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 ERROREvidentlyA= 1001. 3.7B= -0.1501 0.44E-03NOx-errorsy-errors based on formula: SQRT(30*Y)/30

[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:

- Numbers that are added, subtracted or equal to each other must have the same units.
- The standard mathematical functions (sin, cos, exp, log, etc.) produce a unitless output and must operate on a unitless argument.

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.)

In the case of the natural log function *y*=*B* log(*x/A*),
*B* must have the same units as *y* (since log produces
no units) and *A* must have the same units as *x* (since log
must operate on a unitless argument).

The power-law function *y*=*A* *x*^{B}
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* *x*^{B}

can fit the 25 data points with a reduced chi-squared of 1.2

FIT PARAMETER VALUE ERRORA= 1.001 0.12E-02B= 1.999 0.73E-03NOx-errorsy-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. The units of
*A* will then be as normal as the units of *x*.
(If, on the other hand, the fit is intended as an ad hoc, phenomenological
approximation of data ("saving the phenomena"); feel free to continue
with the best possible *B* value.)

Generally folks are careful to avoid nonsense like sin(9.8m/s^{2}), but
people (including me) not uncommonly abuse log. For example:

pH = -log_{10}([H^{+}])

When log is being abused like this switching units just produces a uniform offset:

log_{10}(*x* in cm) = log_{10}(100**x* in m) = log_{10}(100) + log_{10}(*x* in m)
= 2 + log_{10}(*x* in m)

Thus in phenomenological fits using something like *y*=*A*+*B* log(*x*), the *B* parameter
is actually independent of the units used for *x*, while switching *x* units
shifts *A* by a constant. (This can cause all sorts of confusion as (1) usually unit conversion
involves multiplication not subtraction and (2) the units of *x* typically are absent in
*A*, find fact it must be [y]=[A]=[B].)
Similarly the standard deviation of a set of numbers { log(*x*_{i}) }
does not depend on the units of *x* and is itself properly unitless. Confusing but true!