What is the correct method for combining systematic errors in measurements?

Edit: I am assuming that the number of measurements is large enough so that the statistical error is dominant.
  • #1
ahrkron
Staff Emeritus
Science Advisor
Gold Member
760
2
Say you have two measurements of the same quantity:

t = 500 +- 20 +- 3
t = 600 +- 90 +- 10

where the first error is statistical and the second is systematic.

What is the correct way of combining the systematic errors?
i.e., if I want to put both measurements together into
t = x +- dx1 +- dx2

How should I obtain dx2?
 
Mathematics news on Phys.org
  • #2
I assume that you Statistical error is a measurement average. It would seem to me that the systematic errors would be lost in the much larger measurement error, so in essence would be folded in with no effect.

I am open to disscussion.
 
  • #3
Let [tex]\partial x_1,\partial x_2,\partial y_1,\partial y_1[/tex] denote the statistical and systematic errors in the variables [tex]x,y[/tex] respectively.

When one calculates [tex]T=x+y[/tex] I think the errors will be:

[tex]\partial T_1 = \sqrt{(\partial x_1)^2+(\partial y_1)^}[/tex]

and

[tex]\partial T_2 = \sqrt{\left(\frac{\partial x_2}{x}\right)^2+\left(\frac{\partial y_2}{y}\right)^2}[/tex]


However, I am doing this without a textbook so I hope I am remembering correctly... :wink:


Edit: TeX-error corrected.
 
Last edited:
  • #4
If I were to combine the errors that is the most logical approach, still not clear to me that, that yields a meaningful number.

Perhaps I am mistaken in the nature of the statistical error.

What is the nature of the Systematic error? Is in a consistent measurement error or something else?
 
  • #5
Thanks for the replies.

Originally posted by Integral
I assume that you Statistical error is a measurement average.

It is a measurement of the dispersion of the data you have. If the parent distribution is a gaussian, then the RMS of the data is a good estimator of the standard deviation.

The statistical error decreases as you get more data.

On the other hand, there are other things that affect your measurement do not depend (directly, at least) on the amount of data you have. Examples of these are the numbers you use as input to your calculations, or the decisions you make along the way between different (but reasonable) ways to perform the measurement. These are the systematic errors.

It would seem to me that the systematic errors would be lost in the much larger measurement error, so in essence would be folded in with no effect.

Not necessarily. Sometimes you may have a lot of data at your disposal, hence yielding a rather small statistical error, but there are various models for how to fit the data (say, with either an exponential or a polynomial background), and the choice may have a big effect on your final value (i.e., it may shift it by an amount much larger than the statistical error).
 
  • #6
Originally posted by ahrkron
Thanks for the replies.



It is a measurement of the dispersion of the data you have. If the parent distribution is a gaussian, then the RMS of the data is a good estimator of the standard deviation.

The statistical error decreases as you get more data.

On the other hand, there are other things that affect your measurement do not depend (directly, at least) on the amount of data you have. Examples of these are the numbers you use as input to your calculations, or the decisions you make along the way between different (but reasonable) ways to perform the measurement. These are the systematic errors.



Not necessarily. Sometimes you may have a lot of data at your disposal, hence yielding a rather small statistical error, but there are various models for how to fit the data (say, with either an exponential or a polynomial background), and the choice may have a big effect on your final value (i.e., it may shift it by an amount much larger than the statistical error).



I agree. I used to do Fourier transform spectroscopy, and the statistical errors were miniscule. The systematic errors were about 1000 times as large for some data. We'd measure the index of refraction of a material, and get consistancy for several different pieces out to 6 decimal places. Then we'd compare our numbers with other labs, measuring the same samples, and we'd have inconsistancies of almost 1%.

Njorl
 
  • #7
Originally posted by suyver
Let [tex]\partial x_1,\partial x_2,\partial y_1,\partial y_1[/tex] denote the statistical and systematic errors in the variables [tex]x,y[/tex] respectively.

When one calculates [tex]T=x+y[/tex] I think the errors will be:

[tex]\partial T_2 = \sqrt{\left(\frac{\partial x_2}{x}\right)^2+\left(\frac{\partial y_2}{y}\right)^2}[/tex]

Thanks, ... but we're missing something there, since [itex]T_2[\itex] comes out dimensionless. Maybe multiplying by the final value of T? but then both systematic errors have the same weight, regardless of their statistical significance... I'm not saying this is wrong, but it just seems odd (to me at least)...
 
  • #8
Originally posted by ahrkron
Thanks, ... but we're missing something there, since [itex]T_2[\itex] comes out dimensionless. Maybe multiplying by the final value of T? but then both systematic errors have the same weight, regardless of their statistical significance... I'm not saying this is wrong, but it just seems odd (to me at least)...

You must be right, my equation must be wrong as I could immediately have seen from the dimensions. :frown: Like I said, I did it without a textbook. I'll try to dig up my old textbooks over the weekend and see what they say.

Stay tuned, more to come! :smile:
 
  • #9
OK, I checked the only book that I have on this (Squires - Practical Physics). He states that the two errors are independent and therefore they should be treated seperately. This makes sense. As a result, he uses the same method for combining two errors under addition:

[tex]\partial T_i = \sqrt{(\partial x_i)^2+(\partial y_i)^2}[/tex]

regardless weather [tex]\partial T_i[/tex] refers to the statistical or the systematic error.

So, in your case
t = 500 +- 20 +- 3
t = 600 +- 90 +- 10
the result would be

[tex] 550 \pm 92.2 \pm 10.4[/tex]

Hope this helps.
 

What is "combining measurements"?

"Combining measurements" is the process of taking multiple measurements of the same quantity and combining them to get a more accurate and precise value. This is often done in scientific experiments to reduce error and increase the reliability of results.

Why is combining measurements important in scientific experiments?

Combining measurements is important in scientific experiments because it helps to reduce the impact of random errors and increase the precision and accuracy of the results. By combining multiple measurements, scientists can get a more reliable and consistent value for the quantity being measured.

What are some common methods for combining measurements?

Some common methods for combining measurements include averaging, weighted averaging, and least squares regression. These methods take into account the individual measurements and their uncertainties to calculate a more accurate and precise overall value.

How do you determine the uncertainty of a combined measurement?

The uncertainty of a combined measurement depends on the uncertainties of the individual measurements and how they are combined. Generally, the uncertainty of a combined measurement is calculated using the root mean square (RMS) method, which takes into account the individual uncertainties and their effect on the overall uncertainty.

What are some potential sources of error when combining measurements?

Some potential sources of error when combining measurements include instrument error, human error, and systematic error. Instrument error can occur due to limitations of the measuring instrument, while human error can arise from mistakes in recording or reading the measurements. Systematic error can result from faulty experimental design or incorrect assumptions made during the measurement process.

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
21
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
7
Views
1K
  • Introductory Physics Homework Help
Replies
1
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
21
Views
2K
  • Introductory Physics Homework Help
Replies
4
Views
770
  • Introductory Physics Homework Help
Replies
7
Views
1K
  • Classical Physics
Replies
6
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
22
Views
1K
  • Introductory Physics Homework Help
Replies
11
Views
1K
  • Introductory Physics Homework Help
Replies
15
Views
942
Back
Top