A = h w (1) Error Analysis Physics 141

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1 Introduction In all brances of pysical science and engineering one deals constantly wit numbers wic results more or less directly from experimental observations. Experimental observations always ave inaccuracies. In using numbers tat result from experimental observations, it is almost always necessary to know te extent of tese inaccuracies. If several measurements are used to compute a result, one must know ow te inaccuracies of te individual observations contribute to te inaccuracy of te result. If one is comparing a number based on a teoretical prediction wit one based on experiment, it is necessary to know someting about te accuracy of bot of tese if one is to say someting intelligent about weter or not tey agree. Systematic Errors Systematic errors are errors associated wit te particular instruments or tecniques used to carry out te measurements. Suppose we ave a book tat is 9" wide. If we measure its widt wit a ruler wose first inc as previously been cut off, ten te result of te measurement is most likely to be ". Tis is a systematic error. If a termometer immersed in boiling water at normal pressure reads C, it is improperly calibrated. If readings from tis termometer are incorporated into experimental results, a systematic error results. A voltage meter tat is not properly "zeroed" introduces a systematic error. An important point to be clear about is tat a systematic error implies tat all measurements in a set of data taken wit te same instrument or tecnique are sifted in te same direction by te same amount. Unfortunately, tere is no consistent metod by wic systematic errors may be treated or analyzed. Eac experiment must in general be considered individually and it is often very difficult just to identify te possible sources, let alone estimate teir magnitude, of te systematic errors. Only an experimenter wose skills ave come troug long experience can consistently detect systematic errors and prevent or correct tem. Random Errors Random errors are produced by a large number of unpredictable and unknown variations in te experiment. Tese can result from small errors in judgment on te part of te observer, suc as in estimating tents of te smallest scale division. Oter causes are unpredictable fluctuations in conditions, suc as temperature, illumination, line voltage, any kind of mecanical vibration of te experimental equipment, etc. It is found empirically tat suc random errors are frequently distributed according to a simple law. Tis makes it possible to use statistical metods to deal wit random errors. Propagation of errors - Part I If one uses various experimental observations to calculate a result, te result will be in error by an amount tat depends on te errors made in te individual observations. For example, suppose one wants to determine te area (A) of a seet of paper by measuring its eigt () and its widt (w): A = w () Suppose tat te measured eigt differs from te - -

2 actual eigt by, and te measured widt differs from te actual widt by w (see Figure ). w w + w + Fig.. Propagation of errors in te measurement of area A In tis case te calculated area will differ from te actual area A by A, and A will depend on and w: only on te fractional errors in w and. However, tis is not true in general. Measurement Errors If te errors in te measurements of w and in te previous section were known, one could correct te observations and eliminate te errors. Ordinarily we do not know te errors exactly because errors usually occur randomly. Often te distribution of errors in a set observations is known, but te error in eac individual observation is not known. Suppose one wants to make an accurate measurement of w and to determine te area of te rectangle in Figure. If we make several different measurements of te widt, we will probably get several different results. Te mean of measurements is defined as: w =! w i i = (4)!A = (w +!w)( +!) - A = w +!w + w! +!w! - A "!w + w! = A!w w +! () were w i is te result of measurement # i. In te absence of systematic errors, te mean of te individual observations will approac w. Te deviation d i for eac individual measurement is defined as: d i = w i - w (5) Te fractional error in A, wic is defined as te ratio of te error to te true value, can be easily obtained from equation ():!A A =!w w +! (3) In tis example, te fractional error in A depends Te average deviation of te measurements is always zero, and terefore is not a good measure of te spread of te measurements around te mean. A quantity often used to caracterize te spread or dispersion of te measurements is te standard deviation. Te standard deviation is usually symbolized by σ and is defined as: - -

