Math and Measurement
for Biomed Techs
© D. J. McMahon 140923 rev cewood 2016-01-02
Standard Prefixes
Number
Prefix
Symbol
Number
101
deka-
da
10-1
deci-
d
102
hecto-
h
10-2
centi-
c
103
kilo-
k
10-3
milli-
m
106
mega-
M
10-6
micro-
109
giga-
G
10-9
nano-
n
1012
tera-
T
10-12
pico-
p
1015
peta-
P
10-15
femto-
f
1018
exa-
E
10-18
atto-
a
Most frequently used:
Prefix
Symbol
µ
(mu)
1/100
= 10-2 = centi
x 1,000
= 103 = kilo
1/1 000
= 10-3 = milli
x 1,000,000
= 106 = mega
1/1 000 000
= 10-6 = micro
x 1,000,000,000 = 109 = giga
1/1 000 000 000 = 10-9 = nano
Scientific Notation
Scientific notation is a way of expressing numbers that are too big or too small
to be conveniently written in decimal form.
In scientific notation all numbers are written in the form m × 10n
(m times ten raised to the power of n), where the exponent n is an integer, and
the coefficient m is any real number.
In normalized scientific notation, the exponent n is chosen so that the absolute
value of m remains at least one but less than ten. (1 ≤ |m| < 10)
Thus 350 is written as 3.5×102. This form allows easy comparison of numbers, as
the exponent n gives the number's order of magnitude.
In normalized notation, the exponent n is negative for a number
with absolute value between 0 and 1 (e.g. 0.5 is written as 5×10−1).
The 10 and exponent are often omitted when the exponent is 0.
much more:
https://en.wikipedia.org/wiki/Scientific_notation
Dimensional Analysis (the Unit Factor Method)
Dimensional Analysis is a problem-solving method that uses the fact that any number can be
multiplied by one without changing its value.
Unit factors are made from any two terms that describe the same or equivalent value.
examples:
Because ‘times 1’ is equal to ‘divided by 1’, you can invert these unit factors as you wish.
Your goal is to arrange all your unwanted units so they cancel out.
Solve:
much more: https://en.wikipedia.org/wiki/Dimensional_analysis
With few exceptions, medical
parameters are always expressed
in the metric (SI) system.
SI: (Système International d'Unités)
Physical Units & Constants
Name
Symbol
Unit
Unit Name
Charge
Q
C
Coulomb
Voltage
V
V
Volt
Resistance
R

Ohm
Conductance
G
S
Siemens (mho)
Capacitance
C
F
Farad
Inductance
L
H
Henry
Mag. Induction
B
T
Tesla
Frequency
F
Hz
Hertz
Power
P
W
Watt
Energy
E
J
Joule
SI Units that are especially important
for Biomed Technology:
Mass:
kilogram (kg)
Length:
meter (m)
Time:
second (s)
Force:
newton (N)
(kg.m / s2)
Energy:
joule (J)
(N.m) = (watt.sec)
Power:
watt (W)
(N.m / s)
Pressure:
pascal (Pa)
(N/m2)
and
Light:
Torr (.133 kp) = (mm of Hg column)
candela (cd)
(luminous intensity)
lumen (lm)
(cd.sr)
lux (lx)
(lumens / m2)
Magnetic Flux: Tesla (T)
(light flux)
(illuminance)
Pressure as a column of mercury
barometer
manometer
Manometers, two styles;
Pressure measured as a column of mercury:
barometer manometer
• ‘absolute’ pressure
• closed end (contains vacuum)
• measures atmospheric pressure
• 760 mmHg ‘normal’ (at sea level)
• higher atmospheric pressure =
higher number
• (pressure in outer space vacuum
would be 0 mmHg)
• ‘gauge’ pressure
• open end (contains ambient
air pressure)
• measures applied pressure
• 0 mmHg is ‘unpressurized’ state
• higher applied pressure =
higher number
Pressure Conversions (most useful conversions shown in bold)
PSI
1 PSI =
KiloPascal cm of H2O
mm of Hg atmosphere
millibar
1
6.89
70.3
51.7
0.068
68.9
1 KiloPascal =
0.145
1
10.19
7.5
0.0098
10
1 cm of H2O =
0.014
0.1
1
0.735
0.00097
1
1 mm of Hg =
0.019
0.133
1.36
1
0.0013
1.33
1 atmosphere =
14.7
101
1033
760
1
1013
0.0147
0.1
1.02
0.75
0.0009
1
1 millibar =
remember:
1 atmosphere = 14.7 psi = 760 mmHg = 1033 cmH2O = 101 kPa
Hg: 760 mm = 29.92 inches
H2O: 1033 cm = 33.9 feet
Portland State University:
The Big Barometer
In the atrium of the Engineering Building stands the world’s tallest barometer. At 14.2 m tall (46'8"),
the PSU Barometer is made of repurposed 2" i.d. pyrex glass pipe and has a 14 gallon reservoir. The
thick-walled glass pipes were salvaged during a renovation of Science Building 2. They had
previously been used for over 35 years as chemistry lab drain pipes.
