Physics 4A. Error Analysis or Experimental Uncertainty. Error

Transcription

1 Physcs 4A Error Analyss or Expermental Uncertanty Slde Slde 2 Slde 3 Slde 4 Slde 5 Slde 6 Slde 7 Slde 8 Slde 9 Slde 0 Slde Slde 2 Slde 3 Slde 4 Slde 5 Slde 6 Slde 7 Slde 8 Slde 9 Slde 20 Slde 2 Error n Physcs language, error does not mean a mstake error the dfference between an observed or measured result and the true value no measurement, no matter how carefully made, can be completely free of errors there are two types of expermental error systematc errors and random errors Systematc Error Systematc errors errors that result from mstakes nherent n a partcular apparatus (.e. bad equpment) a slow clock ashort meter stck when systematc errors are present, the measured results are always shfted away from the true results n a gven drecton (.e. too hgh or too low) Systematc Error there s some control over systematc errors (.e. usng better equpment) systematc errors are usually very hard to detect and extremely hard to evaluate n lab, we wll assume that all forms of systematc error have been dentfed and mnmzed

2 Random Error random (statstcal) errors expermental uncertantes assocated wth random fluctuatons of any measurement apparatus random errors usually result from nstrumental uncertantes and/or statstcal fluctuatons random errors can be reduced by repeatng the experment many tmes and averagng the results random errors can be treated mathematcally usng statstcs Mean, σ, and SE n lab, we wll be usng three quanttes to express the expermental result of a set of measurements of some quantty x: Mean (x) the average of all measurements (gves your best estmate of the value of the result) Standard devaton (σ x ) ndcates how spread out your dfferent measurements were Standard Error (SE) gves the best measure of the uncertanty n the mean Mean (Average) Suppose that I gave the same test to two dfferent classes. If I wanted to know how well each class dd, what number would I use to compare the two classes? I would probably use the average or mean from each class. However, there s a major problem wth only lookng at the average. What s t??? Mean (Average) The average (or mean) doesn t gve any ndcaton of how spread out the values (test scores) are: (Class ) (Class 2) Mean = Mean =

3 Devaton from the Mean What we would lke s some ndcaton of how spread out the test scores are from each class. One possblty would be to calculate the average devaton, where the devaton d for each value s defned as: devaton = value - average However, there s a major problem wth calculatng the average devaton. What s t??? Test Scores Devaton Devaton from the Mean Test Scores Devaton The average devaton s always zero!!! The standard devaton σ s the quantty used to descrbe how spread out the values are n a gven set of data. The standard devaton σ s defned as follows: σ = = 2 2 ( d ) ( ) x x = = ) Frst, fnd the mean (average) of all values: x σ = = 2 2 ( d ) ( ) x x = = 2) Then, for each value, fnd the devaton of that value from the mean and square t: 2 ( x x )

4 σ = ( d ) = ( x x) 2 2 = = 3) Then, fnd the average of all of the devatons squared (dvdng by - nstead of ): 2 ( x x ) = 4) Fnally, take the square root of the average of the devatons squared: 2 ( x x ) = Test Scores Devaton Devaton^ The sum of all devatons squared s / (0-) = = 2.5 The standard devaton for class s σ = 2.5 Test Scores Devaton Devaton^ The sum of all devatons squared s / (0-) = = 7 The standard devaton for class 2 s σ = 7 The standard devaton σ gves us an dea of how spread out the values n a gven data set are. small σ values are close together large σ values are spread out Statstcally, ~68% (95%) of the values n a data set should fall wthn σ (2σ) of the mean. Ths means that ~68% of the values should fall wthn the range: mean σ and mean + σ and ~95% of the values should fall wthn mean + 2σ.

5 (Class ) average = σ =2.5 ~68% of the value should fall wthn the range: 2.5 and ~68% of the values should fall between 72.5 and 77.5 (Class 2) average = σ =7 ~68% of the value should fall wthn the range: 7 and + 7 ~68% of the values should fall between 58 and 92 Statstcally, ~68% of the values n a data set should fall wthn σ of the average, ~95% of the values n a data set should fall wthn 2σ of the average, and ~99% of the values n a data set should fall wthn 3σ of the average. Standard Error The standard devaton of a set a means (.e. from all lab groups) s called the standard error. The standard error gves the best estmate of the uncertanty of the mean from any ndvdual lab group. Statstcal theory tells us that even when we have only a sngle set of measurements, we can stll estmate the uncertanty n the mean.

6 Standard Error For a sngle set of measurements, the standard error s defned as: σ = standard devaton SE = σ = number of measurements In lab, we wll report the result of a set of measurements of a quantty x as x = x ± 2SE Are Results Consstent wth Theory? In 4A, we wll use the followng crtera to determne whether an expermental measurement of x s consstent wth the theoretcal predcton x thy : If x 2SE x x + 2SE thy then the experment result and the theoretcal predcton are consstent. That s, f the theoretcal result falls wthn two standard errors of the mean, the expermental result s consstent wth the model.