Physics 114: Lecture 5 Uncertainties in Measurement John Federici NJIT Physics Department The contribution of Prof. Dale Gary’s course notes for this course is gratefully acknowledged.

Nobel Prize Trivia For which discovery was Albert Einstein awarded the Nobel Prize? (a) E=Mc2 (b) Special Relativity (c) General Relativity (d) Stimulated Emission – Basic Laser principles (e) Brownian Motion (f) Photoelectric Effect

Nobel Prize Trivia For which discovery was Albert Einstein awarded the Nobel Prize? The Nobel Prize in Physics 1921 was awarded to Albert Einstein "for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect". In 1905 Einstein published four landmark papers in physics - on the photoelectric effect, Brownian motion, the special theory of relativity and equivalence of matter and energy (E=mc2).

Some Terms Accuracy—How close the measurements are to the “true” value (note that we may not always know the true value). Precision—How close repeated measurements are to each other. A measure of the spread of data points. One can make measurements that are highly accurate (their mean is close to the true value) even though they may not be very precise (large spread of measurements). Conversely, on can make very precise measurements that are not accurate. Errors—Deviations of measurements from the “true” value. Error here does not mean a blunder! Also referred to as uncertainties. Systematic Errors—deviations from the “true” value that are very reproducible, generally due to some uncorrected effect of an instrument or measurement technique. An example is using an oven which is not calibrated properly and ALWAYS produces a temperature which is higher than the control setting. Statistical, or Random Errors—fluctuations in measurements that result in their being both too high and too low, due to how precisely the measurement can be made, and which are amenable to reduction by doing repeated measurements.

Precision Versus Accuracy What do vertical lines through data points mean? Precise but Inaccurate Data Accurate but Imprecise Data “True” values are represented by straight lines February 08, 2010

Systematic Errors Systematic Errors are REPRODUCIBLE discrepancies between the “measured” and “true” values February 08, 2010

Parent and Sample Distributions Imagine a process for manufacturing ball bearings. Although each ball bearing is nominally the same, any process is going to cause slight deviations in shape, size, or other measure. If we measure the weight, say, of an infinite number of such ball bearings, these weight measurements will spread into a distribution around some mean value. This hypothetical, infinite distribution is called the parent distribution. The parent distribution’s spread depends, obviously, on how precise the manufacturing process is. We can never measure an infinite number of ball bearings. Instead, we measure a smaller subset of ball bearings, and from this sample we again find that our measurements spread into a distribution around the sample mean. This finite distribution is called the sample distribution. In the limit of an infinite sample, of course, the sample distribution should become the parent distribution (assuming we have no systematic errors).

Example Sample & Parent Dist. At the left are the sample distributions for a series of 16 sets of 50 measurements: At the right is the sum of these 16 measurements (equivalent to 800 measurements). Apparently 800 is close to infinity, since the sample distribution now is quite close to the parent distribution (red line).

What To Do When the “True” Value is Unknown—The Mean When we do not know the “true” value that we are comparing our sample to, we can take the mean of the measurements as an approximation of the “true” value. Of course, the mean of the parent population is

Probability and Median The spread of values about the mean in the parent population (that is, the histogram) forms a function called a probability density function (PDF). We will be using this term many, many times during the course. Its connection to probability is as follows: if you take the PDF and normalize its area, so that the area under the curve (the integral) is unity, then the integral in a restricted range x1 to x2 is the probability that a given measurement will fall in that range. Notice that we use P(x) for the probability, and p(x) for the probability density (PDF). The median (m1/2) is the point where the probability is equal (i.e. 1/2) on each side:

Most Probable Value (Mode) Most probability density functions (PDFs) have a single peak. The value of x at which they peak is the most probable value, or mode. This is the same as the mean for symmetric PDFs, but they can be quite different for asymmetric ones. The most probable value is called mmax, and obeys Examples of when to use median vs. mean. For a set of measurements (sample distribution) that follows the Gaussian (Normal) distribution, the mean and median are basically the same, so long as the sample is large. However, the median is often preferred over the mean, as an estimate of the true value, in the presence of outliers. Say we have a set of measurements x = 190. + randn(1,100); You can check that the mean and median are nearly identical. Now say there was something wrong with the 42nd measurement (x(42) = 300.;). Now the median is nearly unchanged, but the mean is much higher.

Deviations and RMS If the parent distribution mean is m, the deviations from the mean can be written The average of the deviations, by virtue of the definition of the mean, must vanish: Still, we may want to know what is the average absolution deviation, i.e. not consider the sign of the deviation, just the amount: For computational purposes, it is better to define the square of the deviations (called the variance) Then the standard deviation (also called RMS or root-mean-square deviation) is the square-root of the variance, s. To calculate the variance of the sample distribution, use:

Basic Statistics in MatLab February 08, 2010

NOTE: These functions treat vectors and arrays differently Consider a two-dimensional input array, A. If dim = 1, then mean(A,1) returns a row vector containing the mean of the elements in each column. If dim = 2, then mean(A,2) returns a column vector containing the mean of the elements in each row February 08, 2010

Converting 2D data to 1D data If you have a 2D data set (for example, an image) and want to perform a statistical analysis on the ENTIRE 2D set of data, and NOT just on the columns or just the rows, HOW DO YOU DO IT? You must convert 2D data set into a 1D data set and THEN apply statistical functions “:” operator converts the multidimentional array into a VECTOR >> A(:) ans = 1 2 3 A = 1 2 3 Excellent reference for indexing of arrays: https://www.mathworks.com/company/newsletters/articles/matrix-indexing-in-matlab.html

Matlab Example >> z=randn(1,1000); >> max(z) ans = 2.9095 >> min(z) -3.0790 >> mean(z) 0.0038 >> median(z) 0.0398 >> mode(z) ans = -3.0790 >> std(z) ans = 0.9835 The PARENT distribution of the randn function has a mean of zero. Note that for this SAMPLE distribution, the mean is close to zero The PARENT distribution of the randn function has a Standard Deviation of 1. Note that for this SAMPLE distribution, the STD is close to unity. February 08, 2010

How do these values compare to the histogram plot? Mean, Median 2*STD February 08, 2010

Class Exercise: Use Matlab HELP or DOCUMENATION SEARCH as needed Create a MATLAB program which Creates an array of data whose parent function has an AVERAGE of 15 and a Standard Deviation of 4. HINT: Scale the randn function…. z=Offset+Factor*randn(1000,1); Using the appropriate Matlab functions, calculate the mean, mode, median, and standard deviation of the distribution. Is the Standard Deviation and average of the sample data close to that of the PARENT distribution? February 08, 2010