The PSU barometer uses vacuum pump oil instead of water as the working fluid. Vacuum pump oil
has several advantages in this application such as extremely low vapor pressure and the lowest
available fluid density. Low vapor pressure is important for maintaining a vacuum over the fluid
column. The low density of the vacuum oil allows the PSU Barometer’s fluid column to reach a
nominal height of 12.4m and +/- excursions of 0.4m. A water barometer is limited to a height of
10m due to the density of water.
The construction of the barometer received front page coverage in The Oregonian.
In addition to breaking a world record, the architect of the barometer, Tom Bennett was named the
2013 Outstanding Supporter of Research by the Sigma Xi Columbia Willamette chapter.
The PSU Barometer is primarily as a teaching tool for the Civil and Environmental Engineering fluidrelated lab courses. It is available for other PSU Departments and for science and engineering
education outreach programs.
If you would like to contact Maseeh College about using the PSU Barometer in science education
curriculum or to schedule a tour of the college and see it in person please email
barometer@cecs.pdx.edu
vacuum pump oil: 12.4 m = 40.7 feet
https://www.pdx.edu/profile/big-barometer
Atmospheric Pressure at Increasing Elevations
Elevation
in of
Hg
mm of Hg
35
0
29.92
760
30
500
29.38
746
25
1000
28.86
733
20
3000
26.82
681
15
4000
25.84
656
10
5430 (Denver)
24.40
620
5
14411 (Mt Rainier)
19.80
502
0
29035 (Mt Everest)
8.90
225
0
2000
4000
6000
8000
10000
12000
elevation vs inches of Hg
14000
16000
Significant Figures
> In any measurement, we can’t claim more accuracy
than physical reality allows.
> Always ask:
- Plus or minus how much?
- In digital displays, is there “last digit bobble”?
> The number of significant figures is NOT improved by
multiplying errors.
> Three digits (sometimes four) is usually the best you
can do!
Measurement Errors Static Error –
Misreading displays or limitations of equipment
> parallax reading of an analog meter
> interpolation of the scale on an analog meter
> last-digit “bobble” on a digital meter
Dynamic Error –
> errors caused by changing values during measurement
Instrument Insertion Errors –
> “loading” of the device under test by the tester
Mean, Median, and Mode
Mean:
• Sum of all values divided by the number of quantities.
Usually called the ‘average’.
Median:
• It is the middle value in the data set, that is, the value where
exactly half of the values are above it and half below it.
Mode:
• The most frequently occurring value in a set of data.
Example:
Given these measurements (after you rank them in order):
101
103
104
105
106
mean value = 950 (the sum) / 9 = 105.55
median value = 106
mode = 107
107
107
108
109
Mean, Median, and Mode
Standard Deviation
(σ)
> Widely used measure of variability or dispersion of data.
> Standard deviation serves as a measure of how far the
samples of data are spread out.
> A large standard deviation indicates that the data points
are far from the mean; a small standard deviation indicates
that they are clustered closely around the mean.
The 68–98–99.7 Rule:
• Standard deviation:
σ (sigma)
• 1 standard deviation:
X    68.27%
• 2 standard deviations:
X  2  98.45%
• 3 standard deviations:
X  3  99.73%
The mean is the same, but….

=1

=2

=3
Precision vs Accuracy
Precision:
the closeness of many measurement points to each other
Accuracy:
the closeness of many measurement points to a reference
think of ‘accurate’ as ‘actual’
or Remember “P.A.R.T.”:
“Precision and Accuracy mean Repeatability and Trueness”
Precision vs Accuracy
Which is more important in measurements?
Precise and
accurate
Imprecise but
accurate
Precise but
inaccurate
Imprecise and
inaccurate
Root Mean Square
“RMS”
In electronics, used to express AC current or voltage
as its equivalent DC current or voltage.
VRMS = 0.707 × Vpeak
Vpeak = 1.414 × VRMS
accurate for a sine wave only
So, 110 VAC (rms) = peak voltage of 155.5 volts
RMS voltage is the equivalent “heating voltage” of AC :
sine wave
A sine wave is a curve with this shape:
Signals in the electromagnetic spectrum
(heat, radio, light, x-rays, etc) have a
speed of 3x108 meters/second.
With this standard speed, frequency
and wavelength are reciprocal to each
other.
