difference of y units on the measuring scale represents the same true amount of the attribute being measured regardless of where on the scale the measurements are taken. Consider the following data on how long it takes a runner to finish a race: Runner: Joe Lou Sam Bob Wes Rank: 1 2 3 4 5 (True) Time (sec) 60 60.001 65 75 80 Copyright 2010, Karl L. Wuensch - All rights reserved. Scales.doc 2 These ranks are ordinal data. In terms of ranks, the difference between Lou and Sam is equal to the difference between Joe and Lou, even though the true difference between Lou and Sam is 4999 times as large as the true difference between Joe and Lou. Assume we used God’s stopwatch (or one that produced scores that are a linear function of her stopwatch, which is ultimately a matter of faith) to obtain the TIME (sec) measurements above. As Interval data, the amount of time represented by 65 minus 60, Sam vs Joe, is exactly the same as that represented by 80 minus 75, Wes vs Bob. That is, an interval of 5 sec represents the same amount of time regardless of where on the scale it is located. b m t Note that a linear function is monotonic, but a monotonic function is not necessarily linear. A linear function has a constant slope. The slope of a monotonic function has a constant sign (always positive or always negative) but not necessarily a constant magnitude. In addition to the three criteria already mentioned, a fourth criterion is necessary for a scale to be ratio: 4. a = 0, that is m(Oi ) = bXi, that is, a “true zero point.” If m(O) = 0, then 0 = bX, thus X must = 0 since b > 0. In order words, if an object has a measurement of zero, it has absolutely none of the measured attribute. For ratio data, m(O1 ) m(O2 ) = bX1 bX2 = X1 X2 Thus, we can interpret the ratios of ratio measurements as the ratio of the true magnitudes. For nonratio, interval data, m(O1 ) m(O2 ) = (a + bX1 ) (a + bX2) X1 X2 since a 0. When you worked Gas Law problems in high school [for example, volume of gas held constant, pressure of gas at 10 degrees Celsius given, what is pressure if temperature is raised to 20 degrees Celsius] you had to convert from Celsius to Kelvin because you were working with ratios, but Celsius is not ratio [20 degrees Celsius is not twice as hot as 10 degrees Celsius], Kelvin is. 3 Additional Readings See the two articles in BlackBoard, Articles, Scales of Measurement Ratio versus Interval Scales of Measurement -- a graphical explanation of the difference PSYC 2101: Howell Chapters 1 & 2 – a document from my undergraduate class, includes material on scales of measurement and other very basic concepts. Copyright 2010, Karl L. Wuensch - All rights reserved.