1. Scales of Measurement

2. What is Measurement?
Loose definition: “the assignment of numerals to objects or events according to rules” (presumably any rules). A numeral is any symbol other than a word.
Strict definition: “any method by which a unique and reciprocal correspondence is established between all or some of the magnitudes of a kind and all or some of the numbers...”

3. Nominal Scale of Measurement
numerals (symbols, such as 0, 1, A, Q #, ) are arbitrarily assigned to name objects/events or qualitative classes of objects/events.
Does not meet the criteria of the strict definition of measurement.

4. Write your Banner ID on your money and throw it in this bag.
I pull out two bills. From the ID numbers I can tell whether or not they came from the same person.
The ID numbers do not tell me how much of something you have.

5. Notation
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
Xi = alias for t(Oi )

6. Ordinal Data m(O1 )  m(O2 ) only if t(O1 )  t(O2 )
That is, there is a positive monotonic relationship between m and t

7. Interval Data m(Oi ) = a + bXi b > 0
That is, the measurements are a positive linear function of the true values.
t(O1 ) ‑ t(O2 ) = t(O3 ) ‑ t(O4 ) if m(O1 ) ‑ m(O2 ) = m(O3 ) ‑ m(O4 )

9. Ratio Data a = 0, that is m(Oi ) = bXi
that is, there is a “true zero point.”
m(O1 ) m(O2 ) = bX1  bX2 = X1  X2
we can interpret the ratios of ratio measurements as the ratio of the true magnitudes
Gas law problems

12. Gay-Lussac’s Law
when you increase the temperature of a sealed container the pressure increases by the ratio of the new temperature to the old temperature.
Aerosol can with pressure of 70 psi at 10 degrees Celsius.
Raise temperature to 30 Celsius.
What is the new pressure.

13. Mistake
30 Celsius is three times as hot as 10 Celsius, so the new pressure will be 3(70) = 210 psi
210 psi is the pressure at which standard aerosol cans rupture.
KA-BOOM!

14. Not to Worry
The ratio of the true temperatures can only be obtain with a scale with a true zero point.
Using degrees Kelvin, 70(303)/283 = 74.9 psi, no problem.
But don’t be stupid enough to throw such cans into a fire.