300 MHz (3x108 cycles per second)
signals have a wavelength (λ) of 1
meter.
The sine wave has a pattern that repeats. The
length of this repeating piece of the sine wave
is called the wavelength (λ) . The wavelength
can be found by measuring the length or
distance between one peak of a sine wave and
the next peak.
All periodic waves can be made by adding up
sine waves of different frequencies and
amplitudes. This is called Fourier Analysis,
sometimes calculated by the Fast Fourier
Transform (FFT).
‘Apparent power’ vs
‘True (or Real) Power’
or
VA vs Watts
Watts is Real Power (P) - the power (V x A) that does work.
VA is Apparent Power - the vector sum (S) of real power (P) and reactive power (Q).
Reactive loads such as inductors and capacitors
dissipate zero power, yet the fact that they drop voltage
and draw current gives the deceptive impression that
they actually do dissipate power.
This “phantom power” is called reactive power.
http://www.allaboutcircuits.com/textbook/alternating-current/chpt-11/truereactive-and-apparent-power/
Apparent Power is used when sizing wiring and components.
Real Power is what accomplishes useful work in the device.
Apparent Power is always > Real Power if there is any reactive factor.
Power Factor = W / VA ( 0.60 is typical)
Logarithmic Units: Decibels
Bel was used in the telephone industry (named after Alexander Graham
Bell).
The Bel is usually too large for most applications, so it is rarely if ever used.
Decibel (dB) is one-tenth of a Bel. It is simply a means of logarithmically
expressing the ratio between two signal levels.
Used mainly in amplifier comparisons
special cases:
audio Volume Unit (VU): 0 VU = 1 mV at 1 KHz through 600 
radio dBm: 0 dBm = 1 mV of RF through 50 
Three General Categories of Measurement
Direct measurement:
• Compare the parameter to some calibrated standard.
Indirect measurement:
• Measure something other than the actual parameter
that we want.
Null measurement:
• Compare a calibrated source to an unknown value
and adjust the unknown value until the difference
between them is zero.
Measurement
Standards:
> International References: at the ISI
> Primary Standards:
at the NIST
> Working Standards:
“NIST Traceable”
> Secondary Standards:
on-site references
> Gauges & Instruments: routine equipment
Resolution
(“Definition”)
The degree to which we can distinguish
the individual elements of an output.
eg: the lines in a video display test pattern,
or the change of pitch in an audio signal
pH:
The measure of acidity or alkalinity of any liquid
Water always has a small amount of hydrogen (H+) and hydroxide
(OH-) ions.
pH is the numeric value from 0 to 14,
taken from the negative of the exponent of the concentration of hydrogen ion.
So if a solution has a hydrogen concentration of 1 x 10-8, then its pH is 8.
pH = - log10 [H+]
**smaller pH number is more acidic **
pH range and values of common substances
Poiseuille's Law :
( pwah-zwee )
defines the flow (Q) of fluid passing a point along the tube
in terms of:
> the fluid's viscosity (η)
> the tube's radius
(r)
> the tube’s length
(L)
> the pressure difference along the tube (ΔP)
4
π r ΔP
Q = -----------8ηL
π r 4 ΔP
Q = -----------8ηL
In other words,
The flow in a tube is directly proportional to the fourth power of
the radius. This means that doubling the radius of the tube
increases the fluid flow by a factor of 16.
Poiseuille's Law :
Example from vascular physiology:
Poiseuille's Law :
Example from pulmonary physiology:
Instrumentation Amplifier
Advantages for physiological monitoring:
> High Common Mode Rejection Ratio (CMRR):
The capability of an instrument to reject a signal that
is common to both input leads.
[ CMRR = Differential Gain / Common Mode Gain ]
> High input Z
> Wide bandwidth
> Low noise
Instrumentation Amplifier
Analog to digital conversion
A process in which a continuously variable (analog) signal is
changed into a multi-level signal without altering its essential
content.
The input is a voltage that varies among a theoretically infinite
number of values (sine waves, speech, ECG, etc). The output
has defined levels or states. The simplest digital signals are
in binary values.
Digital signals propagate more efficiently than analog signals,
because digital impulses, which are well-defined and orderly,
are easier for electronic circuits to distinguish from noise,
which is chaotic.
Quantization Error:
Error resulting from trying to represent a continuous
analog signal with discrete, stepped digital data.
When the analog value being sampled falls between
two digital “steps.” the analog value must be
represented by the nearest digital value, resulting in a
very slight error.
The difference between the continuous analog
waveform and the stair-stepped digital representation
is quantization error. For a sine wave, quantization
error will appear as extra harmonics in the signal. For
music or program material, the signal is constantly
changing and quantization error appears as wideband
noise, cleverly referred to as “quantization noise.”