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

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286 Measurement and Data Analysis for Engineering and Science

= ±1.00, linearity = ±2.00, and repeatability and hysteresis = ±0.25.

Estimate (a) the transducer’s instrument uncertainty in the pressure

in units of inches of water and (b) the % instrument uncertainty in a

pressure reading of 1 in. H

O. (c) Would this be a suitable transducer

to use in an experiment in which the pressure ranged from 0 in. H

O to

2 in. H

O and the pressure reading must be accurate to within ± 10 %?

10. The mass of a golf ball is measured using an electronic balance that has

a resolution of 1 mg and an instrument uncertainty of 0.5 %. Thirty-one

measurements of the mass are made yielding an average mass of 45.3

g and a standard deviation of 0.1 g. Estimate the (a) zero-order, (b)

design-stage, and (c) ﬁrst-order uncertainties in the mass measurement.

What uncertainty contributes the most to the ﬁrst-order uncertainty?

11. A group of students wish to determine the density of a cylinder to be

used in a design project. They plan to determine the density from mea-

surements of the cylinder’s mass, length, and diameter, which have in-

strument resolutions of 0.1 lbm, 0.05 in., and 0.0005 in., respectively.

The balance used to measure the weight has an instrument uncertainty

(accuracy) of 1 %. The rulers used to measure the length and diameter

present negligible instrument uncertainties. Nominal values of the mass,

length, and diameter are 4.5 lbm, 6.00 in., and 4.0000 in., respectively.

(a) Estimate the zero-order uncertainty in the determination of the den-

sity. (b) Which measurement contributes the most to this uncertainty?

density.

12. The group of students in the previous problem now perform a series

of measurements to determine the actual density of the cylinder. They

perform 20 measurements of the mass, length, and diameter that yield

average values for the mass, length, and diameter equal to 4.5 lbm,

5.85 in., and 3.9924 in., respectively, and standard deviations equal to

0.1 lbm, 0.10 in., and 0.0028 in., respectively. Using this information

and that presented in the previous problem, estimate (a) the average

density of the cylinder in lbm/in.

, (b) the systematic errors of the mass,

length, and diameter measurements, (c) the random errors of the mass,

length, and diameter measurements, (d) the combined systematic errors

of the density, (e) the combined random errors of the density, (f) the

uncertainty in the density estimate at 95 % conﬁdence (compare this to

the design-stage uncertainty estimate, which should be smaller), and (g)

an estimate of the true density at 95 % conﬁdence.

13. Given King’s law, E

= A + B

√

U, and the fractional uncertainties in

A, B, and U of 5 %, 4 %, and 6 %, respectively, determine the percent

fractional uncertainty in E with the correct number of signiﬁcant ﬁgures.

Uncertainty Analysis 287

14. The resistivity ρ of a piece of wire must be determined. To do this, the

relationship R = ρL/A can be used and the appropriate measurements

made. Nominal values of R, L, and the diameter of the wire, d, are 50 Ω,

10 ft, and 0.050 in., respectively. The error in L must be held to no more

than 0.125 in. R will be measured with a voltmeter having an accuracy

of ±0.2 % of the reading. How accurately will d need to be measured if

the uncertainty in ρ is not to exceed 0.5 %?

15. The tip deﬂection of a cantilever beam with rectangular cross-section

subjected to a point load at the tip is given by the formula

δ =

P L

3EI

, where I =

Here, P is the load, L is the length of the beam, E is the Young’s modulus

of the material, b is the width of the cross-section, and h is the height

of the cross-section. If the instrument uncertainties in P , L, E, b, and h

are each 2 %, (a) estimate the fractional uncertainty in δ. This beam is

used in an experiment to determine the value of an unknown load, P

, by

performing four repeated measurements of δ at that load under the same

controlled conditions. The resulting sample standard deviation of these

measurements is 8 µm and the average deﬂection is 20 µm. Determine

(b) the overall uncertainty in the deﬂection measurements estimated at

90 % conﬁdence assuming that the resolution of the instrument used to

measure δ is so small that it produces negligible uncertainty.

16. The resistance of a wire is given by R = R

[1 + α(T − T

)] where T

= 20

◦

C, R

= 6 Ω ± 0.3 % is the resistance at 20

◦

C, α = 0.004/

◦

±1 % is the temperature coeﬃcient of resistance, and the temperature

of the wire is T = 30 ±1

◦

C. Determine (a) the normal resistance of the

wire and (b) the uncertainty in the resistance of the wire, u

17. Calculate the uncertainty in the wire resistance that was described in

the previous problem using the ﬁrst-order ﬁnite-diﬀerence technique.

