# Significant Figures and Measurement Uncertainty in Scientific Measurements

• Vibhor
In summary, the homework statement states that the period of oscillation of a pendulum is 2.62 seconds. The estimated standard deviation for this period is 0.15 seconds, which means that the period could be anywhere between 2.62 seconds and 2.82 seconds with an accuracy of 50%.
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 ?

## 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

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.

---
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 ?

BvU said:
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?

haruspex said:
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## Δastat ##\pm## Δasyst (syst) which looks really professional

haruspex said:
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 ?

Vibhor said:
So which of the two , (2.62±0.11)s OR (2.6±0.1)s seems better ?
The first.

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 .

Vibhor said:
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.

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

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BvU

## What is an error in scientific measurement?

An error in scientific measurement refers to the difference between a measured value and the true value. It can occur due to a variety of factors, such as limitations in measurement tools, human error, or natural variability.

## Why is it important to consider significant figures in scientific measurements?

Significant figures are important in scientific measurements because they indicate the precision of a measurement. They help convey the level of confidence in a measurement and ensure that calculations are not overestimated or underestimated.

## How do you calculate the number of significant figures in a measurement?

The number of significant figures in a measurement is determined by counting all the digits from the first non-zero digit to the last digit, including any zeros in between. Zeros at the beginning of a number are not significant, while zeros at the end of a number may or may not be significant depending on the context.

## What is the purpose of rounding in scientific measurements?

Rounding is used in scientific measurements to reduce the number of significant figures in a number to a more appropriate level. This is done to reflect the precision of the measurement and to avoid misleading results in calculations.

## How do you handle error and significant figures in scientific calculations?

When performing calculations, the result should be rounded to the same number of significant figures as the measurement with the fewest significant figures. Additionally, any errors from individual measurements should be propagated through the calculation to determine the overall uncertainty in the final result.

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