1. Physics 114: Exam 2 Review Material from Weeks 7-11
John Federici
NJIT Physics Department

2. Concepts Covered on the Exam
The list of concepts covered on the exam are:
Central Limit Theorem, averages of averages
Combining data with different standard deviations
Confidence intervals, Confidence level
SIGNIFICANT DIGITS in error
Chi-squared test, goodness of fit
Weighted mean and Error
Least-squares fitting, minimizing chi-square, linear least-squares fitting
Degrees of freedom
Fitting a polynomial
Linearization of fitting equation.

3. Suggested materials to Review
Review lecture notes for weeks Pay attention to CONCEPTS and specific examples given in class.
There are no questions on Matlab code. HOWEVER, You should be able to interpret the “results” window of the Curve Fitting App.
Review HW#7, problem 1, Problem 2, Problem 3. NOTE, if you are given a problem similar to problem 3, the number of data points will be small enough that you can use the EQUATIONS at the end of the exam to calculate your answers.
Review HW#8, Problem 1, Problem 3.
Review HW#9, Problem 1, Problem 2.
Review concepts of HW#11, Problem 2.

4. Mean of Means and Standard Error
One can join multiple sets of measurements to refine both the estimated value (mean of means) and the standard error (standard deviation of the mean).
The mean of means is given by, , where the xi are individual measurements of the mean.
If the standard deviations of each of the measurements are all the same (si = s), then they cancel and we have the usual
Likewise, the rule for combining data sets with different errors is
And for equal errors this is
This last is a key result to remember—combining measurements reduces the standard deviation by the square root of the number of measurements.
Do example: x = [10.7, 7.2, 11.2, 9.9, 11.3], s = [2, 2, 2, 1.5, 2.5]. Ans: 10.0±1.4

5. Weighted Mean and Error
Perhaps the errors themselves are not known, but the relative weighting of the measurements is known. For example, say you want to combine means taken with different numbers of measurements (or different integration times). Defining the weights as proportional to the variances kwi = si2, the proportionality constant cancels and we have
We can then define an average standard deviation:
After obtaining that average standard deviation, the standard error (standard deviation of the mean) is, as before, decreased by the square-root of the number of measurements:

6. Probability Distribution
The Gaussian distribution (bell curve) shows the expected distribution of measurements about the mean. This can be interpreted as a probability.
Thus, ~68% of measurements should fall within 1s of the mean, i.e.
Likewise, ~95% of measurements should fall within 2s of the mean.
In science, it is expected that errors are given in terms of ±1s. Thus, stating a result as 3.4±0.2 means that 68% of values fall between 3.2 and 3.6.
In some disciplines, it is common instead to state 90% or 95% confidence intervals (1.64s, or 2s). In the case of 90% confidence interval, the same measurement would be stated as 3.4± To avoid confusion, one should say 3.4±0.37 (90% confidence level).

7. Chi-Square Probability
Chi-square is a criterion for the goodness of fit of a function, e.g. y(x), and is defined as
In other words, it is just the sum of the squared deviations of points from the function, normalized by the variances. When the fit is good, we normally expect the squared deviations to average around s 2, so each term is about 1 and the total chi-square is about equal to n, the number of degrees of freedom.
For the special case of a linear fit to a set of points (y(x) = a + bx),
We can find the best fit straight line by minimizing chi-square.
Generally, we can find the best fit of any function by replacing y(x) with another equation representing that function.

8. Linear Least Squares Fitting
Minimizing chi-square, we found that we could solve for the parameters a and b that minimize the difference between the fitted line and the data (with errors si) as:
where
In the case of equal errors, they cancel and we can drop the s and replace with N.
The uncertainties in the parameters are:

9. Matlab Commands
Remember, we also had used the CURVE FITTING APP to fit data

10. Reduced Chi-Square Recall that the value of c2 is
It is often easier to consider the reduced chi-square,
which is about unity for a good fit.
If we compare points to a fit of a sine function, changing parameters changes c2, and obviously the minimum chi-square is the best fit.
change amplitude
change frequency

11. Degrees of Freedom n
The number of degrees of freedom represent the ways in which things can be varied independently. It is generally the number of independent data points (the number of measurements), reduced by the number of parameters deduced from the measurements.
Thus, if the data points are used to determine a mean, then the data points can be varied, but are constrained to have the given mean. This constraint must be subtracted from the number of points, so in this case the number of degrees of freedom is n = N – 1.
If we use the data points to define a line (i.e. solve for two parameters a and b for the line), then n = N – 2.
You should learn to recognize the number of parameters needed to fully describe a function. The sine wave of the previous example can be adjusted in three ways (has three parameters). We showed two (amplitude and frequency). Can you guess the third? For this fit, we have n = N – 3.
We need to know this in order to use the reduced chi-square as a measure of when we have an acceptable fit.

12. Practice Problem

13. Practice Problem

14. Practice Problem

15. Sample Problem … Cont.

16. Sample Problem … Cont.

17. Sample Problem … CONT

18. Sample Problem 2