# Taring a digital scale improves precision?

Suppose you need to weight something that weights $$x$$ in a digital scale that reads a non zero value a when empty. There are two options:

1. Tare the scale and weight, giving $$x \pm \Delta x_1$$.
2. Weight and substract a from final reading, giving $$x \pm \Delta x_2$$.

When subtracting, error propagation must be performed, so the final uncertainty should increase: $$\Delta x_2 = \sqrt{\Delta x_1^2 + \Delta x_1^2} = \sqrt{2} \; \Delta x_1$$

But when a scale is tared and shows 0.00 g, for example: Is the zero value taken into account in error propagation or when taring the scale it internally fixes an "absolute zero" with negligible uncertainty compared to its precision? In other words: is the zero reading 0.00 g $$\pm \; \Delta x_0$$ with $$\Delta x_0$$ lower than $$\Delta x_1$$?

EDIT: I'm assuming negligible uncertainty related to accuracy and linearity in the range of interest and a stable reading to the last digit.

The answer is probably that it depends on the specific model of the scale. If you care strongly about precision, you should obtain a set of reference weights and conduct a series of experiments to establish that your protocol of poweron-container-tare-weigh satisfies your requirements for precision, accuracy, and linearity in the region that you're interested.

All devices with a digital readout have a minimum uncertainty of $$\pm1$$ unit in the final displayed digit. That's because the analog-to-digital converter that talks to the display must have some hysteresis (usually some version of a Schmitt trigger) so that electronic noise doesn't sometimes make the last digit flicker between adjacent values. That hysteresis means there is a "dead zone" within each displayed value where an adjacent value would be a better approximation of the signal, but the design engineers have decided not to trust that value tell the display to change. It's impossible to know the width of this dead zone without getting into the nitty-gritty of the electronics.

Of course a device can have a worse uncertainty than one unit in the least significant digit. I have a bathroom scale that displays my weight to the nearest tenth of a pound — but it only displays even tenths digits (0.2, 0.4, but never 0.3). Furthermore, that particular scale has terrible reproducibility: if I get off and get back on, I may see the same value or I may see a value that's two or three pounds away. (That problem hit me this summer when I was trying to determine whether a geriatric cat was losing weight. Twenty years ago with an analog scale, using myself as a tare was a robust way to weigh a small animal. But while the two-pound uncertainty in my digital scale is a 1% uncertainty for me, it was a 40% uncertainty for the cat — a problem of catastrophic cancellation.)

In a high-quality scale where digitization noise is the primary source of uncertainty, it's possible that an internal tare could improve your precision by storing the tare value (whether digital or analog) with a higher precision than is displayed. In that case the discretization noise would contribute to your signal once, when you weigh your object-plus-container, rather than contributing separately when you weigh your object and when you weigh your container. But as with so many error-analysis problems, it can be a lot of work to convince yourself that you've correctly identified what you don't know.

• What does the tare button do? Changing the ouput digits would not improve precision, while changing the analog voltage would Sep 20 '20 at 14:15
• I guess that for a stable reading, the Scmitt trigger in ADC suggests an uncertainty of +- 0.5 in the final reading. If there is flickering between two numbers, I would average and keep the uncertainty. Sep 20 '20 at 14:17
• You can only assume an uncertainty of half the least significant digit if the width of the hysteresis loop is zero. Choosing the most optimistic assumption is poor experimental hygiene. As to what the tare button does: I would assume different manufacturers do different things.
– rob
Sep 20 '20 at 17:10

I've been reading some digital scale manuals and convinced myself that taring is the correct procedure in terms of improving precision.

Digital scale manufacturers provide the instrument uncertainty (see for example this model) with a specific protocol: first tare, then measure. In this conditions, instrument uncertainty is $$\pm$$ 1 in the last digit (although the exact number may vary between scales). This means that the zero reading must not be considered in error propagation.

The more specific question of whether taring resets display or changes an internal reference is still open, but exceeds the purpose of my original question. Getting into the nitty-gritty of the electronics is not necessary as long as the manufacturer provides the technical limitations.