Patrick F. Dunn, Measurement, Data Analysis, and Sensor Fundamentals for Engineering and Science, 2nd Edition
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476 Measurement and Data Analysis for Engineering and Science
P r[ ] probability of an event
S
2
x
sample variance
Sk skewness
x
0
true mean of x or population mean
µ
m
m-th central moment
σ
2
true variance
Chapter 6:
p(x) probability density function
P (x) probability distribution function
p(z
1
) normal error function
S
x
sample standard deviation
±t
ν,P
S
x
/
√
N precision interval of the true mean
¯x sample mean
x
0
mean value of population
{¯x} pooled mean
{S
x
} pooled standard deviation
{S
¯x
} pooled standard deviation of the means
{S
2
x
} pooled variance
{¯x} pooled weighted mean
{S
x
}
w
pooled weighted standard deviation
{S
¯x
}
w
pooled weighted standard deviation of the means
α level of significance
Γ gamma function
ν degrees of freedom
σ true standard deviation
σ
2
true variance
Symbols 477
χ
2
chi-square
˜
χ
2
reduced chi-square
Chapter 7:
B
i
source bias limit
B
x
systematic uncertainty
C confidence level
∆B
x
/B
x
relative systematic uncertainty
e
i
elemental instrument error (see text for specific types)
e
I
overall instrument error
F SO maximum or full-scale output
N number of measurements in a sample
P
i
source precision index
P
x
random uncertainty
P
¯x
random uncertainty in the average value of the measurand x
r result
t
ν,P
Student’s t factor
u
0
zeroth-order instrument uncertainty
u
I
instrument uncertainty
u
c
combined standard uncertainty
u
2
c
combined estimated variance
u
d
design-stage uncertainty
u
N
N-th order uncertainty
u
2
r
combined estimated variance
u
r
uncertainty in a result
u
x
uncertainty in x
U
x
overall uncertainty
478 Measurement and Data Analysis for Engineering and Science
x
0
true mean value or population mean of x
X
true
true value of x
¯x sample mean of x
ν degrees of freedom
ν
B
x
degrees of freedom for the systematic uncertainty
ν
P
x
degrees of freedom for the random uncertainty
θ
i
absolute sensitivity coefficient
Chapter 8:
D the sum of the squares of all the vertical distances between the measured
and calculated values used in regression analysis
d
i
the i-th vertical distance between the i-th measured and calculated values
e
i
the error between the i-th calculated and measured values
G a coefficient matrix used in multi-variable linear analysis
R a coefficient matrix used in multi-variable linear analysis
r sample correlation coefficient
R
xx
autocorrelation function
R
xy
, R
yx
cross-correlation functions
e
S
2
x
variance of the x errors
e
S
2
y
variance of the y errors
S
yx
standard error of the fit
SSE sum of the squares of the error
SSR sum of the squares of the regression
SST sum of the squares of the total error
t
ν,P
S
yx
precision interval
¯x sample mean
¯
x
2
sample mean of the sum of the squares
¯y
i
estimate of the sample mean
Symbols 479
y
0
i
true mean of y
i
α true intercept
β true slope
Γ gamma function
ρ population correlation coefficient
ρ
xx
population autocorrelation coefficient
ρ
xy
, ρ
yx
population cross-correlation coefficients
σ
xy
population covariance
σ
x
population variance of x
σ
2
y
i
population variance of y
i
Chapter 9:
A
0
, A
n
, B
n
, C
n
Fourier coefficients
A
i
imaginary or complex amplitude
A
r
real amplitude
A(ω) real component of the Fourier transform
B(ω) imaginary component of the Fourier transform
c wave speed
C amplitude
f cyclic frequency
i imaginary number =
√
−1
n integer
R
x
(τ) temporal-averaged autocorrelation coefficient
R
x
(t
1
, t
1
+ τ) ensemble-averaged autocorrelation coefficient
S
x
sample standard deviation of x
S
2
x
sample variance of x
t time
480 Measurement and Data Analysis for Engineering and Science
T fundamental period
¯x sample mean value of x
x
rms
root-mean-square of x
Y (ω) complex Fourier coefficient
z complex number
z
∗
complex conjugate
α phase angle resulting from complex amplitude
κ wave number
λ wavelength
µ
x
temporal-averaged mean value
µ
x
(t
1
) ensemble-averaged mean value
ω circular frequency
φ phase angle
Chapter 10:
A
0
, A
n
, B
n
, C
n
Fourier coefficients
a
0
, a
k
, b
k
, c
k
discrete Fourier coefficients
f cyclic frequency
f
a
aliased frequency
f
1
fundamental frequency
f
N
Nyquist frequency
f
s
sampled frequency
k folding diagram factor
k
a
folding diagram aliased factor
m positive integer
N number of sample points
T
1
fundamental period
Symbols 481
T
r
time-record length
u(t) windowing function
U(f) Fourier transform of the windowing function
Y
k
discrete Fourier transform
Y (f) Fourier transform
y(rδt) discretized signal
δf frequency increment
δt time increment
Chapter 11:
g gravitational acceleration
g
o
local gravitational acceleration
m mass
MW molecular weight
p pressure
R gas constant
R
b
radius of a body
R universal gas constant
S Sutherland constant
T temperature
T
o
reference temperature
w weight
z distance
µ absolute viscosity
µ
o
reference absolute viscosity
ρ density
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C
Review Problem Answers
The following lists the answers to the odd-numbered Review Problems.
