Patrick F. Dunn, Measurement, Data Analysis, and Sensor Fundamentals for Engineering and Science, 2nd Edition

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276 Measurement and Data Analysis for Engineering and Science

is referred to the report by Ferson et al. [25] and the monograph by Salicone

[26] for more detailed information.

Treating epistemic uncertainty, as explained in detail by Ferson et al.

[25], can be illustrated by examining a data set consisting of N outcomes,

each with a diﬀerent range of possible values. The range for one outcome

possibly may overlap that of another. How the values are distributed within

a range is not speciﬁed. The outcomes can be represented as







, y







140

150

160

170

180

190

200

210

220

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

height (cm)

P(height)

FIGURE 7.8

The p-box representation of the probability distribution function of the height

estimates (solid) with 95 % conﬁdence limits (dashed).

where y

denotes the lower value of the range of the i-th estimate and y

the upper value, with i = 1, ..., N.

Consider the example in which the heights of six diﬀerent people are

estimated visually by an observer as each person runs past a height scale

located 10 m from the observer. The height estimates in centimeters made

by the observer in ascending order are

Uncertainty Analysis 277







160, 168

162, 174

165, 177

172, 187

185, 193

191, 200







These results are displayed graphically in a p-box representation in Figure

7.8. Here, y represents the measure (magnitude) of the height and P (y) the

empirically established probability distribution function of the height. The

width of each box is the range for each height estimate and the height of

each box is 1/N (here, 1/6), as indicated by the solid-line bounds in the

ﬁgure. For example, based upon the observer’s estimates, there is a 50.0 %

probability that the observed heights were in the range from 160 cm (the

minimum height observed) to a height of from 172 cm to 177 cm (172 cm if

the lower bound at 50 % is used, and 177 cm if the upper bound at 50 % is

used), and 83.3 % probability that they were in the range from 160 cm to a

height of from 187 cm to 193 cm.

Conﬁdence limits for P (y), making no assumption of the speciﬁc distribu-

tion function that governs the possible values, can be determined based upon

methods developed by Kolmogorov [27] and Smirnov [28]. These are termed

Kolmogorov-Smirnov (distribution-free) conﬁdence limits. The lower, B

(y),

and upper,

B(y), limits for each estimate are given by

(y),

B(y)] =

[min(1, S

(y) + D

max

(α, N )), max(0, S

(y) − D

max

(α, N ))], (7.88)

where S

is the fraction of the left end points of the estimate values that

are at or below the magnitude y, S

is the fraction of the right end points

of the estimate values that are at or above the magnitude y, and D

max

is the

Smirnov statistic that is a function of α and N. Statistically, 100(1 − α) %

of the time, the conﬁdence limits will enclose the distribution function of the

population from which the samples were drawn. When N is greater than

∼50 and α = 5 % (95 % conﬁdence limits), D

max

(0.05, N) ≈ 1.36/

√

For smaller values of N, the value of D

max

(α, N ) can be obtained from a

statistical table [29], [30]. These conﬁdence limits do not depend upon the

type of the distribution function, only that the distribution is continuous

and that the samples have been drawn randomly and independently from

the distribution.

Applying these methods to the above example of the height estimates,

assuming 95 % conﬁdence and with D

max

(0.05, 6) = 0.525, yields the con-

ﬁdence limits shown by the dashed-line bounds in Figure 7.8. For example,

95 % of the time when height estimates are made in the manner described

above, 50 % of the heights could be anywhere in the range from the mini-

mum possible height (theoretically 0 cm) to 200 cm. Clearly, these are very

278 Measurement and Data Analysis for Engineering and Science

extensive bounds. Their extent results from the small sample size, here N =

6, and the distribution-free interval assumption. Larger sample sizes and/or

a speciﬁcation of the interval distribution will reduce these limits. For ex-

ample, if N = 100, D

max

(0.05, 100) = 0.134, 95 % of the time when height

estimates are made, 50 % of the heights could be anywhere in the range

from the minimum possible height to ∼187 cm.

