data domains and introduction to statistics
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Data Domains and Introduction to Statistics. Chemistry 243. Photons are modulated by sample. Electromagnetic methods. Electrical methods. Instrumental methods and what they measure. Instruments are translators. - PowerPoint PPT PresentationTRANSCRIPT
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
PTP
A bc TPP
cb
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
Quanti
ty to
be m
easu
red
Interm
ediat
e
quan
tity 2
Numbe
r
Interm
ediat
e
quan
tity 1
PMT Resistor Digital voltmeter
Emission
Volta
ge (V
= iR
)
Inten
sity
Curren
t
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 1000About a millionAbout 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 Nm
1/ 2 mdm m m
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
m
s
22 1lim
N
ii
N
x
N
ms
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 em
s
s
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 xs
N
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 ns s s s s
m Nss
Confidence intervals In most situations m cannot be determined
Would require infinite number of measurements Statistically we can establish confidence interval
around in which m is expected to lie with a certain level of probability.
x
Calculating confidence intervals We cannot absolutely
determine s, 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 drp q r
x x xs s s sp q r