# Combining measurements

1. Dec 16, 2003

### ahrkron

Staff Emeritus
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?

2. Dec 16, 2003

### Integral

Staff Emeritus
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. Dec 16, 2003

### suyver

Let $$\partial x_1,\partial x_2,\partial y_1,\partial y_1$$ denote the statistical and systematic errors in the variables $$x,y$$ respectively.

When one calculates $$T=x+y$$ I think the errors will be:

$$\partial T_1 = \sqrt{(\partial x_1)^2+(\partial y_1)^}$$

and

$$\partial T_2 = \sqrt{\left(\frac{\partial x_2}{x}\right)^2+\left(\frac{\partial y_2}{y}\right)^2}$$

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

Edit: TeX-error corrected.

Last edited: Dec 16, 2003
4. Dec 16, 2003

### Integral

Staff Emeritus
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. Dec 18, 2003

### ahrkron

Staff Emeritus
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 ammount much larger than the statistical error).

6. Dec 18, 2003

### Njorl

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. Dec 18, 2003

### ahrkron

Staff Emeritus
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. Dec 19, 2003

### suyver

You must be right, my equation must be wrong as I could immediately have seen from the dimensions. 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!

9. Dec 22, 2003

### suyver

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:

$$\partial T_i = \sqrt{(\partial x_i)^2+(\partial y_i)^2}$$

regardless weather $$\partial T_i$$ refers to the statistical or the systematic error.

So, in your case
the result would be

$$550 \pm 92.2 \pm 10.4$$

Hope this helps.