# Uncertainties in a set of measurements

• KDPhysics
In summary, the speaker is discussing how to calculate and express uncertainties in measurements. They mention using the sample standard deviation with 1 degree of freedom to compute uncertainties, and how there are different conventions for expressing the results. They also discuss the issue of different significant figures in measurements and how it is important to be clear and consistent with the chosen method. They also mention that the halving of the uncertainty does not necessarily result in an increase in precision by a factor of 10, and that it should be rounded up to the nearest significant digit.
KDPhysics
Homework Statement
Error analysis for standard introductory physics experiment.
Relevant Equations
Arithmetic mean, standard deviation, half range.
Suppose I measure the distance between two objects for three trials. The two objects then get farther away, and I measure the distance between them again for three trials. I repeat this for 3 more different distances, getting a total of 15 measurements (3 trials for 5 distances).

I then compute the average and the half-range (approximates the uncertainty) for each distance. For example, the first distance will have average 3.3m and uncertainty 0.15m . In this, case, since the uncertainty is fairly small, I have been told to keep the two significant figures and simply make the number of digits after the comma in the average to coincide with the number of digits after the comma of the uncertainty. So, i would write 3.30±0.15. That is clear to me.

The problem is, what if the second distance is 5.3±0.7. In this case, I only keep one significant figure for the uncertainty. But then, the measurements will have a different number of significant figures. Is this ok? Or should I report the first measurement as 3.3±0.2?

Also, to compute the uncertainty, should I use the half range or the standard deviation (i'm in high school)?

Well, I'm not an expert on how to treat uncertainties, but in the lab courses I've been, there seem to be different conventions on how to express them. First of all, I think that uncertainty is always computed using the sample standard deviation with 1 degree of freedom:
$$\sigma_{\bar{x}}=\frac{1}{\sqrt{N^2-N}}\sqrt{\sum_{i=1}^N(x_i-\bar{x})^2}$$ with $$\bar{x}=\frac{1}{N}\sum_{i=1}^Nx_i$$.

To then express the results I've seen different conventions, I usually give my results with the uncertainty having 2 significant digits, but in some labs, they prefer to give 2 significant digits if the first digit is 1 or 2, but only one significant digit if it's 3 or more.
I think that the reason for that is Benford's law that says that almost 50% of the time you uncertainty will start by 1 or 2, but I don't know very much.

I don't know if this helps you very much.

So even in high school I should use the standard deviation?
Also, is it ok if my measurements have different significant figures e.g. 5.3±0.7 and 3.30±0.15 in the previous post?

KDPhysics said:
So even in high school I should use the standard deviation?
Also, is it ok if my measurements have different significant figures e.g. 5.3±0.7 and 3.30±0.15 in the previous post?

No, I don't think there's any problem, and nothing of this depends on whether you are in High School or not. As long as you are clear and consistent with your method (and you explain it properly) it should be okay. Another issue would be that your professor wants it in another way, then of course you should always adapt your conventions to those imposed to you.

alright, thank you!

KDPhysics said:
if the second distance is 5.3±0.7
Your problem arises from the two digit precision (0.15) quoted for the error in the first distance. That is an artefact of the division by 2.
Now, that halving does represent an increase in precision, just not by an entire factor of 10. Using the standard deviation methods you would reduce the uncertainty by a factor ##\sqrt{N-1}##, so in this case ##\sqrt 2##. That's not enough to be adding a precision digit, so round the .15 up to .2.

## What is the definition of "uncertainty" in a set of measurements?

Uncertainty in a set of measurements refers to the estimated or calculated range of values within which the true value of a measurement is expected to lie. It takes into account any potential errors or limitations in the measurement process.

## What factors contribute to uncertainties in a set of measurements?

There are many factors that can contribute to uncertainties in a set of measurements. These can include the precision and accuracy of the measuring instruments, the skill and technique of the person conducting the measurements, environmental conditions, and any inherent variability in the quantity being measured.

## How is uncertainty calculated in a set of measurements?

There are several methods for calculating uncertainty in a set of measurements, including the use of statistical analysis, propagation of errors, and the use of standard deviation and confidence intervals. The specific method used will depend on the type of measurement and the available data.

## Why is it important to consider uncertainties in a set of measurements?

Considering uncertainties in a set of measurements is important because it allows for a more accurate and reliable representation of the true value of a quantity. It also allows for a better understanding of the limitations and potential errors in the measurement process, which can help to improve future measurements.

## How can uncertainties in a set of measurements be minimized?

There are several ways to minimize uncertainties in a set of measurements. These include using high-quality measuring instruments, following proper measurement techniques, conducting multiple measurements and taking an average, and identifying and addressing any potential sources of error. Collaborating with other researchers and using established protocols can also help to minimize uncertainties.

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