Significant Figures and Measurement Uncertainty in Scientific Measurements

[Vibhor]

Homework Statement

We measure the period of oscillation of a simple pendulum .In successive measurements ,the readings turn out to be 2.63 s , 2.56s , 2.42s , 2.71s and 2.80s . What is the period of oscillation taking into account appropriate significant figures ?

Homework Equations

The Attempt at a Solution

Mean period of oscillation T = ( 2.63 + 2.56 + 2.42 + 2.71 + 2.80 ) / 5 = 13.12/5 =2.62s

Arithmetic mean of absolute error ΔT = ( 0.01 + 0.06 + 0.20 + 0.09 + 0.18 ) /5 =0.11 s

This gives time period = 2.62 ± 0.11s

But , since the arithmetic mean is 0.11 s there is already an error in the tenth of a second . Hence there is no point in giving the period to a hundredth .

Now , my doubt is whether time period should be expressed as (2.62 ± 0.11s) OR (2.6 ± 0.1s) .

Thanks

 

[BvU]

You have five observations; average is 2.624, estimated standard deviation $\sigma$ is 0.15 (excel stdev.s), so the estimated standard deviation in the 2.62 is 0.065 ($\sigma_m = \sigma/\sqrt N$) .

The relative accuracy of $\sigma$ is $1/\sqrt 5$ so about 50%. Your best shot is $\ 2.62 \pm 0.07$ seconds.

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Aside from that:
Numerically, you ended up with an error estimate with first digit 1. I learned that in such a case one gives two digits since the step (0,1,2) is too big. But: With only five observations there is a lot to be said in favor of giving only one digit anyway.

And another aspect: if you are going to do something with this result (plot T as a function of some length, calculate g, etc.), the remainder of your expose will be easier to follow with 2.62 +/- 0.11 than with 2.6 +/- 0.1 . And if you don't leave out mentioning the number of observations, a good reader understands.

Somewhat a matter of taste; I'd love to read other arguments. Any takers ?

 

[haruspex]

I'd love to read other arguments

My view is that rounding to some number of digits in order to indicate the accuracy is merely a way to imply the range when no other indication is provided. If you are going to specify a range explicitly, you can, and should, show more digits for the best estimate (usually the middle of the range).
I didn't understand the remark about relative accuracy. Isn't it just a question of deciding how many standard deviations constitute a +/- error range? Two or three, maybe?

 

[BvU]

I didn't understand the remark about relative accuracy. Isn't it just a question of deciding how many standard deviations constitute a +/- error range? Two or three, maybe?

I learned (yes, long ago...) that if you report only a +/- Δa then that's the sigma. If you want to report a range then that can be 2 or 3 sigma, but then you include a confidence level (e.g. 95% C.L) .

The relative error remark was on the relative error in the estimate of $\sigma$ from a sample. It is $\approx 1/\sqrt N$. You need a sample size of 10000 to get an error of 1% in the $\sigma$ !

Life becomes even more inteesting if you have a systematic error to report; you can fold it into the statistical error, but you can also write a $\pm$ Δa_stat $\pm$ Δa_syst (syst) which looks really professional

 

[Vibhor]

My view is that rounding to some number of digits in order to indicate the accuracy is merely a way to imply the range when no other indication is provided. If you are going to specify a range explicitly, you can, and should, show more digits for the best estimate (usually the middle of the range).

So which of the two , (2.62±0.11)s OR (2.6±0.1)s seems better ?

 

[haruspex]

So which of the two , (2.62±0.11)s OR (2.6±0.1)s seems better ?

The first.

 

[Vibhor]

From the OP :

But , since the arithmetic mean is 0.11 s there is already an error in the tenth of a second . Hence there is no point in giving the period to a hundredth .

Do you mind explaining why you didn't like the above reasoning ?Please explain as simply as possible .

 

[haruspex]

From the OP : Do you mind explaining why you didn't like the above reasoning ?Please explain as simply as possible .

It doesn't hurt to show the extra .01s, and it is closer to the truth. If you write +/-0.1s then you are not quite encompassing the range. But to write +/-0.2s would be unnecessarily cautious. Maybe you think that +/-0.1s should be interpreted as per the sig figs convention, i.e. as really meaning +/-(0.1+/-.05), but that just gets silly.
Suppose you had computed the error range as +/-0.123. In that case, I think it would be reasonable to write it as +/-0.13. The extra 0.007 is insignificant, and it does encompass the computed error range.

 

[Tom.G]

Since you are trying to get a handle on measurement uncertainity, here's a link to a somewhat dated version of the international standard.
I haven't checked lately, there may be a later version. NB. It's about 7 megabytes zipped.

GUM - Guide to expression of uncertainity in measurement2008 & Intro & Supplement 1

URL: http://www.bipm.org/utils/common/documents/jcgm/JCGM_pack_2010-03.zip
(My download Date was: 2011/02/14 15:00)