Gaining Perspective on Valid Digits in Physics Problems

[johann1301h]

TL;DR Summary

Discussion about number of valid digits

I am trying to gain some perspective into the topic of «number of valid digits» when doing physics problems.

I have been doing physicsproblems for å long time, but i have never really understood why the rules regarding number of digits to include in the final answer - is the way that it is.

I think the core principle behind it all is this;

DONT OVERREPRESENT DATA

And i think this again is a principle sprung out of emperical idealism/philosophy.

But then i ask, why is it so wrong to «OVERREPRESENT DATA»

Take for example the rating system at IMdB.com, for rating movies. Each visitor has a chouse between 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 stars to rate a movie. Not for instance 9.2. But yet the rating «of shawshank redemtion» is 9.2, even though not one single visitor gave it 9.2 stars. This is clearly(?) an overrepresentation of data. Why is it right or acceptable to use more digits then is in the data in this case, but not when you are for example measuring the magnetic field strength around the earth, or any other emprical measurment?

 

[Vanadium 50]

This is clearly(?) an overrepresentation of data.

Why? Movie A gets 100 9's and Movie B gets 50 9's and 50 10's. Which is the better regarded movie?

 

[scottdave]

This is the way I understand it: When people select stars, that is an exact quantity, so theoretically there are an infinite number of zeros after the decimal place. Where significant figures is important is when something is measured.
So there is precision & accuracy. I'm on my phone, so I won't look them up at the moment, but I believe accuracy has to do with how well a measuring device, such as a scale behaves, and Precision has to do with what level (how many digits) you can read from the device.
If you weigh something on a scale - Suppose the scale has a level of precision of 1 gram. If it reads 50 grams, you don't really know if it is 50.49 or 49.51 do you? If you decide it is 50.000 and do some calculations, then report a bunch of digits, those extra digits are meaningless.

 

[gmax137]

If you really want to get into it, find a copy of John Taylor's book, Introduction to Error Analysis.

 

[Tom.G]

Or if you want to dig really deep, try "Guide to the Expression of Uncertainty in Measurement"
https://www.bipm.org/en/publications/guides/gum.html
Have fun. It will either keep you awake at night or put you to sleep!

 

[olgerm]

the more digits you use to describe a quantity the more accurate your description of result of measurement is and the smaller the mean of error is. the maximum number of digits is only for readability.

 

[Redbelly98]

Take for example the rating system at IMdB.com, for rating movies. Each visitor has a chouse between 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 stars to rate a movie. Not for instance 9.2. But yet the rating «of shawshank redemtion» is 9.2, even though not one single visitor gave it 9.2 stars. This is clearly(?) an overrepresentation of data. Why is it right or acceptable to use more digits then is in the data in this case, but not when you are for example measuring the magnetic field strength around the earth, or any other emprical measurment?

In your example, "9.2" is not necessarily an overrepresentation of data.

Suppose there are 10 raters, with 8 of them rating the movie as 10 and 2 of them rating it as a 9. Now let's compute the average rating, using the standard rules for significant figures.

Step 1: Add the individual ratings, which are all accurate to the "1s" place. The sum is 92, and we consider that accurate to the 1s place because, when we add numbers, we keep the precision -- e.g. 10s, 1s, 0.1s, etc. place -- of the least precise individual number being added.

Note that "92" has 2 significant figures, even though some ratings (the 9s) each had just 1 sig fig.

Step 2: Divide 92 by 10, the number of raters. We keep the 2 sig figs in 92, and note that the 10 is an exact number, the number of raters, and report the result as 9.2.