It is important to emphasize that these conﬁdence limits, which are based

on a lack of knowledge about the speciﬁc values of the data, do not consider

additional uncertainties, such as instrument resolution and measurement ac-

curacy and precision. Ferson et al. [25] present a hybrid approach in which

the methods of interval statistics are combined with the standard uncer-

tainty method, thereby overcoming this limitation. Their results show that

the conﬁdence limits are the most extensive (widest) when distribution-free

intervals are assumed. When normal-distribution intervals are assumed, the

conﬁdence limits become less extensive (narrower). The current ISO method

gives the least extensive (narrowest) conﬁdence limits because it assumes a

normal distribution and neglects epistemic uncertainty.

Uncertainty Analysis 279

7.14 Problem Topic Summary

Topic

Review Problems

Homework Problems

Basic Uncertainty

1, 3, 4, 8, 10, 12, 13,

8, 21, 23

14, 18, 20, 22, 23, 24

Result Uncertainty

5, 9, 19

1, 2, 7, 13, 14, 15, 16,

19, 21, 26, 27, 28

General Uncertainty

2, 6, 7, 9, 11, 15, 16,

3, 5, 9, 10, 11, 18,

17, 21

19, 23, 24, 25

Detailed Uncertainty

4, 6, 12

Finite-Diﬀerence Approach

TABLE 7.2

Chapter 7 Problem Summary

7.15 Review Problems

1. A researcher is measuring the length of a microscopic scratch in a mi-

crophone diaphragm using a stereoscope. A ruler incremented into ten-

thousandths of an inch is placed next to the microphone as a distance

reference. If the stereoscope magniﬁcation is increased 10 times, what

property of the distance measurement system has been improved? (a)

Sensitivity, (b) precision, (c) readability, (d) least count.

2. A multimeter, with a full-scale output of 5 V, retains two decimal dig-

its of resolution. For instance, placing the multimeter probes across a

slightly used AA battery results in a readout of 1.35 V. Through cali-

bration, the instrument uncertainties established are sensitivity, 0.5 %

of FSO, and oﬀset, 0.1 % of FSO. What is the total design stage un-

certainty in volts based on this information to 95 % conﬁdence? (Note:

The readout of the instrument dictates that the uncertainty should be

expressed to three decimal places.)

3. Three students are playing darts. The results of the ﬁrst round are shown

in Figure 7.9, where the circle in the center is the bullseye. Circles =

Player 1; squares = Player 2; triangles = Player 3. In terms of hitting the

bullseye, which player best demonstrates precision, but not accuracy?

(a) Player 1 (circles), (b) Player 2 (squares), (c) Player 3 (triangles).

280 Measurement and Data Analysis for Engineering and Science

FIGURE 7.9

Dartboard.

4. Compare the precision of a metric ruler, incremented in millimeters,

with a standard customary measure ruler, incremented into 16ths of an

inch. How much more precise is the more precise instrument? Express

your answer in millimeters and consider the increments on the rulers to

be exact.

5. For a circular rod with density, ρ, diameter, D, and length, L, derive an

expression for the uncertainty in computing its moment of inertia about

the rod’s end from the measurement of those three quantities. If ρ =

2008 ± 1 kg/m

, D = 3.60 mm ± 0.05 mm, and L = 2.83 m ± 0.01

m, then compute the uncertainty in the resulting moment of inertia to

the correct number of signiﬁcant ﬁgures in the SI units of kilograms per

square meter. Signiﬁcant ﬁgures are based on the measured quantities.

The formula for the moment of inertia of a circular rod about its end is

I = ρπ/12D

6. The velocity of the outer circumference of a spinning disk may be mea-

sured in two ways. Using an optical sensing device, the absolute uncer-

tainty in the velocity is 0.1 %. Using a strobe and a ruler, the uncertainty

in the angular velocity is 0.1 rad/s and the uncertainty in the diameter

of the disk is 1 mm. Select the measurement method with the least un-

certainty for the two methods if the disk is 0.25 m in diameter and is

spinning at 10 rpm.

