# Do we lose significant figures with digital instruments?

With digital scales, we get an answer to the hundredths or thousandths place, but we do not have the ability to estimate "between the lines". Is the real significance of a measurement different when it comes from a digital scale, or thermometer or other device than when it is read with the possibility of visual interpolation?

As an example, I might have a digital scale that gives a reading of 25.36 grams. It could be anywhere between 25.355 and 25.365 grams. If I read a sliding scale to 25.36 grams, the 6 is an estimate and the true value could range by a few hundredths of a gram while the digital scale at most would have a range of +/- 5 thousandths of a gram.

I realize that you won't have a sliding scale that can match a digital scale anyway, but the same could occur with digital thermometers reading to 0.1 degrees and indicating a range of true value of +/-0.05 while a reported value to the nearest 0.1 degree on a thermometer could be several tenths of a degree off. So is there any acknowledgement that the reported values have different meanings with regard to significance?

• As far as I know, we aren't supposed to interpolate visually, but take the complete reading. No +/- estimation. The provision to compensate for this is to do an error analysis - regarding the least count of the instruments you use and all. – Eashaan Godbole Dec 12 '16 at 4:09
• Don't forget that digital instruments record the number in binary anyway. How would you represent $0.1$ in binary? $0.0001100110011\ldots$ And if you were to truncate because there's not enough memory in the balance to represent the infinite number of repeating figures, then it's not $0.1$ is it? – Zhe Dec 12 '16 at 4:35
• Related answer concerning interpolation: Using sigfigs when measuring with an instrument with marks other than powers of ten? – Loong Dec 12 '16 at 13:05
• @Loong, I thought there was something like that-a quantum model of graduated cyllinders. We are really just recognizing that the application of gradation divisions are an estimate. I remember having seen the terms "readability" on a few graduated cylinders and the readability was in values like those in the third column on the table. I have had some 10 mls with 0.1 ml gradations though. I assume no estimation allowed between the lines though they are class A. Also some 100s with 1 ml gradations. – Joseph Hirsch Dec 12 '16 at 15:23
• One thing I will mention is that the ability to estimate to 0.5 ml with 1 ml gradations only causes the retention of even more sig figs with such a tool versus a digital tool. We retain an additional sig fig even though our precision is only to the nearest .0 or .5. So hypothetically, 25.35 on a sliding scale would be anywhere between 25.325 and 25.375 (a range of 0.05) while 25.35 on a digital scale would be between 25.345 and 25.355 (a range of 0.01). – Joseph Hirsch Dec 12 '16 at 15:36

As far as I know the visual interpolation of analogous measurements is bad scientific practice anyway. We assume that rounding errors work in both directions and compensate each other in the long run. On the other hand visual interpolation by a human can introduce systematic errors.

But putting this discussion away one has to note that even if you visually interpolate, you still digitalise your data (just with a finer grid). By digitalising I do not mean that you write them into a computer, but the abstract mapping of continous data points of $\mathbb{R}$ into a finite subset.

In the end it just is a question of how fine the grid is you are measuring with. And yes, if you don't introduce systematic errors a visual interpolation makes your grid a little bit finer. On the other hand switching to electronic devices usually gives you orders of magnitude better results.

Some examples:

Let's assume that someone wants to measure lengths of $\mu$m with a normal ruler that has 1 mm spacing. It is intuitively clear that this won't work. (Even if you can interpolate.)

Now let's look at another example that was brought up in the comments by @Zhe. The problem that every number in your computer is an element of a finite set. This introduces rounding errors, when saving real numbers (real as in $\mathbb{R}$). If you execute the following python code or something similar in your language of choice, you will see this rounding error.

import sys
print(sys.float_info.epsilon)
# for python2:
# print sys.float_info.epsilon


On my 64bit machine this is in the order of $10^{-16}$. The standard deviation of the meter is in the order of $10^{-11}$ (wikipedia). Floating point noise will rarely be a problem, when dealing with experimental lengths.

This kind of analysation has to be done for every experimental measurement device you are using. Regardless of electronics vs. non electronic measurements. The advantage of "old analogous" measurements is that you can intuitively see from the scale how fine the grid is, that you are working with.

• Just wondering, are digital scales designed to "round" to the nearest final decimal place on the display or do they trigger the next value when a threshold is passed. For example, a stopwatch will read 0.01 s only after 0.01 s, and before 0.02 s. How is a scale designed so that it reads 0.01 grams between 0.005 and 0.015 grams rather than when the 0.01 gram threshold is passed? – Joseph Hirsch Dec 13 '16 at 0:59
• Also, while the description of the problem of interpolation makes sense when viewed from the point of view that the tools (like a graduated cylinder) are imprecise, not just the reading of the tool. Still, I am surprise because MANY college level chem texts describe the method of estimating between the lines-taking the size of smallest increment on the device, dividing it by 10, and then multiplying that by the fractional distance of the reading between the two increments. – Joseph Hirsch Dec 13 '16 at 1:08
• Your first comment: That is implementation dependent, but rounding would make more sense. Your second comment: I can just say what I learned in my lab practicals about visual interpolation. The interpolation scheme you mentioned makes definetely sense to me, but still it may open up the possibility of beautifying data by a human. – mcocdawc Dec 13 '16 at 15:07
• It has some base 10 bias (people don't visually divide a span into 10ths very naturally-maybe quarters) and also I think there is a tendancy NOT to chose a value that coincides with an increment mark if the visual does not put the value indiscernibly close to the line. For example, if one increment is 0.1 and the next is 0.2, observers will tend to avoid reading a value as being 0.10 (or 0.20) because they can detect that it is not precisely on the line, but they can not detect that it is still closer to the increment line than the next 0.01. "Off the line" readings may dominate. – Joseph Hirsch Dec 13 '16 at 15:23