3 ! = -! i = w i - w (6) Te square of te standard deviation σ is called te variance of te distribution. A small value of σ indicates a small error in te mean. It can be sown tat te error in te mean obtained from measurements is unlikely to be greater tan σ/ /. Tus, as we would expect, more measurements result in a more reliable mean. Te Gaussian Distribution probability tat te result of a measurement lies between - and is (wic is of course obvious). Te sape of te Gaussian distribution for various values of σ is sown in Figure. A small value of σ obviously indicates tat most measurements will be close to µ (small fractional error).! =. Te Gaussian distribution plays a central role in error analysis since measurement errors are generally described by tis distribution. Te Gaussian distribution is often referred to as te normal error function and errors distributed according to tis distribution are said to be normally distributed. Te Gaussian distribution is a continuous, symmetric distribution wose density is given by: P(x)! =.! =.5 P(x) =! " exp - x - µ! (7) Te two parameters µ and σ are te mean and te variance of te distribution. Te function P(x) in equation (7) sould be interpreted as follows: te probability tat in a particular measurement te measured value lies between x and x+dx is P(x)dx. Te normalization factor in eq. (7) is cosen suc tat:! -! P(x) dx = (8) Tis relation is equivalent to stating tat te - - Fig.. Te Gaussian distribution for various σ. Te standard deviation determines te widt of te distribution. In many applications te measurement errors are given in terms of te full widt at alf maximum (FWHM). Te FWHM of a Gaussian distribution is somewat larger tan σ: FWHM =! ln =.35! (9) Te Gaussian distribution can be used to estimate te probability tat a measurement will fall witin specified limits. Suppose we want to compare te x - 3 -

4 result of a measurement wit a teoretical prediction. If te measurement tecnique as a variance σ te probability tat te result of a measurement lies between µ - nσ and µ + nσ is given by: P(n) =! " µ + n! µ - n! exp - x - µ! () Te following table sows equation (9) evaluated for several values of n. n P(n) 68.3 % 95.4 % % For example, te oscillation period of a pendulum is measured to be 5.4 s ±.6 s. Based on its lengt one predicts a period of 7. s. Te table sows tat te probability on suc a large difference between te measured and predicted value to be.3 %. It is terefore very unlikely (altoug not impossible) tat te large difference observed between te measured and predicted value is due to a random error. Propagation of Errors - Part II Te determination of te area A discussed in "Propagation of Errors - Part I" from its measured eigt and widt was used to demonstrate te dependence of te error A on te errors in measurements of te eigt and widt. Te calculated error A is an upper limit. In most measurements te errors in te individual observations are uncorrelated and normally distributed. Te probability tat te errors in te measurement of te widt and te eigt collaborate to produce an error in A as large as A is small. Te teory of statistics can be used to calculate te variance of a quantity tat is calculated from several observed quantities. Suppose tat te quantity Q depends on te observed quantities a, b, c,... : Q = f(a,b,c,...) () Assume σ a, σ b, σ c, etc. are te variances in te observed quantities a, b, c, etc. Te variance in Q, σ Q, can be obtained as follows:! Q = "f "a! a + "f "b! b +... () Applying tis formula to te measurement of te area A, te standard deviation in A is calculated to be:! A = w! +! w (3) Te fractional standard deviation in A can be easily obtained from equation 3:! A A =! w w "! w w +! +! Example: Propagation of Errors (4) Suppose we want to calculate te force F using te following relation: - 4 -

5 F = 4! m R T! F = 4 " mr T! m m +! R R + 4! T T Te following values ave been obtained for m, R and T: = F! m m +! R R + 4! T T m =.4 ±. kg R =.53 ±. m T =. ±. s Wat is te calculated force F, and wat is its standard deviation? Te force F can be easily calculated: F = 7.9. Te standard deviation of te force can be obtained using te following formula:! F =!m! m +!R! R +!T Differentiating te formula for F we obtain:!m = 4! R T!R = 4! m T!T = - 8! m R T 3! T Substituting tese expressions in te formula for te standard deviation, we obtain: Substituting te numbers into tis equation, we obtain te following value for te standard deviation: σ F =.7 Te final answer is terefore: F = 7.9 ±.7 Example: Measuring a spring constant A spring was purcased. According to te manufacturer, te spring constant k of tis spring equals.3 /cm. For a spring, te following relation olds: F = k x, were F is te force applied to one end of te spring and x is te elongation of te spring. A series of measurements is carried out to determine te actual spring constant. Te results of te measurements are sown in Figure 3 and in Table. Since F/x = k, te last column in Table F () x (cm) F/x (/cm) Table. Results of a series of measurements of te spring constant