7. Using a pair of calipers with 0.001 in. resolution, a machinist measures

the diameter of a pinball seven times with the following results in units

of inches: 1.026, 1.053, 1.064, 1.012, 1.104, 1.098, and 1.079. He uses

the average of the measurements as the correct value. Compute the

uncertainty in the average diameter in inches to one-thousandth of an

inch.

Uncertainty Analysis 281

8. Match the following examples of experimental error (numbered 1

through 4) and uncertainty to the best categorization of that example

(lettered a through d): (1) temperature ﬂuctuations in the laboratory,

(2) physically pushing a pendulum through its swing during one exper-

imental trial, (3) the numerical bounds of the scatter in the distance

traveled by the racquet ball about the mean value, (4) releasing the

pendulum from an initial position so that the shaft has a small initial

angle to the vertical for each trial; (a) uncertainty, (b) systematic error,

9. A geologist ﬁnds a rock of unknown composition and desires to measure

its density. To measure the volume, she places the rock in a cylinder,

which is graduated in 0.1 mL increments, half-ﬁlled with water, so that

the rock is submerged in the water. She removes the rock from the

cylinder and directly measures the rock’s mass using a scale with a

digital readout resolution of 0.1 g. No information is provided from the

manufacturer about the scale uncertainty. She records the volume, V,

and mass, m, as follows: V = 40.5 mL, m = 143.1 g. Determine the

percent uncertainty in the density expressed with the correct number of

signiﬁcant ﬁgures.

10. In addition to the digital display, the manometer described in the pre-

vious problem has an analog voltage output that is proportional to

the sensed diﬀerential pressure in units of in inches of water. A re-

searcher calibrates the analog output by applying a series of known

pressures across the manometer ports and observing both the analog

and digital output for each input pressure. A linear ﬁt to the data yields

P = 1.118E + 0.003, where P is pressure (in. H

O) and E is the analog

voltage (V). If for zero input pressure, the manufacturer speciﬁes that

no output voltage should result, what magnitude of systematic error has

been found?

11. The mean dynamic pressure in a wind tunnel is measured using the

manometer described in the last two problems and a multimeter to mea-

sure the output voltage. The recorded voltages are converted to pres-

sures using the calibration ﬁt presented in Problem 10. The resolution

of the multimeter is 1 mV and the manufacturer speciﬁes the following

instrument errors: sensitivity = 0.5 mV and linearity = 0.5 mV. If 150

multimeter readings with a standard deviation of 0.069 V are acquired

with a mean voltage of 0.312 V, what is the uncertainty in the resulting

computed mean pressure in inches of water? Express the answer to the

precision of the digital readout of the manometer.

12. A technician uses a graduated container of water to measure the volume

of odd-shaped objects. The changes in the density of the water caused

by ambient temperature and pressure ﬂuctuations directly contribute

282 Measurement and Data Analysis for Engineering and Science

to (a) systematic error, (b) redundant error, (c) random error, or (d)

multiple measurement error.

13. A graduate student orders a set of very accurate weights to calibrate a

digital scale. The desired accuracy of the scale is 0.1 g. The manufac-

turer of the weights states that the mass of each weight is accurate to

0.04 g. What is the maximum number of weights that may be used in

combination to calibrate the scale?

14. A test engineer performs a ﬁrst-run experiment to measure the time

required for a 2010 prototype car to travel a fourth of a mile beginning

from rest. When the car begins motion, a green light ﬂashes in the

engineer’s ﬁeld of vision, signaling him to start the time count with a

hand-held stopwatch. Similarly, a red light ﬂashes when the car reaches

the ﬁnish line. The resulting times from four trials are 13.42 s, 13.05 s,

12.96 s, and 12.92 s. Outside of the test environment, another engineer

measures the ﬁrst test engineer’s reaction time to the light signals. The

results of the test show that the test engineer over-anticipates the green

light and displays slowed reaction to the red light. Both reaction times

were measured to be 0.13 s. Compute the average travel time in seconds,

correcting for the systematic error in the experimental procedure.

