3.6.3 Total error when combining
multiple measurements
The final case to be covered is where
the final measurement is calculated from several measurements that are combined
together in a way that involves more than one type of arithmetic operation. For
example, the density of a rectangular-sided solid block of material can be
calculated from measurements of its mass divided by the product of measurements
of its length, height and width. The errors involved in each stage of arithmetic
are cumulative, and so the total measurement error can be calculated by adding
together the two error values associated with the two multiplication stages
involved in calculating the volume and then calculating the error in the final
arithmetic operation when the mass is divided by the volume.
Example 3.10
A rectangular-sided block has edges
of lengths a, b and c, and its mass is m. If the values and possible errors in
quantities a, b, c and m are as shown below, calculate the value of density and
the possible error in this value.
a = 100 mm ± 1%, b =
200 mm ± 1%, c = 300 mm ± 1%, m = 20 kg ± 0.5%.
Solution
Value of ab = 0.02 m2 ± 2%
(possible error = 1% + 1% = 2%)
Value of (ab)c = 0.006 m3 ±
3% (possible error = 2% + 1% = 3%)
Value of m/abc = 20/0.006 = 3330 kg/m3
± 3.5% (possible error = 3% + 0.5% = 3.5%)
3.7 Self-test questions
3.1 Explain the difference between
systematic and random errors. What are the typical sources of these two types
of error?
3.3 Suppose that the components in
the circuit shown in Figure 3.1(a) have the following values:
R1 = 330 Ω ; R2 =
1000 Ω ; R3 = 1200 Ω ; R4 = 220 Ω ; R5 = 270 Ω .
If the instrument measuring the
output voltage across AB has a resistance of 5000 Ω , what is the measurement
error caused by the loading effect of this instrument?
3.4 Instruments are normally
calibrated and their characteristics defined for partic[1]ular
standard ambient conditions. What procedures are normally taken to avoid
measurement errors when using instruments that are subjected to changing
ambient conditions?
3.5 The voltage across a resistance R5
in the circuit of Figure 3.10 is to be measured by a voltmeter connected across
it.
(a) If the voltmeter has an internal
resistance (Rm) of 4750 Ω , what is the measure[1]ment error?
(b) What value would the voltmeter
internal resistance need to be in order to reduce the measurement error to 1%?
3.6 In the circuit shown in Figure
3.11, the current flowing between A and B is measured by an ammeter whose
internal resistance is 100 Ω . What is the measure[1]ment
error caused by the resistance of the measuring instrument?
3.7 What steps can be taken to reduce
the effect of environmental inputs in measurement systems?
3.8 The output of a potentiometer is
measured by a voltmeter having a resistance Rm, as shown in Figure
3.12. 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.)
3.9 In a survey of 15 owners of a
certain model of car, the following figures for average petrol consumption were
reported.
25.5 30.3 31.1 29.6 32.4 39.4 28.9
30.0 33.3 31.4 29.5 30.5 31.7 33.0 29.2
Calculate the mean value, the median
value and the standard deviation of the data set.
3.10 (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 µ and standard
deviation α and show how you would obtain the best estimate of these two
quantities from a sample of population n.
(c) The following ten measurements
are made of the output voltage from a highgain amplifier that is 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 ten measurements. If one thousand measurements were taken, instead
of ten, but α remained the same, by how much would the accuracy of the
calculated mean value be improved?
3.11 The following measurements were
taken with an analogue meter of the current flowing in a circuit (the circuit
was in steady state and therefore, although the measurements varied due to
random errors, the current flowing was actually constant):
21.5 mA, 22.1 mA, 21.3 mA, 21.7 mA,
22.0 mA, 22.2 mA, 21.8 mA, 21.4 mA, 21.9 mA, 22.1 mA
Calculate the mean value, the
deviations from the mean and the standard deviation.
3.12 The measurements in a data set
are subject to random errors but it is 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 ±1.5 α, where α is the standard
deviation of the measurements.
3.13 The thickness of a set of
gaskets varies because of random manufacturing distur[1]bances
but the 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.
3.14 A 3 volt d.c. power source
required for a circuit is obtained by connecting together two 1.5 V batteries
in series. If the error in the voltage output of each battery is specified as ±1%,
calculate the likely maximum possible error in the 3 volt power source that
they make up.
3.15 In order to calculate the heat
loss through the wall of a building, it is necessary to know the temperature
difference between the inside and outside walls. If temper[1]atures of 5°C and
20°C are measured on each side of the wall by mercury-in-glass thermometers
with a range of 0°C to +50°C and a quoted inaccuracy figure of ±1% of
full-scale reading, calculate the likely maximum possible error in the
calculated figure for the temperature difference.
3.16 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 the possible errors in the measured
voltage and current values are š1% and ±2% respectively, calculate the likely
maximum possible error in the power value deduced.
3.17 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.
3.18 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:
d = m/(a × b × c)
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).
3.19 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:
Volume = (h2 - h1)π(d/2)2
where h1 and h2 are the starting and
finishing surface heights and d is the measured diameter of the tank.
(a) If h1 = 2 m, 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.
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