Data Domains and Introduction to Statistics

1. Data Domains and Introduction to Statistics Chemistry 243

2. Photons are modulated by sample Electromagnetic methods Electrical methods Instrumental methods and what they measure

3. Instruments are translators • Convert physical or chemical properties that we cannot directly observe into information that we can interpret.

4. 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. • Thermostat: Displacement used to activate switch http://upload.wikimedia.org/wikipedia/commons/d/d2/Bimetaal.jpg http://upload.wikimedia.org/wikipedia/commons/2/26/Bimetal_coil_reacts_to_lighter.gifhttp://static.howstuffworks.com/gif/home-thermostat-thermometer.jpg

5. Components in translation

6. 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

7. Voltage (V = iR) Quantity to be measured Intermediate quantity 1 Intermediate quantity 2 Emission Current Number Intensity Initial conversion device PMT Intermediate conversion device Resistor Readout conversion device Digital voltmeter Data domains Often viewed on a GUI (graphical user interface)

8. 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

9. 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

10. 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.

11. Byte prefixes About 1000 About a million About a billion

12. Serial and parallel binary encoding Slow – not digital; outdated (serial) Fast – between instruments “serial-coded binary” data Binary Parallel: Very Fast – within an instrument “parallel digital” data

13. 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.

14. 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

15. Statistical distribution • Often follows a Gaussian functional form

16. Introductory statistics: Standard deviation and variance • Standard deviation (s): • Variance (s2):

17. 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

18. Statistical distribution • 50 Abs measurements of an identical sample • Let’s go to Excel Table a1-1, Skoog

19. But no one hasan infinite data set …

20. Standard deviation and variance, continued • s is a measure of precision (magnitude of indeterminate error) • Other useful definitions: • Standard error of mean

21. 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.

22. 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

23. 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

24. Propagation of errors • How do errors at each set contribute to the final result?