Measurement Uncertainty -- Summary; Quiz / Problems

This section introduced the subject of measurement uncertainty, and the length of the section gives testimony to the great importance attached to this subject. Measurement errors are a fact of life and, although we can do much to reduce the magnitude of errors, we can never reduce errors entirely to zero. We started the section off by noting that measurement uncertainty comes in two distinct forms, known respectively as systematic error and random error. We learned that the nature of systematic errors was such that the effect on a measurement reading was to make it either consistently greater than or consistently less than the true value of the measured quantity. Random errors, however, are entirely random in nature, such that successive measurements of a constant quantity are randomly both greater than and less than the true value of the measured quantity.

In our subsequent study of systematic measurement errors, we first examined all the sources of this kind of error. Following this, we looked at all the techniques available for reducing the magnitude of systematic errors arising from the various error sources identified. Finally, we examined ways of quantifying the remaining systemic measurement error after all reasonable means of reducing error magnitude had been applied.

Our study of random measurement errors also started off by studying typical sources of these.

We observed that the nature of random errors means that we can get close to the correct value of the measured quantity by averaging over a number of measurements, although we noted that we could never actually get to the correct value unless we achieved the impossible task of having an infinite number of measurements to average over. We found that how close we get to the correct value depends on how many measurements we average over and how widely the measurements are spread. We then examined the two alternative ways of calculating an average in terms of the mean and median value of a set of measurements. Following this, we looked at ways of expressing the spread of measurements about the mean/median value. This led to the formal mathematical quantification of spread in terms of standard deviation and variance. We then started to look at graphical ways of expressing the spread. Initially, we considered representations of spread as a histogram and then went on to show how histograms expand into frequency distributions in the form of a smooth curve. We found that truly random data are described by a particular form of frequency distribution known as Gaussian (or normal).We introduced the z variable and saw how this can be used to estimate the number of measurements in a set of measurements that have an error magnitude between two specified values. Following this, we looked at the implications of the fact that we can only ever have a finite number of measurements. We saw that a variable called the standard error of the mean could be calculated that estimates the difference between the mean value of a finite set of measurements and the true value of the measured quantity (the mean of an infinite data set). We went on to look at how this was useful in estimating the likely error in a single measurement subject to random errors in the situation where it’s not possible to average over a number of measurements. As an aside, we then went on to look at how the z variable was useful in analyzing tolerances of manufactured components subject to random variations in a parallel way to the analysis of measurements subject to random variations. Following this, we went onto look at Χ² distribution. This can be used to quantify the variation in the variance of a finite set of measurements with respect to the variance of the infinite set that the finite set is part of. Up until this point in the section, all analysis of random errors assumed that the measurement set fitted a Gaussian distribution. However, this assumption must always be justified by applying goodness of fit tests, so these were explained in the following section, where we saw that a Χ² test is the most rigorous test available for goodness of fit. A particular problem that can affect the analysis of random errors adversely is the presence of rogue data points (data outliers) in measurement data. These were considered, and the conditions under which they can justifiably be excluded from the analyzed data set were explored. Finally, we saw that yet another problem that can affect the analysis of random errors is where the measurement set only has a small number of values. In this case, calculations based on z distribution are inaccurate, and we explored the use of a better distribution called t distribution.

The section ended with looking at how the effects of different measurement errors are aggregated together to predict the total error in a measurement system. This process was considered in two parts. First, we looked at how systematic and random error magnitudes can be combined together in an optimal way that best predicts the likely total error in a particular measurement. Second, we looked at situations where two or more measurements of different quantities are combined together to give a composite measurement value and looked at the best way of dealing with each of the four arithmetic operations that can be carried out on different measurement components.

Quiz / Problems

1. Explain the difference between systematic and random errors. What are the typical sources of these two types of errors?

