Physics 310 Errors in Physical Measurements Error definitions Measurement distributions Central measures

Physics 310 Errors are uncertainties... Every physical measurement has an uncertainty, i.e., it has an error. Every physical measurement has an uncertainty, i.e., it has an error. FRandom uncertainties (errors) þ Can be reduced but not eliminated. þ No measurement is infinitely precise. þ Determines the precision of the data. FSystematic uncertainties (errors) þ Can be reduced and eliminated þ Produces systematic shifts in the data þ Determines the accuracy of the data. The problem: What is the best way to find the values for which most likely represent the parent population from which our sample is obtained? The problem: What is the best way to find the values for a k which most likely represent the parent population from which our sample is obtained?

Physics 310 Errors are uncertainties... Errors are not blunders or mistakes. Errors are not blunders or mistakes. FBlunders or mistakes - þ Must be found and corrected. þ Are not quoted in error estimates on measurements. þ “Human Error” is not a valid error. Errors are quoted as  Errors are quoted as x ±  x   needs to be estimated -   x needs to be estimated - þ From the data. þ From the measurement. þ May have asymmetric values about þ May have asymmetric values about x

Physics 310 The parent population... If you are measuring a physical quantity e.g., the distance a neutron penetrates into a given material, repeating the experiment times produces slightly different values for, i.e., :where goes from to. If you are measuring a physical quantity (x) e.g., the distance a neutron penetrates into a given material, repeating the experiment N times produces slightly different values for (x), i.e., (x i ) :where i goes from 1 to N. If goes to, then this is the total of all possible measurements of this quantity - and this set is called the Parent Population. If N goes to ∞, then this is the total of all possible measurements of this quantity - and this set is called the Parent Population.

Physics 310 The sample population... If is finite, this will be a sample of the total number of all possible measurements of this quantity - and this smaller set of measurements is called the If N is finite, this will be a sample of the total number of all possible measurements of this quantity (x) - and this smaller set of measurements is called the Sample Population. Sample Population. We will make notational distinctions between these populations. We will make notational distinctions between these populations.

Physics 310 The sample population... If is large, the results obtained from the sample population of measurements will approach those of the parent population, but-- If N is large, the results obtained from the sample population of measurements will approach those of the parent population, but-- We will never know the actual values from the parent population, even though we seek them. We will never know the actual values from the parent population, even though we seek them. Our goal is to find the best estimates of the parent population values - Our goal is to find the best estimates of the parent population values -

Physics 310 The sample population... …and to estimate the precision and accuracy of our estimate of. …and to estimate the precision and accuracy of our estimate of (x). This latter exercise is called error analysis. This latter exercise is called error analysis. Results are often reported as Results are often reported as x ±  x ±  x where is the best representative value of, is the estimate of the random error, and is the estimate of the systematic error. where x is the best representative value of (x),  x is the estimate of the random error, and  x is the estimate of the systematic error.

Physics 310 Deviation... Because we cannot know from the parent population (the “true” value) we cannot formally compare our value with a “true” measurement of. Because we cannot know (x) from the parent population (the “true” value) we cannot formally compare our value with a “true” measurement of (x). There are few quantities whose value is predicted exactly from theory. For example: There are few quantities whose value is predicted exactly from theory. For example: F The charge on 1 electron is 1.6 x C. F The speed of light is x 10 8 m/s. F The gravitational constant is 6 x N m 2 /kg 2 All measured! All measured!

Physics 310 Deviation... Therefore, our measurement of can only be compared with the another measurement of, each of which has an uncertainty (experimental error), and neither of which is the “true” value. Therefore, our measurement of (x) can only be compared with the another measurement of (x), each of which has an uncertainty (experimental error), and neither of which is the “true” value. Such a comparison results in a deviation between the two measured quantities - or between a measured quantity and a theoretically expected quantity. Such a comparison results in a deviation between the two measured quantities - or between a measured quantity and a theoretically expected quantity.

Physics 310 Deviation... However, on the basis of measurement theory, we may postulate what the expected form of the distribution of measurements should be expected to be. However, on the basis of measurement theory, we may postulate what the expected form of the distribution of measurements (x i ) should be expected to be. F A plot of vs for a freely falling object. F A plot of v vs t for a freely falling object. F A plot of the distribution (histogram) of measurements of neutron depth in indium. F A plot of the angular distribution of photons from annihilation. F A plot of the angular distribution of photons from e + e - annihilation.

Physics 310 Deviation... It is therefore useful to compare not only the best estimate of, but also the distribution of measured values of. It is therefore useful to compare not only the best estimate of (x), but also the distribution of measured values of (x). If the distributions do not appear to agree, what does this mean? If the distributions do not appear to agree, what does this mean? H A problem with the experiment? H A problem with the theory? H Both?

Physics 310 Quantitative representations... Given a set of measurements, what are quantitative ways of expressing results? Given a set of N measurements, what are quantitative ways of expressing results?  The mean,  The mean,  = F The deviation, F The deviation, d = (x - )  The variance,  The variance,  2 = ) 2 >  The standard deviation =  The standard deviation  = √ ) 2 > n Each quantity has physical units! Don’t forget to include them! n Know how to compute each.

Physics 310 Computatons: the mean _ The sample mean is defined as: _ The parent population mean is then:

Physics 310 Computatons: the deviation _ The sample deviation is defined as: _ The parent population deviation is then:

Physics 310 Computatons: the variance _ The sample variance is defined as: _ The parent population variance is then:

Physics the standard deviation...the standard deviation _ The sample standard deviation is defined as: _ The parent population standard deviation is then:

Physics 310 Distributions... Take a set of measurements. Take a set of N measurements. Form a histogram of the measurements. (This gives the distribution of the measurements.) Form a histogram of the measurements. (This gives the distribution of the measurements.)  This gives the number of measurements of between and as, where is the total number of bins. ( is the fixed bin width.)  This gives the number of measurements of between x and x +  x as n j, j = 1,k where k is the total number of bins. (  x is the fixed bin width.)  Now, with this you can estimate and because represents a distribution function for the measurements. How do you do it?  Now, with this you can estimate  and  because n j represents a distribution function for the measurements x i. How do you do it?

Physics 310 Distributions... The mean and standard deviation from a distribution are: The mean and standard deviation from a distribution are:

Physics 310 Normalized Distributions... From a histogram of the measurements, you can form a normalized distribution of the measurements. From a histogram of the N measurements, you can form a normalized distribution of the measurements. F Take each value, and divide it by. This will give the fractional number of all measurements in the bin. The sum of all will be, and hence the distribution function is a normalized (discrete) distribution. F Take each value n j, j = 1,k and divide it by N. This will give the fractional number f j of all measurements in the bin j. The sum of all f j will be 1, and hence the distribution function f j is a normalized (discrete) distribution. F If is very large, his concept can be extended to a continuous probability function F If N is very large, his concept can be extended to a continuous probability function P(x).

Physics 310 Normalized distributions... The mean and standard deviation from a normalized discrete distribution are: The mean and standard deviation from a normalized discrete distribution are:

Physics 310 Normalized distributions... The mean and standard deviation from a continuous distribution are: The mean and standard deviation from a continuous distribution are:

Physics 310 Normalized distributions... The denominator is just: The denominator is just:

Physics 310 Normalized distributions... The standard deviation is then: The standard deviation is then:

Physics 310 Normalized distributions... …or in terms of a probability function …or in terms of a probability function P(x):

Physics 310 Normalized distributions... Or for any function: Or for any function: