1 Random and systematic errors

Transcription

1 1 ESTIMATION OF RELIABILITY OF RESULTS Such a thing as an exact measurement has never been made. Every value read from the scale of an instrument has a possible error; the best that can be done is to say that it lies within certain limits. The more accurately the reading is made, the closer together lie the limits. The possible error arises for two reasons: (a) inability of the observer to make an exact reading of the scale, and (b) in inaccuracies inherent in the instrument itself. The former introduces an error of observation, the latter an instrumental error. When the results of a measurement are given, the final answer at least, should be accompanied by an error estimate. It is not necessarily a virtue to produce an answer with a small error. Accurate answers are impossible to obtain from some crude instruments. The error is a measure of your confidence in the answer, and too low an error is, if anything, worse than too high a value. It would amount to your making a higher claim than is warranted. 1 Random and systematic errors Some errors appear consistently each time a reading is taken, for example, if a metre scale was drawn with slightly wrong spacings, this would influence all repeated readings in the same way. We call this a systematic error. Such errors are hard to detect and difficult to treat in any general way. Each experiment has to be examined individually for systematic errors. Random errors are those which come up differently each time a reading is taken. They are statistical in origin and can be treated using statistical methods. Repeated readings of the same quantity will give a statistical sample and this serves both to provide a better answer and to estimate the random error. Rules for combining random errors The final answer is usually obtained by an algebraic calculation from several independent instrument readings, for example, a resistance may be calculated as the ratio of a voltmeter reading to an ammeter reading. Both these

2 Reliability of Results readings will contain random errors which can be estimated and it is then required to calculate the random error on the resistance value. By successive application of the following relations, any algebraic formula can be manipulated. For two independent quantities, the errors may be combined to give Sum and Difference (A ± A) + (B ± B) = (A + B) ± (A ± A) (B ± B) = (A B) ± Product and Ratio (A ± A)(B ± B) = (AB)[1 ± ( A A ) + ( B B ) ] ( A) + ( B) ( A) + ( B) A ± A B ± B = A B [1 ± ( A A ) + ( B B ) ] Notice that products and ratios involve fractional errors so percentage may here be used conveniently, while sums and differences involve the absolute error, and percentage errors should not be used. The rules are special applications of a more general error formula. If the result R is a function of measurements A, B and C etc. where R = f(a, B, C,...) then R = ( f A ) ( A) + ( f B ) ( B) + In some algebraic relations, for example, log, exp, sin, etc., it may be quicker to use this relation directly to combine the errors. Example Voltmeter reading = 19.5 ± 1 volts. Ammeter reading = 4.3 ± 0.5 amps. If the resistance is accurately given by the ratio of the voltage to the current (which may not be so if there are internal resistance effects), then

3 Reliability of Results 3 Resistance = 19.5 ± ± 0.5 Ω = ± ( ) + ( ) Ω = 4.53 (1 ± 0.13) Ω and clearly the ammeter error is more serious than the voltmeter error. This could have been given as 4.5(1 ± 0.1) ohms. A quicker way to calculate product and ratio errors is to use percentage. Taking the sample example, Voltmeter reading = 19.5 ± 1 volts = 19.5 ± 5% volts Ammeter reading = 4.3 ± 0.5 amps = 4.3 ± 1% amps You see immediately that the ammeter error is the more serious Resistance = ± (5 + 1 )(%) Ω = 4.53 ± ( )(%) Ω = 4.53 ± 169(%) Ω = 4.5 Ω ± 13% In fact, the effect of V turns out to be negligible in R, in this example! 3 Use common sense in applying error formulas 1. Do not give more figures in the answer than are justified by the error.. Usually only one figure is needed in the estimate of the error. 3. Do not forget about systematic errors. If you feel that the random error, as obtained by applying the rules is much smaller than is reasonable, look around for systematic errors and mention them in the final results.

