# Tolerance value as part of formula

I work for a quartz resonator manufacturer, and we specify the parabolic temperature coefficient as

Δf/f_o = -0.0340±0.006ppm/°C²

The units on $$\Delta f$$ (the absolute parabolic coefficient) are Hz ppm/$$^o$$C$$^2$$. The units on $$f_o$$ (the resonance frequency) are Hz.

Recently somebody in our organisation has expressed a view that the formula doesn't make mathematical sense due to orders of operation. And we should place brackets around the constant and the tolerance value.

I am of the opinion that the tolerance value doesn't form part of the formula and is recognised to only apply to the constant.

This has now grown to involve far to many people and is taking far to much of my time.

What are peoples opinion on this from an engineering point of view. Should the parabolic coefficient be changed to Δf/f_o = (-0.0340±0.006ppm)/°C²

Chris

• Comments are not for extended discussion; this conversation has been moved to chat.
– hazzey
Dec 2, 2021 at 22:58
• I guess I don't understand the issue. How is this different than having a unit like acceleration $m/s^2$ ? My vote is that the tolerance makes sense as is, unless I see more explanation of the alleged difficulty. Can you show a formula where the quantity is used? Dec 4, 2021 at 4:09

Let's try this:

$$(A)\Delta f = (B)f_o*(-0.00340 \pm 0.006)ppm/^oC^2$$

$$(Constant)HZ ppm/^oC^2 = (Variable)HZ(Range)ppm/^oC^2$$

Where A = Coefficient; B is variable.

I agree with the change. Or $$\Delta f/f_o = -0.00340 \pm 0.006$$, $$ppm/^oC^2$$ & $$\Delta f/f_o = -0.00340 \pm 0.006$$ $$(ppm/^oC^2)$$ should both be fine too, though none of the expressions will change the result, but more clear.

Accepting the units on the individual terms, we are left with the following approaches:

$$\Delta f/f_o = -0.00340 \pm 0.006$$ ppm / $$^o$$C$$^2$$

$$\Delta f/f_o = -0.00340 \pm 0.006$$ (ppm / $$^o$$C$$^2$$)

$$\Delta f/f_o$$ (ppm / $$^o$$C$$^2$$) $$= -0.00340 \pm 0.006$$

Enclosing () on the number and only one term on the units is incorrect. Enclosing () on the number itself may not be incorrect but is redundant. You must put a space between a value and its units except as noted here.