From Physics 111-Lab Wiki

All pages in this lab. Note To print Full Lab Write-up click on each link below and print separately

I. Error Analysis Notes

II. Error Analysis Exercise

III. MatLab Scripts for curve fitting. If inside the Physics111 Library site GoTo Advanced Lab Reprints then click on MatLab folder.

---

Used in Beta Ray Spectroscopy, Compton Scattering, Gamma-ray Spectroscopy, Muon Lifetime, Optical Pumping, Rutherford Scattering.

Contents 1 Introduction 2 Error of a Single Parameter 2.1 Measurement statistics 2.2 Counting statistics 3 Curve Fitting with Two Parameters (Linear Regression)

Introduction

View the video introduction to error analysis.

First run through one of the several good texts on error analysis, such as the one by J. R. Taylor, "An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, 2nd Edition", University Science Books (1996). [A good introduction that is easy to follow.] This will give you a background on the statistical model on which we base our statements, and define many of the terms used, such as average, true mean, estimated mean for a sample, variance, standard deviation, random error, systematic errors, and others. There is no substitute for this reading. To give all details here would require us to write a whole book. We are only going to skim the subject of statistical errors.

Error of a Single Parameter

Measurement statistics

As we read about laboratory results, we often see the value of a measured parameter as written something like 2.10 ± 0.05. The value 2.10 in this example is usually an average of several measurements, because successive measurements of any particular quantity are never exactly the same. Where does ± 0.05, called the standard deviation, come from, and what does it mean?

When we make a measurement and get a value for a single quantity or parameter, we want to know how likely it is to be correct. So we measure it again and again, say N times in all, and look at the scatter in the measurements. For example, we might have y = 2.10 as the average of the 5 measurements y_i = 2.18, 2.03, 1.95, 2.24, 2.10,

We then calculate the standard deviation sigma, with the equation

Loosely speaking, this means that if we make another set of N measurements, the average of this new set has a 67% likelihood of falling within a range from 2.05 to 2.15, or more compactly written as 2.10 ± 0.05

Counting statistics

In experiments like Muon Lifetime, we record the number of events or counts in a given time interval, where the events are random, as in radioactive decay. Suppose we get N counts in a fixed time interval. If we take other samples for the same interval, we get some close but different values of N. After many such measurements, we can construct the standard deviation, as described above. We don't need to do this, because statistical considerations specify that if the counts come at truly random times, then the standard deviation is . So, we can make a single record of counts and write N ± for the counts in a given interval of time. The larger the N, the larger the standard deviation, but the fractional error grows smaller as the number of counts grows larger. The fractional error is . For example, to get a fractional error of 1%, N must be 10 000!

Curve Fitting with Two Parameters (Linear Regression)

In many laboratory experiments we have several parameters, rather than just one, and we want to fit a smooth analytical curve to the datum points and extract the relevant parameters from this curve. In cases where the curve is a straight line, we want to fit a data set to determine the slope and intercept of the line, each of which has some physical significance.

We start with the equation y = mx + b = f(x). We assume that the independent variable is x, that successive values of x are equally spaced, and that we record a datum point yi at each value xi. We then select a straight line and adjust its slope "m" and intercept "b" to get a "best fit" to the experimental points yi. By definition we get a best fit when the sum of the squares of the deviations of the fitted curve from the datum points at each xi is a minimum. In mathematical terms, adjust m and b so as to get a minimum of the sum:

We do this by taking the derivatives of the sum with respect to m and b, setting the equations equal to zero, and solving the simultaneous equations for the "best" values of m and b. This is called a "least-square fit". These values will have error bars - standard deviations - that can be calculated with techniques described in the texts. Lyons gives a worked-out example.