18. An experiment is conducted to verify an acoustical theory. Sixty-one

pressure measurements are made at a location using a pressure measure-

ment system consisting of a pressure transducer and a display having

units of kilopascals. A statistical analysis of the 61 measurand values

yields a sample mean of 200 kPa and a sample standard deviation of 2

kPa. The resolution of the pressure display is 6 kPa. The pressure trans-

ducer states that the transducer has a combined hysteresis and linearity

error of 2 kPa, a zero-drift error of 2 kPa, and sensitivity error of 1

kPa. (a) What classiﬁcation is this experiment? Determine the system’s

(b) zero-order uncertainty, (c) instrument uncertainty, (d) uncertainty

arising from pressure variations, and (e) combined standard uncertainty.

Assume 95 % conﬁdence in all of the estimates. Express all estimates

with the correct number of signiﬁcant ﬁgures.

288 Measurement and Data Analysis for Engineering and Science

19. A hand-held velocimeter uses a heated wire and, when air blows over the

wire, correlates the change in temperature to the air speed. The reading

on the velocimeter, v

std

, is relative to standard conditions deﬁned as

std

= 70

◦

F and P

std

= 14.7 psia. To determine the actual velocity,

act

, in units of feet per minute, the equation

act

= v

std

[(460 + T )/(460 + T

std

)][P

std

/P ]

must be applied, where T is in

F and P is in psia. The accuracy of the

reading on the velocimeter is ±5.0 % or ±5 ft/min, whichever is greater.

The velocimeter also measures air temperature with an accuracy of ±1

◦

F. During an experiment, the measured air velocity is 400 ft/min and

the temperature is 80

◦

F. The air pressure can be assumed to be at

standard conditions. Determine (a) the actual air velocity, (b) the un-

certainty in the actual air velocity, u

act

, and (c) the percent uncertainty

in the actual air velocity.

20. Compute the random uncertainty (precision limit) for each of the fol-

lowing and explain the reasoning for which equation was used.

(a) An engineer is trying to understand the traﬃc ﬂow through a partic-

ularly busy intersection. In 2008, every oﬃcial business day during the

month of September (excluding holidays and weekends) he counts the

number of cars that pass through from 10 AM until 1 PM. He found

that the number of cars averaged 198 with a variance of 36. He wishes

to know the uncertainty with 90 % conﬁdence.

(b) A student wishes to determine the accuracy of a relative humidity

gauge in the laboratory with 99 % conﬁdence. He takes a reading every

minute for one hour and determines that the mean is 48 % relative

humidity with a standard deviation of 2 %.

to determine the relative humidity in the room with 99 % conﬁdence

prior to running his experiments. He also takes a reading every minute

for one hour and determines that the mean is 48 % relative humidity

and the standard deviation is 2 %. (Assume he knows nothing about

what his partner has done.)

(d) An engineer designing cranes is working with a manufacturing en-

gineer to assess whether as-manufactured beams will be able to satisfy

a ten-year guarantee for normal use at which time they will need re-

furbishment or replacement. On the drawing he speciﬁed an absolute

minimum thickness of 3.750 in. The manufacturer measures 200 beams

oﬀ the assembly ﬂoor and they have an average thickness of 4.125 in.

with a standard deviation of 0.1500 in. Does the manufacturer have 99

% conﬁdence that the as-manufactured beams will meet the ten-year

guarantee?

Uncertainty Analysis 289

21. The resistance of a wire is given by R = R

[1 + α (T − T

)] where T

and R

are ﬁxed reference values of 20

◦

C and 100 Ω ± 2.5 %, re-

spectively. The temperature coeﬃcient is α = 0.004/

◦

C ± 0.1 %. The

development engineer is checking the resistance of the wire and mea-

sures the temperature to be T = 60

◦

C. When measuring the wire and

reference temperatures, the engineer used the same thermocouple that

had a manufacturer’s accuracy of ± 1

◦

C. (a) Determine the nominal

resistance of the wire and the nominal uncertainty. (b) Assess whether

the certainty was positively or negatively aﬀected by using the same

thermocouple rather than two separate thermocouples with the same

nominal accuracy. Note: When calculating percentages of temperatures,

an absolute scale needs to be used.