2. In what ways can the act of measurement cause a disturbance in the system being measured?

3. In the circuit shown in Fgr. 17, the resistor values are given by R1 = 1000 Ohm ;

R2 = 1000 O ;V = 20 volts. The voltage across AB (i.e., across R2) is measured by a voltmeter whose internal resistance is given by Rm = 9500 Ohm.

(a) What will be the reading on the voltmeter? (b) What would the voltage across AB be if the voltmeter was not loading the circuit (i.e., if Rm = infinity)? (c) What is the measurement error due to the loading effect of the voltmeter?

4. Suppose that the components in the circuit shown in Fgr. 1a have the following values:

If the instrument measuring the output voltage across AB has a resistance of 5000 Ohm, what is the measurement error caused by the loading effect of this instrument?

5. (a) Explain what is meant by the term "modifying inputs." (b) Explain briefly what measures can be taken to reduce or eliminate the effect of modifying inputs.

6. Instruments are normally calibrated and their characteristics defined for particular standard ambient conditions. What procedures are normally taken to avoid measurement errors when using instruments subjected to changing ambient conditions?

7. The voltage across a resistance R5 in the circuit of Fgr. 18 is to be measured by a voltmeter connected across it.

(a) If the voltmeter has an internal resistance (Rm) of 4750 Ohm, what is the measurement error? (b) What value would the voltmeter internal resistance need to be in order to reduce the measurement error to 1%?

8. In the circuit shown in Fgr. 19, the current flowing between A and B is measured by an ammeter whose internal resistance is 100 Ohm. What is the measurement error caused by the resistance of the measuring instrument?

9. What steps can be taken to reduce the effect of environmental inputs in measurement systems? 10. (a) Explain why a voltmeter never reads exactly the correct value when it’s applied to an electrical circuit to measure the voltage between two points.

(b) For the circuit shown in Fgr. 17, show that the voltage Em measured across points AB by the voltmeter is related to the true voltage Eo by the following expression:

(c) If the parameters in Fgr. 17 have the following values, R1 = 500 Ohm;R2 = 500 Ohm; Rm = 4750 Ohm, calculate the percentage error in the voltage value measured across points AB by the voltmeter.

11. The output of a potentiometer is measured by a voltmeter having resistance Rm, as shown in Fgr. 20. Rt is the resistance of the total length Xt of the potentiometer, and Ri is the resistance between the wiper and common point C for a general wiper position Xi. Show that the measurement error due to the resistance, Rm, of the measuring instrument is given by....

Hence show that the maximum error occurs when Xi is approximately equal to 2Xt/3.

(Hint: differentiate the error expression with respect to Ri and set to 0. Note that maximum error does not occur exactly at Xi = 2Xt/3, but this value is very close to the position where the maximum error occurs.)

12. In a survey of 15 owners of a certain model of car, the following values for average fuel consumption were reported:

Calculate mean value, median value, and standard deviation of the data set.

13. The following 10 measurements of the freezing point of aluminum were made using a platinum/rhodium thermocouple:

658.2 659.8 661.7 662.1 659.3 660.5 657.9 662.4 659.6 662.2 Find (a) median, (b)mean, (c) standard deviation, and (d) variance of the measurements.

14. The following 25 measurements were taken of the thickness of steel emerging from a rolling mill:

3.97 3.99 4.04 4.00 3.98 4.03 4.00 3.98 3.99 3.96 4.02 3.99 4.01 3.97 4.02 3.99 3.95 4.03 4.01 4.05 3.98 4.00 4.04 3.98 4.02 Find (a) median, (b)mean, (c) standard deviation, and (d) variance of the measurements.

15. The following 10 measurements were made of output voltage from a high-gain amplifier contaminated due to noise fluctuations:

1.53, 1.57, 1.54, 1.54 ,1.50, 1.51, 1.55, 1.54, 1.56, 1.53 Determine the mean value and standard deviation. Hence, estimate the accuracy to which the mean value is determined from these 10measurements. If 1000measurements were taken, instead of 10, but s remained the same, by how much would the accuracy of the calculated mean value be improved?

