1 APPENDIX 1 ERROR ESTIMATION Measurements are always subject to some uncertainties no matter how modern and expensive equipment is used or how careully the measurements are perormed These uncertainties are oten called errors although they are not errors in that way the word error is used in everyday lie where it reers to mistakes or even blunders In the laboratory measurements the errors mean the estimated uncertainties due to the limitations o the equipment and the observer The possible uncertainties are divided into: 1) Systematic errors, which aect all readings in the same way or in some regular manner They are usually caused by the instrument and i they are recognized they can be avoided For example, the zero error o the micrometer is a systematic error ) Random errors, which cause the measurable values o a quantity to be inconsistent when the measurement is repeated several times They are caused either by the instrument or by the observer and they are unpredictable When the measurement results are reported error estimation concerning the random errors should always be included In act, the measurement result without any inormation on its uncertainty is rarely useul Thereore, it is important to learn to estimate the uncertainties rom the very beginning The error estimation has two important purposes: 1) To ind out which actors have the maximum eect on the uncertainty o the result For example, i you had weighed the studied ring with a beam balance, you would probably have noticed that the large uncertainty o the mass causes large uncertainty to the density o the ring Then, you could improve the measurement equipment by using the analytical balance instead o the beam balance ) To know the suitable numerical precision o the result When the measurement results are reported their numerical precisions are indicated by writing as many digits as are signiicant The signiicance o digits can be determined by using the estimated uncertainties There are two various levels in the error estimation: Level 1: Error estimation o a directly measurable quantity This can be done either by estimating the uncertainty o the measured quantity directly rom the measurement precision o the instrument or by measuring the quantity several times when the uncertainty can be calculated or example by using the maximum deviation rom the mean value I the quantity is measured just once its absolute experimental uncertainty is given by the measurement precision o the instrument For example, i you had measured the height o the cylindrical ring in Exercise 1 just once, the absolute uncertainty o the height would be either the measurement precision o the micrometer, eg 001 mm or
APPENDIX 1 ERROR ESTIMATION one hal o it eg 0005 mm I you are not sure which one o these two uncertainties to use choose the larger one, because in the error estimation the maximum uncertainty is determined So, in Fig 14 in the instruction o the Exercise 1 the height o the ring is o the orm (159 ± 001) mm In the Exercise 1 the ring was weighed just once and so the absolute uncertainty o the ring s mass is equal to the measurement precision o the analytical balance I it is possible the quantity is measured several times and then the actual measurement result is the mean value o all the observed values The absolute uncertainty o the mean value is either the maximum deviation rom the mean value, the standard deviation or the standard error o the mean value In this exercise the diameters as well as the height o the ring are measured ten times and the maxima deviations rom the mean values are calculated However, sometimes the maximum deviation is smaller than the measurement precision o the instrument used In this case the absolute uncertainty is given by the measurement precision Level : Quantity cannot be measured directly but it is a unction o two or more quantities which are directly measurable In this case the absolute or the ractional error o the quantity can be calculated with a total dierential method In the next the ollowing symbols are used: is a quantity to be determined and it depends on the independent quantities according to the equation = ( x, ), x, is the absolute uncertainty (or the absolute error) o the quantity, is the ractional or relative uncertainty (or the ractional error) o the quantity and it is oten given in per cents when it can be called the percentage uncertaint x, are the absolute uncertainties o the measured quantities x, and they are given or example by the measurement precisions o the instruments used or by the maximum deviations rom the mean values, x x, y z z,are the ractional uncertainties o the measured quantities and they have been calculated by dividing the absolute uncertainties with the measured values o the quantities, a, b, c, are constants which can be assumed to be precise
1 Total dierential method or estimating absolute uncertainty Determining o the absolute uncertainty o the quantity = ( x, ) is based on a calculation o a total dierential d o the quantity The total dierential d is o the orm d = dx dy dz, (L11) x y z where x, y and z are the partial derivatives o with respect to the quantities x, As an application o the total dierential we get an equation which is important in error estimation According to this equation the maximum value o the absolute uncertainty o the quantity is given by x y z (L1) x y z It is worthwhile to notice that the worst possible case, eg the maximum possible absolute uncertainty is considered in Eq (L1) The absolute values are used in the right side because we think that the errors do not cancel each other Example 11 Consider a quantity o the orm = bx cy az Derive using the total dierential method an equation or the maximum possible absolute uncertainty o the quantity and derive with it an expression or the maximum possible ractional uncertainty Solution: For Eq (L1) the partial derivatives o the unction ( x, z) with respect to the quantities x, y and z must be calculated These are obtained in the same way as the common derivatives by keeping one quantity at a time as a variable and by assuming all the others to be constants The partial derivatives are o the orm x = bx ; = cy; = a, y z and so Eq (L1) gives or the maximum possible uncertainty bx x cyy az = bx x cyy az The maximum ractional uncertainty is thus
