How Do You Properly Calculate Uncertainty in Measurements?

[jk494]

I've come across it it at least half a dozen classes in my life but I never really learned how to do uncertainty calculations properly. Right now I am torn between what is taught and what makes intuitive sense. In every book or website there is a different explanation of this concept with different rules of thumb and every time I look up a new source I am confused more.

First of all, I am still not sure what a significant figure is. Some places have told me it is a digit that doesn't change over repeated measurements, others say it's just the number of digits you report in a measurement, then the fractional uncertainty says how many sig figs you should have but the absolute uncertainty almost always changes digits that are "significant".
a.)How far can you guess the measurement? If my ruler only goes to 1mm and I measure something between 1.1 and 1.2cm can I guess a smaller digit?
b.)In general do you estimate past the smallest division on an instrument? Is the absolute uncertainty half of that? How does uncertainty as a standard deviation come in from here? How do you know how many sig figs to keep in the standard deviation?
c.)How does the fractional uncertainty relate to the number of significant figures in a measurement? How does it relate to the size of the round off error? For example you have a measurement of 1.00 which the rules say has 3 sf. It has an uncertainty of 0.05. The fractional uncertainty of this is 5% which says it should only have 2sf. The range of this measurement would be [0.95 1.05] which changes all of the digits in the measurement, so are any of them are actually "significant"?

 

[BvU]

Hello JK,

Handling errors in science is both a science and a craft -- almost an art. Uncertainty analysis is only part of the job, gut feeling and intuition play an important role too.

Significant figures -- just what it says. 6.6 in itself means I believe the 0.6. Somewhere between 6.55 and 6.65. But don't bite my head off if it turns out to be 6.48. After all, only 64% of the Gauss distribution is within $\pm$ 1 $\sigma$

If I report 6.6 $\pm$ 2.2 that means I have done a great number of measurements: I report 2.2 as standard deviation. Now the relative standard deviation of the estimated standard deviation (2.2) is $\approx {1\over \sqrt N}$. So somewhere between 2.15 and 2.25 (2%) would mean 2500 independent measurements -- never happens. That's why we usually report only one digit of the standard deviation (unless that is a 1, sometimes).But when I find $6\pm2$ doesn't reflect my effort properly, I will report $6.6 \pm 2$ but others might not agree.

I could go on for hours about this. Let me first try to answer:

a.)How far can you guess the measurement?

With a good ruler with fine marks about 0.2 divisions. So pretend 0.1 division and report 45.6 mm instead of 46.
Check a few rulers with each other to see how good or bad a ruler can be.
Realize you always do two readings: one to match the 0 with one point and one to match the second point.

b.)In general do you estimate past the smallest division on an instrument? Is the absolute uncertainty half of that? How does uncertainty as a standard deviation come in from here? How do you know how many sig figs to keep in the standard deviation?

You do the best you can. Practice with a ruler, a vernier calliper and a micrometer to check how good you are.

That way you can estimate your own standard deviation for measurements with a ruler. -- provided you avoid systematic errors. Yet another chapter.

Running out of time-gotta go.