15. A digital manometer measures the diﬀerential pressure across two in-

puts. The range of the manometer is 0 in. H

O to 0.5 in. H

O. The LED

readout resolves pressure into 0.001 in. H

O. Based on calibration, the

manufacturer speciﬁes the following instrument errors: hysteresis error

= 0.1 % of FSO; linearity = 0.25 % of FSO; sensitivity = 0.1 % of

FSO. Determine the design stage uncertainty of the digital manometer

in inches of water to the least signiﬁcant digit resolved by the manome-

ter.

16. The smallest division marked on the dial of a pressure gage is 2 psi. The

accuracy of the pressure gage as stated by the manufacturer is ±1 psi.

Determine the design-stage uncertainty in psi and express it with the

correct number of signiﬁcant ﬁgures.

17. A student conducts an experiment in which the panel meter display-

ing the measurement system’s output in volts ﬂuctuates up and down

in time. Being a conscientious experimenter, the student decides to es-

timate the temporal random error of the measurement. She takes 100

repeated measurements and ﬁnds that the standard deviation equals a

whopping 1.0 V! Determine the temporal random error in volts at 95 %

conﬁdence and express the answer with the correct number of signiﬁcant

ﬁgures.

18. Standard measurement uncertainty is (a) the error in a measurement,

(b) the probability of a measurement being correct, (c) the probability

Uncertainty Analysis 283

of a measurement not being correct, (d) an estimate of the range of

probable errors in a measurement, or (e) the sum of the systematic and

random errors.

19. If the uncertainties in the length and diameter of a cylinder are 2 % and

3 % respectively, what is the percent uncertainty in its volume expressed

with the correct number of signiﬁcant ﬁgures?

20. Sixty-four pressure measurements have a sample mean equal to 200

N/m

and a sample variance equal to 16 N

. What is the percent

uncertainty in the pressure measurement if the only contributor to its

uncertainty is the random error?

21. A voltmeter having three digits displays a reading of 8.66 V. What per-

cent instrument uncertainty must the voltmeter have to yield a design-

stage uncertainty of 0.01 V at 8.66 V?

22. Determine the temporal precision error, in V, in the estimate of the aver-

age value of a voltage based upon nine measurements at 95 % conﬁdence

and a sample variance of 9 V

23. Match the following examples of experimental error and uncertainty (1

through 4) to the type of uncertainty they correspond to (a through

d): (1) ﬂuctuations in the humidity of the air during the summer while

conducting a month-long series of experiments, (2) holding a ruler at an

angle to the measurement plane for a series of measurements, (3) while

taking data, bumping into a table that holds a pendulum experiment, (4)

the numerical bounds of the scatter in the height a ball bounces during

measurements of its coeﬃcient of restitution; (a) systematic error, (b)

experimental mistake, (c) random error, (d) uncertainty.

24. A student records a small sample of three voltage measurements: 1.000

V, 2.000 V, and 3.000 V. Determine the uncertainty in the population’s

true mean value of the voltage estimated with 50 % conﬁdence. Express

your answer with the correct number of signiﬁcant ﬁgures.

7.16 Homework Problems

1. The supply reservoir to a water clock is constructed from a tube of

circular section. The tube has a nominal length of 52 cm ± 0.5 cm, an

outside diameter of 20 cm ± 0.04 cm, and an inside diameter of 15 cm

± 0.08 cm. Determine the percent uncertainty in the calculated volume.

284 Measurement and Data Analysis for Engineering and Science

2. A mechanical engineer is asked to design a cantilever beam to support

a concentrated load at its end. The beam is of circular section and has

a length, L, of 6 ft and a diameter, d, of 2.5 in. The concentrated load,

F , of 350 lbf is applied at the beam end, perpendicular to the length of

the beam. If the uncertainty in the length is ±1.5 in., in the diameter

is ±0.08 in., and in the force is ± 5 lbf, what is the uncertainty in the

calculated bending stress, σ? [Hint: σ = 32F L/(πd

).] Further, if the

uncertainty in the bending stress may be no greater than 6 %, what

maximum uncertainty may be tolerated in the diameter measurement if

the other uncertainties remain unchanged?

