More on Fitting and Statistics:†
Quality Factor, Chi Squared, Estimating Parameters such as the Slope
A statistic is any function of your data.† For example, suppose you measure the temperature outside ten times. You decide that you want to report what the temperature was at that moment.† You could just use your first measurement, T1 or perhaps you decide to use the last measurement T10.† You have many options:
Each one of the above choices is a function of the data and therefore represents a statistic that one could use to estimate the true value of the temperature.† Hopefully, most physics students would have some basic intuition and their common sense might suggest that the average value should be a good choice and that it should be a better choice than most of the other suggestions above.† For a moment ask yourself if you actually performed the ten measurements and wanted to report the correct temperature what value you would report.† Also ask if there are any arguments that you could use to justify your choice.
Statistics is a field which addresses the question of which potential functions of a data are the best for estimating quantities of interest.† The above problem was straightforward in that one directly measures the quantity of interest. However, often the measurement is more complex.† For example, a student might measure the position and the time of a moving object but be interested in the objectís velocity.† If the object is moving at a constant velocity one could plot the data and draw a straight line that passes as close to the data as possible and then use the slope of this line to estimate the velocity.† It is not obvious that this approach is related to the method described in the temperature example.† Is this slope in any way a function of the position-time data recorded ?† Is the slope a statistic as defined above that represents an estimator for velocity?† The answer is yes. Curve fitting, although complex, is just a method for finding an estimator and in principle that estimator will always be some function of the data used in the fit.
Before pursing the relationship between parameters extracted using curve fitting and explicit functions of the data such as an average, let us consider how one judges a statistic.† Let us return to the temperature measurement and attempt to evaluate the possible choices.† First we introduce some qualities that one might require a good statistic or estimator should have:
Tends to be close to the true value
Rating as to how close the statistic is on average to the true value
The statistic is just as likely to be above as it is below the true value
Here we take the reasonable approach that there is a TRUE value and that one can imagine many experiments done in the same manner.† By examining the methods used in an experiment one can build a theory that predicts the likelihood or probability of getting an experimental result.† Let us imagine a simpler problem.† How likely is it to role a die and record five sixes in a row?† Knowing that a die has an equal likelihood of landing on any of its six faces we can calculate this probability.† As a matter of fact we can calculate the probability for any combination and therefore predict how many times five sixes will appear if we role five die ten thousand times.† We donít need to perform the experiment to predict this result.† We also know that if we did perform the experiment the results will vary because a die produces random results. In the same manner we can evaluate results from an experiment and evaluate whether an average value or an individual measurement has superior qualities as an estimator. We can evaluate the properties of each statistic chosen to estimate the temperature.
The definitions given in the table are paraphrased definitions for quantities that are evaluated in the field of statistics to judge the choices used to estimate quantities of interest. So if you want to provide your best guess for the temperature you need to evaluate your choices as to consistency, efficiency, bias and other characteristics. Informally, you can probably argue that the average is better than any one measurement of the temperature.† This of course is not based on a single experiment because it just might be the case that T7 is exactly the true temperature when you performed the experiment. And therefore, for your measurement, T7 would be the best choice (of course you do not know this). However the recommend statistic is the average which was evaluated based on the expected outcome of many temperature experiments.† Over many trials you will find that the average temperature is better that T7. It is very unlikely that if your classmate performs the same experiment as you that their T7 will duplicate your result and be very close to the exact true temperature.† This is the sense in which we evaluate performance of statistics in terms of quality factors such as bias. †One needs to imagine many experiments of the same type performed and characterize each potential statistics by seeing how it performs.
Hopefully, you are convinced that once can formally introduce qualities that can be used to evaluate the choice of a statistic and that a good experimenter will choose estimators that have optimal properties.† In P140L and P150L the student is usually given the method to use to estimate values.† We typically use averages and fitting.† This discussion only serves to provide some background and a bit more insight into the problems of estimation and statistics so that the student has the sense that methods employed are justified by more rigorous mathematics.
As stated most students do not need to be convinced that using an average value is a reasonable choice for an estimator. Let us return to the question of fitting data as a method for extracting values or estimators.
Given a set of data a student can draw a line through the points.† To accomplish this formally we define chi squared.
You take each data point †and evaluate your fitting function at the corresponding point.† If the data and fit agreed perfectly then .† If the data has uncertainty then there will not be perfect agreement for every data point but some data points may be very close in value to the fit. Dividing these differences by the uncertainties allows you to put more weight on data with smaller uncertainty.
Curve fitting Ť minimize
Find those parameters that make† †as small as possible.† Actually if the fit function is
Then the process of finding the smallest value can be carried out mathematically. For the case where all of the data have the same uncertainty,
So one can simply plug their data into the above formula to find the value for the slope that minimizes .
If the fitting function is not linear it is usually not possible to solve the equations for an exact formula but the parameters can be determined using a computer process.
Further analysis of the curve fitting method has shown that the resulting values for the parameter that minimize †are good, well behaved estimators.†
Before concluding it is useful to look more carefully at the definition of† .† One can argue that on average a data point will typically be about †away from the fit.† For the student this should seem plausible. Therefore .† This is called the normalized value of chi squared and we expect it to be close to 1.† If this value is too small then the fit is too good and you have overestimated your errors.† If it is too large then there is a problem.† If the best curve that you can find misses the data by a large amount then something is wrong.† It might be that the computer failed to find the correct set of parameters. The minimization algorithm is reporting a bad result.† For complicated functions a false minimum is sometimes returned. †It might be that your choice of fitting function is incorrect.† You canít in general put a straight line through data that follow an exponential curve and have all of the data close to the line.
 In most cases the experimenter doesnít know the true value.† It is understood that at the moment you make your measurement and at the location where you make your measurement there is indeed an exact value for the temperature.† The TRUE VALUE is therefore based on the premise that in principle one could carry out a measurement to an accuracy that would reveal a value with an uncertainty approaching zero.†