22. An instrument has a stated accuracy of q %. An experiment is conducted

in which the instrument is used to measure a variable, z, N times under

controlled conditions. There are some temporal variations in the instru-

ment’s readings, characterized by S

. Determine the overall uncertainty

in z.

23. A design criterion for an experiment requires that the combined standard

uncertainty in a measured pressure be 5 % or less based upon 95 %

conﬁdence. It is known from a previous experiment conducted in the

same facility that the pressure varies slightly under ‘ﬁxed’ conditions,

as determined from a sample of 61 pressure measurements having a

standard deviation of 2.5 kPa and mean of 90 kPa. The accuracy of the

pressure measurement system, as stated by the manufacturer, is 3 %.

Determine the value of the smallest division (in kPa) that the pressure

indicator must have to meet the design criterion.

24. A calibration experiment is conducted in which the output of a

secondary-standard pressure transducer having negligible uncertainty is

read using a digital voltmeter. The pressure transducer has a range of

0 psi to 10 psi. The digital voltmeter has a resolution of 0.1 V, a stated

accuracy of 1 % of full scale, and a range of 0 V to 10 V. The calibration

based upon 61 measurements yields the least-squares linear regression

relation V (volts) = 0.50P (psi), with a standard error of the ﬁt, S

equal to 0.10 V. Determine the combined standard uncertainty in the

voltage at the 95 % conﬁdence level.

25. An inclined manometer has a stated accuracy of 3 % of its full-scale

reading. The range of the manometer is from 0 in. H

O to 5 in. H

O. The

smallest marked division on the manometer’s scale is 0.2 in. H

O. An

experiment is conducted under controlled conditions in which a pressure

diﬀerence is measured 20 times. The mean and standard deviation of

the pressure-diﬀerence measurements are 3 in. H

O and 0.2 in. H

respectively. Assuming 95 % conﬁdence, determine (a) the zero-order

uncertainty, u

, (b) the temporal precision uncertainty that arises from

290 Measurement and Data Analysis for Engineering and Science

the variation in the pressure-diﬀerence during the controlled-conditions

experiment, and (c) the combined standard uncertainty, u

26. Determine the uncertainty (in ohms) in the total resistance, R

, that is

obtained by having two resistors, R

and R

, in parallel. The resistances

of R

and R

are 4 Ω and 6 Ω, respectively. The uncertainties in the

resistances of R

and R

are 2 % and 5 %, respectively.

27. A student group postulates that the stride length, L, of a marathon

runner is proportional to a runner’s inseam, H, and inversely propor-

tional to the square of a runner’s weight, W . The inseam length is to be

measured using a tape measure and the weight using a scale. The esti-

mated uncertainties in H and W are 4 % and 3 %, respectively, based

upon a typical inseam of 70 cm and a weight of 600 N. Determine (a)

the percent uncertainty in L, (b) the resolution of the tape measure (in

cm), and (c) the resolution of the scale (in N).

28. Given that the mass, M , of Saturn is 5.68 × 10

kg, the radius, R,

is 5.82 × 10

m, and g (m/s

) = GM/R

, where G = 6.6742 × 10

−11

N·m

/kg

, determine the percent uncertainty in g on Saturn, assuming

that the uncertainties in G, M , or R are expressed for each by the place

of the least-signiﬁcant digit (for example, u

= 0.01 × 10

m).

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Regression and Correlation

CONTENTS

8.1 Chapter Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

8.2 Least-Squares Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

8.3 Least-Squares Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

8.4 Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

8.5 Regression Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

8.6 Conﬁdence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

8.7 Linear Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

8.8 Uncertainty from Measurement Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

8.9 Determining the Appropriate Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

8.10 *Signal Correlations in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

8.10.1 *Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

8.10.2 *Cross-Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

8.11 *Higher-Order Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

8.12 *Multi-Variable Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

8.13 Problem Topic Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

8.14 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

8.15 Homework Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

Of all the principles that can be proposed for this purpose, I think there is

none more general, more exact, or easier to apply, than that which we have

used in this work; it consists of making the sum of the squares of the errors a

minimum. By this method, a kind of equilibrium is established among the

errors which, since it prevents the extremes from dominating, is appropriate

for revealing the state of the system which most nearly approaches the truth.

Adrien-Marie Legendre. 1805. Nouvelles m´ethodes pour la d´etermination des

orbites des com`etes. Paris.

Two variable organs are said to be co-related when the variation of the one is

accompanied on the average by more or less variation of the other, and in

the same direction.

Sir Francis Galton. 1888. Proceedings of the Royal Society of London. 45:135-145.

295