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.I \ 4.1 Introduction : Measurement is a process of comparing an unknown quantity with a standard quantity. The knowledge of any parameter depends on the nieasurement. The indepth knowledge the parameter oan be easily understood by the use of measurement and'iri-odiflcation *uJJ, is a of i= process of monitorinji'''our.*ing. a parameter" using,lnrtru,o.rtruno expressing the parameter in terms of meaningful numbers. Measuring The measuring instrument is a device for determining the magnitude of the parameter being measured. Y ff.2 Definitions : 1' Accuracy : It is the degree of correctness or closeness with which the measured reading approaches the true (expected) va1ue. It denotes of the quantity and also indicates the ability of the inslrument to measure the true value" 2' 3' Precision : It is the measure of consistency or repeatability of measurement" It rei'ers to the degree of closeness within a group of measuiement" Iiis usually expressed in terms of deviations in measurement. Resolution : It is the smallest increment of quantity being measurement be detected" The smallest change in a measured variable respond. 4' Significant Figures rvhich can to lvhich an instrument will : The significant figure conveys the actual information magnifude and precision of the quantity. about the For example : A voltage of I l0V, specifled by an instrument may be closer to l09V or I i I v. ll0.0V is closer to 110.1 V oi toq.sV. T'hus there are row Thus are 3 signiflcant figure while four signifi cant figures" p.3 Types of Emor : The difference between the measured value and the true value of the quantity is known as static error" The static errors are categorised as 1" Gross errors (or) Human errors. 2. Systematic errors 3. Random errors. Gross Errors : These errors are mainly due to human mistakes in observing and recording the quantity. These error occur due to incorrect adjustments of instruments and computational mistakes. These errors cannot be treated mathematically. The elirnination of gross error is not possible, but can be minimised. Sunslar ENam Sra,aaw L :

I Elprh,e,u-oIn*&.wrweatf,a.{uotu IItr SetwEC/TC A constant deviation of the operation of the instrument is known as 'Ihe systematic errors are mainly resulting due to the short comings of systematic errors. the instrument and the characteristics of the instrument, such as wom parts, ageing efrect, Systematic Errors : environmental effects etc. There are 3 types of systematic errors L lnstrumentai Errors 2" Envirorunental Errors 3. Observed Errors. Random Errors : The random errors are accidental, small and independent" These errors cannot be predicted and cannot be determined in the ordinary process of taking the measurement. These errors are generally small. I-lence these errors are of real concern only u'hen high degree of accuracy is required" 'fhe only way to reduce these errors is by increasing the number of obsen'ations and using the statistioai method to obtain the best approximation of the reading' t4.4 Absolute and Relative Errors : Absolute Error : When the error is specified in terms of an absolute quantity and not as a percentage, then it is cailed an absolute error^ * 0.5V as an absolute elror. Relative Error : When the en"or is expressed as a percentage or as a fraction of the toral quantity to be measured, then it is called relative error' Eg : 10 + 0.5V indicates Eg:100f) t /. 5% then + 5%o \ or + { } ]it tt . r.lative error [ 20, Relative error is also called fractional error. t,{"5 Statistical Analysis ' : The objective of the statistical method is to achieve consistencl'of the measured value and not their accuracy. To make statistical analysis meaningful a large number of measurements is usually required. 1. Arithmetic mean : The most probable value of measured variable is the arithmetic mean of the number of readings taken. The best approximation of the quantity is possible when the number of readings of the quantity is very large' The arithmetic mean of n measurements of the variable x is given by the expression. Xl + X: t -. -"-, Where i *""""* Xn lxn -+ fuithmeticmean xr toxn -+ n X, i" tonth readingtaken -> Total number of readings 2. Deviation from the mean : This is the deviation of a given reading from arithmetic mean of the group ofvalues. ' &nsfar Facal,,.Sr,z.nnw

lr/lod'yilp, - 1 M ea.++w et4ae.ant o"aqd-, E r r o-r The deviation of the first reading x I is d i and that of second reading x2 is d2 and so on lhe deviation fiom mean can be expressed 1'he algebraic sum as d, : x, - x, d, :az - x etc - of deviation is zero. Deviations : It is def,ned of an infinite number of data is defined as the square root of the sum of individual deviations squared divided by the number of readings. 3" Average Thus standard deviation is expressed as o= Practically the number of reading is finite, when the number of reading is small (n < 30) the denominator is (n - 1) and the standard deviation is represented by S. di +d?r+d]+......+d: S- The standard deviation is also called as mean square deviation. 5" Variance : The square ofstandard deviation is called variance. v = (o)' = [,8.l= n \r ) " Forsmallnumberof rea drng t 9.6 n Id' V=5"=n_l Problems on Statistical Analysis : For the given readings 1.34, 1.38 , 1.56, 1.47, 1.42, 1.43, 1.54, 1.48, 1.49, 1. Calculate. L."Arithmetic Mean 3. Standard deviation Sol rd' : 1.50. 2. Average deviation 4. Variance Given: Readings ; 1.34, 1.38, 1.56, L47, L43,1.54,1"48,1"49, 1.50 n: 1. 10 [ 10 readings] Arithmetic Mean : n Y" - i=, =a[x, + x2 + x, +,..... + xn] n n' ,(-J'^t x I 1.34 + 1.38 + 1.56 + 1.47 + 1.42 + I "43 + I.54 + 1.48 + 1.49 + 1.50 10 ii. Averagc deviation dr = Xr -x : = I .34 -1.461 = -0"121 d, -- \, - x = 1.38 - I 461= -0.081 Sfl,1star Exo,m Sr,z,nar.r 3

I III Electrotnbllfi ,*ru,tnrueacf,o,fi.ox^v Sanq/EC/TC d: = X: - x = 1.56 -1.461= +0.099 do = X, - x = 1.47 - 1.461= +0.009 ds = Xs -i=1.42-1.461= +0.041 do = Xe -i =1.43 -1.461= +0.031 dt =x, - x = 1.54 -1.461= +0.079 - x = 1.48 -1.461= +0.019 ds = Xs -i=1.49 -I.461= -0.029 dr = Xs d,o = X,o -f = 1.50 -1.461= 0.039 -.-- f ldl ---..- - r---:-a: Averagedevlatlon ^ n 0.121 + 0.081 + 0.099+0.009 + 0.041 + 0.031 + 0.079 + 0.019 + 0.029 + 0.039 10 iii" Standard Deviation : Since the number of readings is < 30. Standard eviation is S = (o.ztz)' + (o"os /=-; ^lLo' V n-l r)' + (o.oee)'? + (o.ool ' + (o.o+t)' + (o.o: t ls = 01688l iv. Varia nce : V=Sj= (o.ooss)' = 0.004733 iy oro4d = 2. For the following readings calculate Arithmetic mean. ii. Deviation of each value iii. Algebraic sum of deviations i. 49.7 Sol : , 50.1, 50.3, 49.5, 49.7 Given : Readings : n: 5 12{ t!9Ji{'3 49"7,50.1, 50.3, 49.5,49.7 (i)e'ritirmeticMean:x = ,: xra x' I:::::-xo n5 = + 49'5 + 49'7 +q.ae (ii) Deviations from each vaiue d, = X, -i du = xz 4 = 49.7 - - * = SO.i - 49 "86 = -0^ 16 49.86 = +0.24 &ns*ar Fn<a,v.Su,naw

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