16. The following measurements were taken with an analogue meter of current flowing in a circuit (the circuit was in steady state and therefore, although measurements varied due to random errors, the current flowing was actually constant):

21.5, 22.1, 21.3, 21.7, 22.0, 22.2, 21.8, 21.4, 21.9, 22.1 mA

Calculate mean value, deviations from the mean, and standard deviation.

17. Using the measurement data given in Problem 14, draw a histogram of errors (use error bands 0.03 units wide, i.e., the center band will be from _0.015 to +0.015).

18. (a) What do you understand by the term probability density function? (b) Write down an expression for a Gaussian probability density function of given mean value m and standard deviation s and show how you would obtain the best estimates of these two quantities from a sample of population n.

19. Measurements in a data set are subject to random errors, but it’s known that the data set fits a Gaussian distribution. Use standard Gaussian tables to determine the percentage of measurements that lie within the boundaries of _1.5s, where s is the standard deviation of the measurements.

20. Measurements in a data set are subject to random errors, but it’s known that the data set fits a Gaussian distribution. Use error function tables to determine the value of x required such that 95%of the measurements lie within the boundaries of _xs, where s is the standard deviation of the measurements.

21. By applying error function tables for mean and standard deviation values calculated in Problem 14, estimate (a) How many measurements are <4.00? (b) How many measurements are <3.95? (c) How many measurements are between 3.98 and 4.02? Check your answers against real data.

22. The resolution of the instrument referred to in Problem 14 is clearly 0.01.

Because of the way in which error tables are presented, estimations of the number of measurements in a particular error band are likely to be closer to the real number if boundaries of the error band are chosen to be between measurement values. In part c of Problem 21, values >3.98 are subtracted from values >4.02, thus excluding measurements equal to 3.98. Test this hypothesis out by estimating: (a) How many measurements are <3.995? (b) How many measurements are <3.955? (c) How many measurements are between 3.975 and 4.025? Check your answers against real data.

23. Measurements in a data set are subject to random errors, but it’s known that the data set fits a Gaussian distribution. Use error function tables to determine the percentage of measurements that lie within the boundaries of_2s , where s is the standard deviation of the measurements.

24. A silicon-integrated circuit chip contains 5000 ostensibly identical transistors.

Measurements are made of the current gain of each transistor. Measurements have a mean of 20.0 and a standard deviation of 1.5. The probability distribution of the measurements is Gaussian.

(a) Write down an expression for the number of transistors on the chip that have a current gain between 19.5 and 20.5.

(b) Show that this number of transistors with a current gain between 19.5 and 20.5 is approximately 1300.

(c) Calculate the number of transistors that have a current gain of 17 or more (this is the minimum current gain necessary for a transistor to be able to drive the succeeding stage of the circuit in which the chip is used).

25. In a particular manufacturing process, bricks are produced in batches of 10,000.

Because of random variations in the manufacturing process, random errors occur in the target length of the bricks produced. If the bricks have a mean length of 200 mm with a standard deviation of 20 mm, show how the error function tables supplied can be used to calculate the following:

(a) number of bricks with a length between 198 and 202 mm.

(b) number of bricks with a length greater than 170 mm.

26. The temperature-controlled environment in a hospital intensive care unit is monitored by an intelligent instrument that measures temperature every minute and calculates the mean and standard deviation of the measurements. If the mean is 75_ C and the standard deviation is 2.15, (a) What percentage of the time is the temperature less than 70_ C? (b) What percentage of the time is the temperature between 73 and 77_ C?

27. Calculate the standard error of the mean for measurements given in Problem 13.

Hence, express the melting point of aluminum together with the possible error in the value expressed.

28. The thickness of a set of gaskets varies because of random manufacturing disturbances, but thickness values measured belong to a Gaussian distribution. If the mean thickness is 3 mm and the standard deviation is 0.25, calculate the percentage of gaskets that have a thickness greater than 2.5 mm.