4 4 Reliability of Results 4 Repeated measurements of the same quantity This is quite a different situation from the previous discussion on combining independent quantities. Suppose you repeat the same measurement N times. Each time you get a slightly different value because of random errors. Suppose that the measurements are A 1 A A 3...A i...a N, and that the random error on each measurement is A. Then the best estimate for the value of A is given by the mean or average value of all the readings. Because it is a better estimate than any one reading we expect that the error will be smaller than A, and it is, but how much smaller? The best value for A 1 A A 4... A N repeated measurements is A ± A = A 1 + A + A 3 + A A N N ± A ( ) 1 i=n = A ± A N N i=1 N This can be derived from the error formula for a sum. The error on the mean value A is decreased by the square root of the number of times the measurement is repeated. It is well worthwhile repeating the measurements on the most critical readings, but it may not be worth doing a large number of repeats. For example, four repeats give twice the accuracy, but you need 16 measurements to double the accuracy again. 5 Use of computers and programmable calculators Data reduction is now much faster than in the old days of logarithmic tables and slide rules, because now all the arithmetic can be done by computer or hand calculator. Mostly the algebraic methods are unchanged, but in some parts of the analysis it is now worthwhile to use more elaborate methods even if these have much more complicated algebra, if in doing so a better estimate of the result can be obtained. Certain mathematical procedures are the same for the data reduction of many types of experiment and standard mathematical techniques have been developed for these procedures. Some computer graphics programs and hand calculators have built-in algorithms for analysing data, which can be run by pressing a single key. The use of

5 Reliability of Results 5 computers is encouraged. But remember that there is no advantage if it takes longer to use the computer than to do the same analysis without it. Also the device does not relieve you of the responsibility of knowing what the numbers mean. It merely does the arithmetic for you. A program for doing a linear least squares fit is commonly available in graphics programs, and also on hand calculators. In statistics this process is called linear regression. However, statisticians use different methods to estimate the reliability of their answers and do not estimate errors in the same way. 6 Straight line fit to experimental points The method of looking at the points and estimating by eye the best line is still good. If you use a calculator method it is important to verify by eye that the results are sensible. y b a x Suppose we have a series of n measurements y i each of which is obtained for a setting x i of the apparatus. The calculator uses the ordered pairs (x i, y i ) where i = 1...n, as input data. These represent the n points on a graph. The equation for the best straight line

6 6 Reliability of Results y = ax + b contains the value of the slope a, and the intercept b which are to be estimated from the data. We make the assumptions 1. The errors in x are neglected.. The errors in y are all the same, σ 1 = σ = σ 3 etc. 3. There is a Gaussian probability distribution for the values of y i on either side of the mean of many repeated measurements y i. If all these conditions are satisfied then the best straight line is such that the sum of the squares of the vertical distances of each point from the line is minimised. That is ni=1 (y i y) = minimum The mathematical condition for a minimum is that the derivative of the left side should be zero. Differentiating with respect to a Therefore 0 = a (y yi ) = (ax a i + b y i ) = (ax i + b y i ) (ax a i + b y i ) = (ax i + b y i )x i a x i + b x i = x i y i (1) Similarly differentiating with respect to b gives 0 = b (y yi ) = (ax b i + b y i ) = (ax i + b y i ) (ax b i + b y i ) = (ax i + b y i ) Therefore a x i + bn = y i ()

7 Reliability of Results 7 These simultaneous equations can be solved to give the best values. Krammer s rule gives immediately a = 1 1 y x xy and b = 1 y x xy x 1 x where = x x. We use the notation for the averages y = 1 yi, so that x n = 1 n (x i ) and xy = 1 (xi y n i ). Notice that x is not x nor is (xy) the same as (x)(y). These best values are called least squares fit values and they are programmed in several hand calculators. They are calculable with a non-programmable calculator, though it takes more time. The danger with such a programme is that it is easy to find values of a and b to 8 digit accuracy by just entering the data. But how accurate are these values? 7 Error estimates on least squares fit There are two alternatives. You should choose the most appropriate for your particular experiment. 7.1 Method 1 Estimate of error from quality of fit of points to line Here nothing is assumed about the precision of measurement on any one point. If we have information on the accuracy of the meters, scale reading accuracy of a micrometer etc., this is not used.