3. An electrical engineer must decide on a power usage measurement

method that yields the least uncertainty. There are two alternatives

to measuring the power usage of a DC heater. Either (1) heater resis-

tance and voltage drop can be measured simultaneously and then the

power computed, or (2) heater voltage drop and current can be mea-

sured simultaneously and then the power computed. The manufactur-

ers’ speciﬁcations of the available instruments are as follows: ohmmeter

(resolution 1 Ω and reading uncertainty = 0.5 %); ammeter (resolution

0.5 A and % reading uncertainty = 1 %); voltmeter (resolution 1 V and

% reading uncertainty = 0.5 %). For loads of 10 W, 1 kW, and 10 kW

each, determine the best method based on an appropriate uncertainty

analysis. Assume nominal values as necessary for resistance and current

based upon a ﬁxed voltage of 100 V.

4. A new composite material is being developed for an advanced aerospace

structure. The material’s density is to be determined from the mass of a

cylindrical specimen. The volume of the specimen is determined from di-

ameter and length measurements. It is estimated that the mass, m, can

be determined to be within 0.1 lbm using an available balance scale, the

length, L, to within 0.05 in., and the diameter, D, to within 0.0005 in.

Estimate the zero-order design stage uncertainty in the determination

of the density. Which measurement would contribute most to the uncer-

tainty in the density? Which measurement method should be improved

ﬁrst if the estimate in the uncertainty in the density is unacceptable?

Use nominal values of m = 4.5 lbm, L = 6 in., and D = 4 in. Next, mul-

tiple measurements are performed yielding the data shown in Table 7.3.

Using this information and what was given initially, provide an estimate

of the true density at 95 % conﬁdence. Compare the uncertainty in this

result to that determined in the design stage.

5. High pressure air is to be supplied from a large storage tank to a plenum

located immediately before a supersonic convergent-divergent nozzle.

The engineer designing this system must estimate the uncertainty in

the plenum’s pressure measurement system. This system outputs a volt-

age that is proportional to pressure. It is calibrated against a transducer

Uncertainty Analysis 285

D = 3.9924 in.

¯m = 4.4 lbm

L = 5.85 in.

= 0.0028 in.

= 0.1 lbm

= 0.10 in.

N = 3

N = 21

N = 11

TABLE 7.3

Composite material data.

standard (certiﬁed accuracy: within ±0.5 psi) over its 0 psi to 100 psi

range with the results given below. The voltage is measured with a volt-

meter (instrument error: within ±10 µV; resolution: 1 µV). The engineer

estimates that installation eﬀects can cause the indicated pressure to be

oﬀ by another ±0.5 psi. Estimate the uncertainty at 95 % conﬁdence in

using this system based upon the following information given in Table

7.4.

E(mv):

0.004

0.399

0.771

1.624

2.147

4.121

p(psi):

0.1

10.2

19.5

40.5

51.2

99.6

TABLE 7.4

Storage tank calibration data.

6. One approach to determining the volume of a cylinder is to measure its

diameter and length and then calculate the volume. If the length and

diameter of the cylinder are measured at four diﬀerent locations using

a micrometer with an uncertainty of 0.05 in. with 95 % conﬁdence,

determine the percent uncertainty in the volume. The four diameters

in inches are 3.9920, 3.9892, 3.9961, and 3.9995; those of the length in

inches are 4.4940, 4.4991, 4.5110, and 4.5221.

7. Given y = ax

and that the uncertainty in a is 3 % and that in x is 2 %,

determine the percent uncertainty in y for the nominal values of a = 2

and x = 0.5.

8. The lift force on a Wortmann airfoil is measured ﬁve times under the

same experimental conditions. The acquired values are 10.5 N, 9.4 N,

9.1 N, 11.3 N, and 9.7 N. Assuming that the only uncertainty in the

experiment is a temporal random error as manifested by the spread of

the data, determine the uncertainty (in ±N) at the 95 % conﬁdence level

of the true mean value of the lift force.

9. A pressure transducer speciﬁcation sheet lists the following instrument

errors, all in units of percent span, where the span for the particular

pressure transducer is 10 in. H

O: combined null and sensitivity shift