# The treatment of errors and uncertainties?

• alexgmcm
In summary, Alex is confused about how to deal with errors and uncertainties in experiments. He has read about different methods, but isn't sure how to use them correctly. He would appreciate any advice or help anyone could provide.
alexgmcm
I am confused over how to work with errors and uncertainties.

So far when dealing with a small number of measurements I have used the partial derivative method to calculate the final error in my result e.g. if my result is
$$E=mgh \text{ then assuming no error in g my uncertainty is } \Delta E = \sqrt{\left|\frac{\delta E}{\delta m} \right| ^2 \cdot \Delta m ^2 + \left|\frac{\delta E}{\delta h} \right| ^2 \cdot \Delta h^2}$$

and when dealing with a large number of measurements (normally when the experiment has computerised data acquisiton) I use the standard error:
$$SE = \frac{s}{\sqrt{n}}$$
where n is the number of measurements and s is the standard error (calculated using Bessel's correction which makes it work for smaller N by some mathematical trickery):
$$s=\sqrt{\frac{1}{N-1} \Sigma^{N}_{i=1} (x_i - \bar{x})^2}$$

This has always seemed strange to me as N is therefore usually the same as n which just seems weird. I guess I'm doing it wrong but I am not sure how?

How should errors be treated both in the case when you have a small number of measurements of each variable a statistical approach is impossible, and when you have a large number of results and a statistical approach is more attractive?

Any help or advice would be greatly appreciated,
Alex.

I can't quite understand what you are asking. However: s is an estimate of the standard deviation of a single measurement, while SE is an estimate of the error in the average. Note that for large n, s -> σ (the theoretical standard deviation), while SE -> 0.

alexgmcm said:
I am confused over how to work with errors and uncertainties.

What you wrote is appropriate for independent errors; that is the error in 'm' is independent of the error in 'h'.

By far the best book to learn from (at least, the best book I have seen so far) is John Taylor's book

https://www.amazon.com/dp/093570275X/?tag=pfamazon01-20

Last edited by a moderator:
mathman said:
However: s is an estimate of the standard deviation of a single measurement, while SE is an estimate of the error in the average. Note that for large n, s -> σ (the theoretical standard deviation), while SE -> 0.

Thank you! This finally made me understand that confusion I had. Now it all makes sense, I think so far in my experiments I am justified in assuming uncorrelated (i.e. independent errors) and so therefore need not worry about covariance.

The official way of dealing with errors/uncertainties in measurements is to use whatever method is recommended by the GUM (which is a free document published by the JCGM). Note that by "official" I really mean "The method you must use in order to comply with X" where X would be most organizations/regulations/guidelines regardless of where in the world you are.

See link 2 on the following wiki page
http://en.wikipedia.org/wiki/Measurement_uncertainty#cite_note-GUM-1

## 1. What is the difference between errors and uncertainties in scientific research?

Errors refer to the difference between the measured value and the true value, while uncertainties refer to the range of possible values that the true value could fall within. Errors are typically caused by mistakes or limitations in the measurement process, while uncertainties are a result of inherent variability or unknown factors in the system being studied.

## 2. How do scientists account for errors and uncertainties in their research?

Scientists use a variety of statistical and analytical methods to quantify and minimize errors and uncertainties in their research. This can include repeated measurements, using control groups, and utilizing appropriate statistical tests and models to account for variability.

## 3. Can errors and uncertainties ever be completely eliminated in scientific research?

No, it is impossible to completely eliminate errors and uncertainties in scientific research. However, scientists can strive to minimize their impact by using rigorous methods and continually refining their techniques.

## 4. How do errors and uncertainties impact the reliability of scientific findings?

Errors and uncertainties can have a significant impact on the reliability of scientific findings. If not properly accounted for, they can lead to incorrect conclusions and flawed research. It is important for scientists to thoroughly address and communicate any potential errors and uncertainties in their work to ensure the validity of their findings.

## 5. Can errors and uncertainties ever have a positive impact on scientific research?

Yes, errors and uncertainties can sometimes have a positive impact on scientific research by leading to unexpected discoveries or prompting scientists to reevaluate their methods and theories. However, it is still important for scientists to acknowledge and address these factors in their research to maintain the integrity of their findings.

• Other Physics Topics
Replies
1
Views
3K
• Other Physics Topics
Replies
8
Views
2K
• Other Physics Topics
Replies
13
Views
3K
• Other Physics Topics
Replies
13
Views
2K
• Introductory Physics Homework Help
Replies
8
Views
1K
• Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
• Set Theory, Logic, Probability, Statistics
Replies
4
Views
1K
• Set Theory, Logic, Probability, Statistics
Replies
22
Views
1K
• Set Theory, Logic, Probability, Statistics
Replies
16
Views
1K
• Classical Physics
Replies
7
Views
648