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Error Analysis :-

Q. I want to know that how error in a measurement is associated with least count of measuring device ?

If I take a measurement with Milimeter scale what the Error I have to assume in measurement :-

1.) one tenth of least count i.e. 0.1mm

2.) equal to least count i.e. 1mm

3.) Half of least count i.e. 0.5mm

4.) Depends on situation

References:-

1.) One tenth of least count : OP writes that "Say you have a ruler with centimeter and millimeter markings. You measure the length of a pencil, and it comes out to somewhere in between 8.6 cm and 8.7 cm. It seems a touch closer to 8.6 than to 8.7. So, you say that the pencil is 8.63 cm long. The last digit implies that it is (±.01). This way, the value could be 8.62, 8.63, 8.64, or anywhere in between. The most that you know is that it is definitely closer to 8.6 than 8.7, and the range from 8.62-8.64 just about covers your uncertainty about the measurement." Here OP assumed error is one tenth of least count.

2.) Equal to least count : In some textbooks it is written that a measurement has a error equal to least count of scale ( Understanding Physics for JEE Main and Advanced Mechanics Part 1 page 19 )

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  1. Half of least count : Some users like @EricTowers assume that error in a measurement is half of the least count as if they look at a measurement let's say 8.6cm they take it in interval of (8.55cm to 8.65cm] assuming error is ±0.05cm ( half of least count.1cm)
Govind Prajapat
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Bharat Prajapat
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1 Answers1

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It really depends on your situation. You cannot simply blindly apply some fixed rule to all situations. You need to apply your understanding of both the measuring device and the experiment to making a statistical model of the measurement process. Often there will be multiple models that could work well, in which case you can choose the easiest one.

For example, suppose that I have a rigid ruler and a rigid straight object that I can confidently measure with the ruler. Then if the object falls about 1/3 of the way between the 56 mm and 57 mm marks I could model it as a uniform distribution between 56 mm and 57 mm. Or I could model it as a normal distribution centered at 56.3 mm with a standard deviation of 0.25 mm or 0.5 mm indicating a 95 % probability that the length is within a 1 mm or 2 mm wide region around the measurement respectively.

If the setup is variable, like the object is flexible or curved, then maybe you increase the width of the distribution accordingly. Or you could add the variance of the above umcertainty with the variance of your setup to get a total variance. Or if the setup variance is much larger than the above variance then you can simply neglect the above variance entirely.

The best resource for these questions is the BIPM’s “Guide to Uncertainty in Measurements”

https://www.bipm.org/documents/20126/2071204/JCGM_100_2008_E.pdf

Dale
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