29. If the measured variance of 25 samples of bread cakes taken from a large batch is 4.85 grams, calculate the true variance of the mass for a whole batch of bread cakes to a 95% significance level.

30. Calculate true standard deviation of the diameter of a large batch of tires to a confidence level of 99% if the measured standard deviation in the diameter for a sample of 30 tires is 0.63 cm.

31. One hundred fifty measurements are taken of the thickness of a coil of rolled steel sheet measured at approximately equidistant points along the center line of its length.

Measurements have a mean value of 11.291 mm and a standard deviation of 0.263 mm.

The smallest and largest measurements in the sample are 10.73 and 11.89 mm.

Measurements are divided into eight data bins with boundaries at 10.695, 10.845, 10.995, 11.145, 11.295, 11.445, 11.595, 11.745, and 11.895. The first bin, containing measurements between 10,695 and 10.845, has eight measurements in it, and the count of measurements in the following successive bins is 12, 21, 34, 31, 25, 14, and 5. Apply the Χ² test to see whether the measurements fit a Gaussian distribution to a 95% confidence level.

32. The temperature in a furnace is regulated by a control system that aims to keep the temperature close to 800_ C. The temperature is measured every minute over a 2-hour period, during which time the minimum and maximum temperatures measured are 782 and 819_ C. Analysis of the 120 measurements shows a mean value of 800.3_ C and a standard deviation of 7.58_ C. Measurements are divided into eight data bins of 5_ C width with boundaries at 780.5, 785.5, 790.5, 795.5, 800.5, 805.5, 810.5, 815.5, and 820.5. The measurement count in bin one from 780.5 to 785.5_ C was 3, and the count in the other successive bins was 8, 21, 30, 28, 19, 9, and 2. Apply the Χ² test to see whether the measurements fit a Gaussian distribution to (a) a 90%confidence level and (b) a 95%confidence level (think carefully about whether the Χ² test will be reliable for the measurement counts observed and whether there needs to be any change in the number of data bins used for the Χ² test).

33. The volume contained in each sample of 10 bottles of expensive perfume is measured.

If the mean volume of the sample measurements is 100.5 ml with a standard deviation of 0.64 ml, calculate the upper and lower bounds to a confidence level of 95% of the mean value of the whole batch of perfume from which the 10 samples were taken.

34. A 3-volt d.c. power source required for a circuit is obtained by connecting together two 1.5-volt batteries in series. If the error in the voltage output of each battery is specified as _1%, calculate the likely maximum error in the 3-volt power source that they make up.

35. A temperature measurement system consists of a thermocouple whose amplified output is measured by a voltmeter. The output relationship for the thermocouple is approximately linear over the temperature range of interest. The e.m.f./temp relationship of the thermocouple has a possible error of _1%, the amplifier gain value has a possible error of _0.5%, and the voltmeter has a possible error of _2%.

What is the possible error in the measured temperature?

36. A pressure measurement system consists of a monolithic piezoresistive pressure transducer and a bridge circuit to measure the resistance change of the transducer. The resistance (R) of the transducer is related to pressure (P) according to R = K1 P and the output of the bridge circuit (V) is related to resistance (R)by V =K2 R. Thus, the output voltage is related to pressure according to V = K1 K2 P. If the maximum error in K1 is _2%, the maximum error in K2 is _1.5%, and the voltmeter itself has a maximum measurement error of _1%, what is the likely maximum error in the pressure measurement?

37. A requirement for a resistance of 1220 Ohm in a circuit is satisfied by connecting together resistances of 1000 and 220 Ohm in series. If each resistance has a tolerance of_5%, what is the likely tolerance in the total resistance?

38. In order to calculate the heat loss through the wall of a building, it’s necessary to know the temperature difference between inside and outside walls. Temperatures of 5 and 20_ C are measured on each side of the wall by mercury-in-glass thermometers with a range of 0 to +50_ C and a quoted inaccuracy of _1% of full-scale reading.