4 APPENDIX 1 ERROR ESTIMATION bx bx x cy az bx cyy cy az bx az cy az Total dierential method or estimating the ractional uncertainty Sometimes it is easier to determine irst the ractional uncertainty o the quantity using the logarithmic total dierential method and then to calculate the absolute uncertainty by multiplying the ractional uncertainty with the observed value o the quantity This method is useul especially in the cases where the unction o the determined quantity includes only multiplications, divisions or/and powers o the measured quantities In this method the natural logarithm ln o the quantity is calculated irst and then the total dierential d (ln ) o the logarithm is determined By using Eq (L11) we get (ln ) (ln ) (ln ) d(ln ) = dx dy dz (L1) x y z Because d (ln ) = d we get or the maximum ractional uncertainty (ln ) (ln ) (ln ) x y z x y z, (L14) where the absolute values are again used because we calculate the maximum possible ractional uncertainty Example 1 Determine the maximum possible ractional uncertainty o the quantity = axy (bz) and calculate using it the maximum possible absolute uncertainty Solution: First we derive an expression or the natural logarithm o the quantity ln axy = ln = ln a ln x ln y lnb ln z bz and then we calculate the partial derivatives in Eq (L14) ( ln ) 1 ( ln ) 1 ( ln ) 1 = ; = ; = x x y y z z According to Eq (L14) the maximum possible ractional uncertainty is
5 (ln ) (ln ) (ln ) x y z x y z x = x y y z z x = x y y z z Thus, the maximum possible absolute uncertainty is = ay bz x y z = x y z ax axy x y z bz bz axy bz Presenting the results The results rom the laboratory measurements are compiled to the Results-section o the report Special attention is paid to the numerical precision o the results Usually the results are given with their absolute and ractional uncertainties Dierent auxiliary and end results may be presented also in Analysis-section o the report where the calculations are reported to the reader This section includes or example dierent model calculations whose purpose is to clariy how the analysis proceeds The rules concerning the calculations, the rounding and the presentation o the inal and the auxiliary results are as ollows: 1) Only the end results are rounded Nowadays all the calculations are usually done by using a computer or at least with a calculator and so the numerical precision o the quantities used and o the results can easily be high It is important not to round the auxiliary results used when the end results are calculated For example, in the irst exercise the rounded value o the volume should not be used when calculating the density o the ring ) When the auxiliary results are presented the numerical precision is determined by the rules: a) When the expression o the calculated quantity includes additions and subtractions the result is presented with as many decimals as in the least precise observed quantity b) I the expression includes multiplications and divisions the result is presented with as many signiicant digits as in the least precise observed quantity ) When the result is presented with its absolute uncertainty using an expression the result and the uncertainty must be in the same orm I the result is
6 APPENDIX 1 ERROR ESTIMATION presented in a scientiic notation (eg with a power o ten) or with a preix and the uncertainty in a decimal notation or with a dierent preix the irst task is to decide which one is the best presentation Then either the uncertainty is presented with the same scientiic notation or with the same preix as the result or vice versa The result and its uncertainty are always given with the same decimal precision Example 1 When weighing an object, the result was m = 75 g and the determined uncertainty o the mass was m = 0 mg In this case the suitable unit in the presentation is g and so the uncertainty is o the orm m = 000 g The result could be presented in the orm m = (75 ± 000) g Furthermore, we must investigate i this orm is consistent with a 15-unit rule considered next 4) I the result and its uncertainty are presented with the same decimal precision, a 15- unit rule can be used when rounding the result and the uncertainty to the right numerical precision The aim is to present the result with as many digits as are signiicant According to the 15-unit rule all the digits with uncertainty less than 15 units are signiicant and must be written to the result When rounding the result and its uncertainty it must be remembered that the result is rounded normally but the uncertainty is always rounded upwards According to the 15-unit rule the maximum uncertainty presented can thus be o the orm 0015, 015, 15 etc but i the uncertainty is or example 016 one number should be extracted and the right numerical precision is in this case 0 The numerical precision o the ractional uncertainty must also be determined with the 15-unit rule It says that the maximum ractional uncertainty could be or instance o the orm 015 %, 15 %, 15 %, but also in this case the ractional uncertainty 16 % should be rounded to the orm % Example 14 When a volume V o a ring was measured in a laboratory exercise the result attained was V = 18476 cm and the absolute uncertainty o the volume was ound to be V =1147 mm Present the result with the right numerical precision with its absolute and ractional uncertainties Solution: In this case the cm -units are suitable or presentation and so the absolute uncertainty must at irst be changed to these units and to be presented with the same decimal precision as the result Then the absolute uncertainty is V = 0115 cm The auxiliary presentation o the result is thusv = ( 18476 0115) cm In this presentation the digit 6, which is the last digit o the result has uncertainty o 115 units which is much more than 15 units So, this presentation is not consistent with the 15-unit rule The next possible presentation is achieved when the rightmost digit is extracted both rom the result and rom the uncertainty Now we get a presentation o the ormv = ( 1847 01) cm The uncertainty o the last digit o the result is 1 units which is still greater than the allowed 15 units So, the rightmost digits o the result and the uncertainty are again extracted and the presentation
7 V = ( 1847 01) cm is achieved Now, the last digit 7 o the result has uncertainty o only 1 units, which is less than 15 units and the 15-unit rule is ulilled The ractional uncertainty is in this case V V = ( 0115 18476) 100 % = 06578 % According to the 15-unit rule the result and the ractional uncertainty must be presented in the orm V = 1847 cm 07 % The right presentation o the ring s volume is thus V = (1847 01) cm = 1847 cm 07 %