WAPP+: Units of Parameters

If you use WAPP+ you will end up with a report that starts:

An analysis of the data submitted indicates that a function of the form:
--Exponential-- y=A exp(Bx)
can fit the 25 data points with a reduced chi-squared of 1.2 

                FIT
 PARAMETER     VALUE        ERROR
   A =          1001.        3.7    
   B =        -0.1501       0.44E-03
 
 NO x-errors
 y-errors based on formula: SQRT(30*Y)/30
Evidently A is about 1000 and B is about -.15, but 1000 what? These reports are meaningless if we don't know the units of the results! Clearly the units of the found parameters depend on the units of the numbers you entered and the selected functional form. Lets denote the units for the entered x and y numbers as (respectively) [x] and [y]. In this particular case, we find below:

[A]=[y]

[B]=[x]-1

So if x was recorded in minutes and y in counts/sec,

A = 1001 ± 4 counts/sec

B = -.1501 ± 0.0004 min-1

For any functional form, you can determine the units of the parameters by applying two rules:

  1. Numbers that are added, subtracted or equal to each other must have the same units.
  2. The standard mathematical functions (sin, cos, exp, log, etc.) produce a unitless output and must operate on a unitless argument.
In the above case, since exp() is unitless by Rule 2a, the units of A must equal the units of y according to Rule 1. By Rule 2b, (Bx) must be unitless, so the units of B must cancel the units of x when the two are multiplied, hence [B]=[x]-1.

The linear function y=A+Bx provides an even simpler example. Since A and y are set equal to each other they must have the same units. (Of course, A is the y-intercept and so must be measured in y units.) Since A and Bx are added together, they must have the same units. SO:

[B] · [x] = [A]

[B] = [A]/[x] = [y]/[x]

(Of course, B is slope and so must have units of rise/run.)

The natural log function y=A+B log(x) provides a tricky example. First, if x is unitless we have an easy case and [y]=[A]=[B]. However, if x is not unitless, we have an impossibility: taking the log of units, and so the functional form must be essentially wrong. (Of course, WAPP+ doesn't know the units you entered so YOU have to re-write the functional form.) There must be something dividing out the units of x before the log occurs. The functional form must really be:

y=B log(x/a)

y=B (log(x) - log(a))

y= -B log(a) + B log(x)

That is:

log(a) = -A/B or

a= exp(-A/B)

and of course, [a]=[x] and [B]=[y].

The power-law function y=A xB provides an interesting example. Consider the below real data:

An analysis of the data submitted indicates that a function of the form:
--Power-- y=A xB
can fit the 25 data points with a reduced chi-squared of 1.2 

                FIT
 PARAMETER     VALUE        ERROR
   A =          1.001       0.12E-02
   B =          1.999       0.73E-03
 
 NO x-errors
 y-errors based on formula: .003*Y

By Rule 2, the exponent B must be unitless, and so we have the unit equation:

[A] · [x]1.999 = [y]

[A] = [y] · [x]-1.999

While there is nothing to rule out this possibility, the laws of physics (as currently known) do not involve such funny powers of units; integer power of units: yes, common fractional powers of units: occasionally, powers of 1.999, not yet (and my bet is on never). So if this functional form is intended as a law of physics your best bet is to re-fit with B held fixed at 2. (If, on the other hand, the fit is intended as an ad hoc, phenomenalogical approximation of data ("saving the phenomena"); feel free to continue with the best possible B value.)