8 8 Reliability of Results y b a x The quantity which has been minimised in the least squares fit is the error σ, often called the standard deviation where σ = 1 (y yi ) = 1 (axi + b y i ) n n We use a factor n, rather than n because we have used up degrees of freedom in finding a and b, so there are only n degrees of freedom left. Obviously if we have only two points, the fitted line goes exactly through both, and (y y i ) would be zero. The effect which the error on one particular point (x i, y i ) has on the best fit values of a and b can be written a b σ ai = σ i and σ bi = σ i y i y i where σ i is the error (y y i ) of the point from the line. A good estimate for the error on each point is to use the root-mean-square error on all the points previously obtained, so σ i = σ. The estimated error on the slope σ a is given by n ( σa = i=1 σ a y i ) = σ ( a y i ) = σ ( 1 y i 1 y x xy )

9 Reliability of Results 9 n = σ 1 1 x x i = σ (xi x) = σ ( x n n i x i x + x ) = σ ( x x + x ) = σ ( x x ) σa = σ Similarly the error σ b on b is given by σb = ( σ b ) ( = σ y i y i y x xy x ) (3) n = σ 1 x x i x = σ ( ) x xx n i = σ ( x ) xx n x i + x x σ ( i = x ) x x + x x σb = σ ( x x ) = σ x x (4) The value of σ is most easily calculated numerically from σ = 1 (y yi ) = n n ( y axy by ) n Not much more calculation is needed. Values of x, y, x, xy, and x are already known from the slope calculation. However, y must be calculated to find the errors on the slope a and the intercept b. These relations are only valid if all the points are assumed to have the same precision. If some points are determined less well than others in measurement, for example in a radioactive decay later points have fewer counts and are less precise, this method not only gives the wrong errors, but also the wrong answers for a and b.

10 10 Reliability of Results 7. Method Estimate of error from errors on individual points Here it is assumed that the instrumental error of the apparatus which was used to measure y i is known beforehand. So that σ i is known for each reading y i. In the special case when all σ i are the same, the analysis reduces to the previous case, with the same best fit values of a and b. However σ a and σ b use the known value σ rather than the fit to the line in the equations 3 and 4. y b a x If some points are more accurate than others, we must weight these more than the others. A weighted least squares fit is required. The weight of each point is ω i = 1, where σ σi i is its error, assumed known beforehand. The formulas for the best values of a and b are the same as before if the average values are redefined as weighted average values y = (yi /σ i ) (1/σ i ) x = (xi /σ i ) (1/σ i ) xy = (xi y i /σ i ) (1/σ i ) x = (x i /σ i ) (1/σ i ) The formulas for the errors on a and b are also the same except that the

11 Reliability of Results 11 error estimate σ now is calculated from σ = 1 / ( 1 ), rather than by using σi the standard deviation from the line. Notice that if the instrumental errors have been underestimated, perhaps because some source of error has been forgotten, the method may give too small an error for the slope and intercept, and method 1 is more valid. On the other hand, in an extreme example, suppose there were only two points on the graph, then the best straight lines goes exactly through both and method 1 gives zero error. For 3 or 4 points it is possible that they may be in line by chance even though individual points have large errors, so that it is dangerous to use method 1 for a small number of points. 7.3 Special case where the line must go through the origin The equation for the straight line is here restricted to y = ax and the function to be minimised is (y i y) = (y i ax i ). Differentiating with respect to a gives the condition for the minimum The best slope is x i (y i ax i ) = 0 a = xi y i (xi ) = (xy) (x ) The error on the slope σ a is calculated as before, and we find σ a = σ /(x ). The same as with method 1. The standard deviation of all the points is obtained from σ = n ( ) y a x n.

12 1 Reliability of Results In Method, the standard deviations are known from the apparatus properties. Reference Data Reduction and Error Analysis for the Physical Sciences. P.R.Bevington, McGraw-Hill 199, 1969.