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