6 sows te measured spring constant k. Since k is independent of F and x, our best estimate for k will be te average of te values sown in te last column of Table : k =.98 /cm Figure 4 sows te ratio of F and x as a function of te applied force F. Te solid line sows te calculated spring constant of.98 /cm. x (cm) Fig F () Measured displacement x as a function of te applied force F. Te line sows te teoretical correlation between x and F, wit a spring constant obtained in te analysis presented below. Te standard deviation of te measured spring constant can be easily calculated: σ k =.6 /cm Statistical teory tells us tat te error in te mean (te quantity of interest) is not likely to be greater tan σ/ /. In tis case, = 5, and te error in k is unlikely to be larger tan.3 /cm. Te difference between te measured spring constant and te spring constant specified of te manufacturer is.5 /cm, and it is terefore reasonable to suspect tat te spring does not meet its specifications. k (/cm) Fig F () Calculated ration of F and x as a function of te applied force F. Te line sows te average spring constant obtained from tese measurements. Te standard deviation of te measured spring constant can be easily calculated: σ k =.6 /cm Statistical teory tells us tat te error in te mean (te quantity of interest) is not likely to be greater tan σ/ /. In tis case, = 5, and te error in k is unlikely to be larger tan.3 /cm. Te difference between te measured spring constant and te spring constant specified of te manufacturer is.5 /cm, and it is terefore reasonable to suspect tat te spring does not meet its specifications. Weigted mean Te calculation of te mean discussed so far assumes tat te standard deviation of eac individual measurement is te same. Tis is a

7 correct assumption if te same tecnique is used to measure te same parameter repeatedly. However, in many applications it is necessary to calculate te mean for a set of data wit different individual errors. Consider for example te measurement of te spring constant discussed in te previous Section. Suppose te standard deviation in te measurement of te force is.5 and te standard deviation in te measurement of te elongation is.5 cm. Te error in te calculated spring constant k is equal to:! k =! F x + F x! x For te first data point (F =. and x = 9.7 cm) te standard deviation of k is equal to.37 /cm. For te last data point (F = 5. and x = 5.5 cm) te standard deviation of k is equal to.7 /cm. We observe tat tere is a substantial difference in te standard deviation of k obtained from te first and from te last measurement. Clearly, te last measurement sould be given more weigt wen te mean value of k is calculated. Tis can also be illustrated by looking at a grap of te measured elongation x as a function of te applied force F (see Figure 5). Te teoretical relation between x and F predicts tat tese two quantities ave a linear relation (and tat x = m wen F = ). Te data points sown in Figure 5 ave error bars tat are equal to ± σ. Te dotted lines in Figure 5 illustrate te range of slopes tat produces a linear relation between x and F tat does not deviate from te first data point by more tan standard deviation. Te solid lines illustrate te range of slopes tat produces a linear relation between x and F tat does not deviate from te last data point by more tan standard deviation. Obviously, te limits imposed on te slope (and tus te spring constant k) by te first x (cm) data point are less stringent tan te limits imposed by te last data point, and consequently more weigt sould be given to te last data point. Te correct way of taking te weigted mean of a number of values is to calculate a weigting factor w i for eac measurement. Te weigting factor w i is equal to w i =! i were σ i is te standard deviation of measurement # i. Te weigted mean of independent measurements y i is ten equal to y =! w i y i i =! w i i = First data point Last data point were y i is te result of measurement # i. Te standard deviation of te weigted mean is equal to 3 F () Fig.5. Measured elongation x as a function of applied force F. Te various lines sown in tis Figures are discussed in te text

8 ! y =! w i i = For te measurement of te spring constant we obtain: and k =.95 /cm σ k =.4 /cm Te results obtained in tis manner are sligtly different from tose obtained in te previous section. Te disagreement between te measured and quoted spring constant as increased