(a) Calculate the likely maximum possible error in the calculated value for the temperature difference.

(b) Discuss briefly how using measuring instruments with a different measurement range may improve measurement accuracy.

39. A fluid flow rate is calculated from the difference in pressure measured across a venturi. Flow rate is given by F = K(p2 _ p1), where p1 and p2 are the pressures either side of the venturi and K is a constant. The two pressure measurements are 15.2 and 14.7 bar.

(a) Calculate the possible error in flow measurement if pressure-measuring instruments have a quoted error of _0.2% of their reading. (b) Discuss briefly why using a differential pressure sensor rather than two separate pressure sensors would improve measurement accuracy.

40. The power dissipated in a car headlight is calculated by measuring the d.c. voltage drop across it and the current flowing through it (P = V I). If possible errors in the measured voltage and current values are _1 and _2%, respectively, calculate the likely maximum possible error in the power value deduced.

41. The resistance of a carbon resistor is measured by applying a d.c. voltage across it and measuring the current flowing (R =V/I). If the voltage and current values are measured as 10_0.1 V and 214_5 mA, respectively, express the value of the carbon resistor.

42. The specific energy of a substance is calculated by measuring the energy content of a cubic meter volume of the substance. If the errors in energy measurement and volume measurement are _1 and _2%, respectively, what is the possible error in the calculated value of specific energy (specific energy = energy per unit volume of material)?

43. In a particular measurement system, quantity x is calculated by subtracting a measurement of a quantity z from a measurement of a quantity y, that is, x = y _ z.

If the possible measurement errors in y and z are _ay and _bz, respectively, show that the value of x can be expressed as x = y _ z _ (ay _ bz).

(a) What is inconvenient about this expression for x, and what is the basis for the following expression for x that is used more commonly?

where e =

(b) In a different measurement system, quantity p is calculated by multiplying together measurements of two quantities q and r such that p = qr. If the possible measurement errors in q and r are _aq and _br, respectively, show that the value of p can be expressed as p = (qr)(1 _[a + b]). The volume flow rate of a liquid in a pipe (the volume flowing in unit time) is measured by allowing the end of the pipe to discharge into a vertical-sided tank with a rectangular base (see Fgr. 21). The depth of the liquid in the tank is measured at the start as h1 meters and 1minute later it’s measured as h2 meters. If the length and width of the tank are l and w meters, respectively, write down an expression for the volume flow rate of the liquid in cubic meters per minute. Calculate the volume flowrate of the liquid if the measured parameters have the following values:

If the possible errors in the measurements of h1, h2, l, and w are 1, 1, 0.5, and 0.5%, respectively, calculate the possible error in the calculated value of the flow rate.

44. The density of a material is calculated by measuring the mass of a rectangular-sided block of the material whose edges have lengths of a, b, and c. What is the possible error in the calculated density if the possible error in mass measurement is _1.0% and possible errors in length measurement are _0.5%?

45. The density (d) of a liquid is calculated by measuring its depth (c) in a calibrated rectangular tank and then emptying it into a mass-measuring system. The length and width of the tank are (a)and(b), respectively, and thus the density is given by ... where m is the measured mass of the liquid emptied out. If the possible errors in the measurements of a, b, c, and m are 1, 1, 2, and 0.5%, respectively, determine the likely maximum possible error in the calculated value of the density (d).

46. The volume flow rate of a liquid is calculated by allowing the liquid to flow into a cylindrical tank (stood on its flat end) and measuring the height of the liquid surface before and after the liquid has flowed for 10 minutes. The volume collected after 10 minutes is given by ... where h1 and h2 are the starting and finishing surface heights and d is the measured diameter of the tank.

(a) If h1 = 2m, h2 = 3 m, and d = 2 m, calculate the volume flow rate in m3 /min.

(b) If the possible error in each measurement h1, h2, and d is _1%, determine the likely maximum possible error in the calculated value of volume flow rate (it is assumed that there is negligible error in the time measurement).

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