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Data Domains and Introduction to Statistics
Chemistry 243
Instrumental methods and what they measure
Electromagnetic methods
Electrical methods
Photons are
modulated by sample
Instruments are translators
Convert physical or chemical properties that we cannot directly observe into information that we can interpret.
0
0
log
log
PT
P
A bc T
P
Pc
b
Sometimes multiple translations are needed
Thermometer Bimetallic coil converts temperature to
physical displacement Scale converts angle of the pointer to an
observable value of meaning
adapted from C.G. Enke, The Art and Science of Chemical Analysis, 2001.
http://upload.wikimedia.org/wikipedia/commons/d/d2/Bimetaal.jpghttp://upload.wikimedia.org/wikipedia/commons/2/26/
Bimetal_coil_reacts_to_lighter.gifhttp://static.howstuffworks.com/gif/home-thermostat-thermometer.jpg
Thermostat: Displacement used to activate switch
Components in translation
Data domains
Information is encoded and transferred between domains Non-electrical
domains Beginning and end of
a measurement Electrical domains
Intermediate data collection and processing
Initial conversion
device
Intermediate conversion
device
Readout conversion
device
Qua
ntity
to
be m
easu
red
Inte
rmed
iate
quan
tity
2
Numbe
r
Inte
rmed
iate
quan
tity
1
PMT ResistorDigital
voltmeter
Emis
sion
Volta
ge (V
= iR
)
Inte
nsity
Curre
nt
Data domains
Often viewed on a GUI(graphical user interface)
Electrical domains Analog signals
Magnitude of voltage, current, charge, or power Continuous in both amplitude and time
Time-domain signals Time relationship of signal fluctuations
(not amplitudes) Frequency, pulse width, phase
Digital information Data encoded in only two discrete levels A simplification for transmission and storage of
information which can be re-combined with great accuracy and precision
The heart of modern electronics
Digital and analog signals
Analog signals Magnitude of voltage, current, charge, or power Continuous in both amplitude and time
Digital information Data encoded in only discrete levels
Analog to digital to conversion Limited by bit resolution of ADC
4-bit card has 24 = 16 discrete binary levels 8-bit card has 28 = 256 discrete binary levels 32-bit card has 232 = 4,294,967,296 discrete binary levels
Common today
Maximum resolution comes from full use of ADC voltage range.
Trade-offs More bits is usually slower More expensive
K.A. Rubinson, J.F. Rubinson, Contemporary Instrumental Analysis, 2000.
Byte prefixes
About 1000
About a million
About a billion
Serial and parallel binary encoding
(serial) Slow – not digital; outdated
Fast – between instruments“serial-coded binary” data
Binary Parallel:Very Fast – within an instrument
“parallel digital” data
Introductory statistics
Statistical handling of data is incredibly important because it gives it significance.
The ability or inability to definitively state that two values are statistically different has profound ramifications in data interpretation.
Measurements are not absolute and robust methods for establishing run-to-run reproducibility and instrument-to-instrument variability are essential.
Introductory statistics:Mean, median, and mode
Population mean (m): average value of replicate data
Median (m½): ½ of the observations are greater; ½ are less
Mode (mmd): most probable value For a symmetrical distribution:
Real distributions are rarely perfectly symmetrical
1 1 2 3 ...lim
N
ii N
N
xx x x x
N N
1/ 2 md
Statistical distribution
Often follows a Gaussian functional form
Introductory statistics: Standard deviation and variance
Standard deviation (s):
Variance (s2):
21lim
N
ii
N
x
N
22 1lim
N
ii
N
x
N
Gaussian distribution
Common distribution with well-defined stats 68.3% of data is within 1s of mean 95.5% at 2s 99.7% at 3s
2221
2
x
y e
Statistical distribution
50 Abs measurements of an identical sample Let’s go to Excel
Table a1-1,Skoog
But no one hasan infinite data set …
21
1
N
ii
x x
sN
22 1
1
N
ii
x x
sN
1
N
ii
x
xN
Standard deviation and variance, continued
s is a measure of precision (magnitude of indeterminate error)
Other useful definitions: Standard error of mean
2 2 2 2 21 2 3 ...total n
mN
Confidence intervals
In most situations cannot be determined Would require infinite number of measurements
Statistically we can establish confidence interval around in which is expected to lie with a certain level of probability.
x
Calculating confidence intervals
We cannot absolutely determine , so when s is not a good estimate (small # of samples) use:
Note that t approaches z as N increases.
2-sided t values
Example of confidence interval determination for smaller number of samples Given the following values for
serum carcinoembryonic acid (CEA) measurements, determine the 95% confidence interval. 16.9 ng/mL, 12.7 ng/mL,
15.3 ng/mL, 17.2 ng/mL
or
Sample mean = 15.525 ng/mL s = 2.059733 ng/mL
Answer: 15.525 ± 2.863, but when you consider sig figs you get: 16 ± 3
Propagation of errors
How do errors at each set contribute to the final result?
2 2 2 2
, , ...
, , ...
...
...
i i i i
vv v
x p q r
x f p q r
dx f dp dq dr
x x xdx dp dq dr
p q r
x x xs s s s
p q r