Chapter 1: [1] independent; [3] a beaker of ice water; [5] to validate a
particular hypothesis and, in the process, to determine the range of validity
of that hypothesis; [7] (a) parameter, (b) independent, (c) independent or
extraneous, (d), (e), and (f) dependent
Chapter 2: [1] 4.0 µF; [3] 1 V; [5] 3.47 V; [7] current = Ampere; charge
= Coulomb; electric field work = Joule; electric potential = Volta; resistance
= Ohm; power = Watt; inductance = Henry; capacitance = Faraday; [9]
25.2 Ω; [11] 200; [13] deflection method; [15] compression on the lower-side
gage causes an increase in resistance; [17] 11 Ω; [19] (a) −50 W, (b) 33 W;
[21] 0.1 Ω/
◦
R; [23] < 100 Ω; [25] (d)
Chapter 3: [1] radiator fluid temperature = physical variable; ther-
mistor = sensor; Wheatstone bridge = transducer; car computer = signal
processor; [3] 0.47 Ω; [5] 34.2 µF; [7] (c)
Chapter 4: [1] the larger-diameter thermocouple; [3] 25 %; [5] zero; [7]
−84.3
◦
; [9] (b); [11] (d); [13] 250; [15] (a) zero, (b) second, (c) second, (d)
first, (e) first, (f) zero; [17] overdamped (R > 2
p
L/C), critically damped
(R = 2
p
L/C), underdamped (R < 2
p
L/C); [19] (a) −6.47 x 10
−5
, (b)
−5.74 × 10
−5
, (c) −4.20 × 10
−5
, thermistor; [21] 2; [23] 49.4
Chapter 5: [1] 31st-of-the-month birthday = 0.0192, August birthday
= 0.0849, February 29, 1979 birthday = 0, month-with-30-days birthday =
0.329; [3] 0.0625; [5] 1; [7] 2.4; [9] 0.4082; [11] (c); [13] the fourth central
moment; [15] 0.49 % with replacement, 0.47 % without replacement; [17] 31
%
Chapter 6: [1] 0.9544; [3] 0.1621; [5] 0.8742; [7] 0.149; [9] 99.94 %; [11]
10.82; [13] 28.6 %; [15] (d); [17] from 27 to 93; [19] 1 %; [21] 0.1359; [23]
0.1331; [25] both are nondimensional
Chapter 7: [1] readability; [3] player 3; [5] 0.004 59 kg m
2
; [7] 0.024 in.;
[9] 0.1 %; [11] 0.013; [13] 6; [15] 0.002; [17] 0.20 V; [19] 6 %; [21] 0.1 %; [23]
(1) with (c), (2) with (a), (3) with (b), (4) with (d)
483
484 Measurement and Data Analysis for Engineering and Science
Chapter 8: [1] -0.50; [3] Legendre; [5] 0.98; [7] 9.1 V
Chapter 9: [1] 2.687; [3] 0.0450; [5] (a) periodic with 1 s period, (b)
aperiodic because of the exponential term; [7] (c), (d) and (e); [9] (b) and
(c); [11] (a) and (d) periodic, (b) and (c) aperiodic
Chapter 10: [1] 5; [3] 0 Hz; [5] (b)
Chapter 11: [1] (a); [3] (c); [5] (d); [7] 73.5 kg; [9] 291.3 N; [11] 12.04
× 10
23
; [13] 12 000 N; [15] (d); [17] (c); [19] 1125 rpm; [21] (b); [23] (c); [25]
A: mV
2
and B: mV
2
/
p
m/s
Chapter 12: [1] (e)
Index
absolute quantization error, 87
absolute sensitivity coefficient, 243
absolute uncertainty, 243
accuracy, 236
active, 61
actuator, 60
aliasing, 381
ammeter, 38
ampere, 17, 417
amplifier
instrumentation, 77
amplitude, 344
mean squared, 366
amplitude ambiguity, 381
analog, 144
anode, 17
anti-alias, 384
atomic mass unit, 418
atomic weight, 418
autocorrelation, 325
coefficient, 325
function, 325
average
ensemble, 345
bandwidth, 116, 380
beat frequency, 349
binary numbers, 85
binomial, 155
bit reversal, 389
calibration, 104, 236
accuracy, 105
curve, 104
dynamic, 104
experiment, 104
precision, 105
random, 104
range, 104
sequential, 104
static, 104
candela, 418
cathode, 17
central moment, 160
first, 161
fourth, 162
m-th, 160
second, 161
third, 162
zeroth, 161
central tendency, 144
circuit
linear, 21
class intervals, 145
coefficient of determination, 315
combination, 172
combined estimated variance, 235
common-mode rejection ratio, 72
complex
Fourier coefficient, 368
complex modulation, 370
conditional probability, 167
conventionalism, 3
convolution, 371, 400
correlation analysis, 313
Coulomb’s law, 17
coverage factor, 241
critically damped, 121
cross-correlation, 328
coefficients, 329
functions, 328
current
alternating, 17
conventional, 17
direct, 17
cutoff frequency, 79
485