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Statistics for Analytical Chemistry

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Statistics for Analytical Chemistry

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  1. Statistics for Analytical Chemistry

  2. Reading –lots to revise and learn • Chapter 3 • Chapter 4 • Chapter 5-1 and 5-2 • Chapter 5-3 will be necessary background for the AA lab • Chapter 5-4 we will use later

  3. Data Analysis • Most data quantitative - derived from measurements • Never really know error • With more measurements you get a better idea what it might be • Don’t spend a lot of time on an answer -where only 20% accuracy is required -or where sampling error is big - although you don’t want to make the error worse

  4. Significant Figure Convention • Final answer should only contain figures that are certain, plus the first uncertain number • eg 45.2% • error less than 1% or we would only write 45% • error larger than 0.05% or would write 45.23%

  5. Remember • Leading zeros are not significant • Trailing zeros are significant • 0.06037 - 4 significant figures • 0.060370 - 5 significant figures • 1200 ???? • 12 x 102 - 2 significant figures

  6. Rounding Off • Round a 5 to nearest even number • 4.55 to 4.6 • Carry an extra figure all through calculations • BUT NOT 6 EXTRA • Just round off at the end

  7. Adding • Absolute uncertainty of answer must not exceed that of most uncertain number • Simple rule: Decimal places in answer = decimal places in number with fewest places 12.2 00.365 01.04 13.605 goes to 13.6

  8. When errors are known • Rr =(A a) + (B b) + (C c) • where r2 = a2 + b2 + c2 • Example: Calculate the error in the MW of FeS from the following atomic weights: • Fe:55.847 0.004 S:32.064 0.003 • r = (0.0042 + 0.0032)1/2 • MW = 87.911 0.005

  9. Multiplication and Division • Simplest rule: Sig figs in answer = smallest number of sig figs in any value used • This can lead to problems - particularly if the first digit of the number is 9. • 1.07400 x 0.993 = 1.07 • 1.07400 x 1.002 = 1.076 • Error is ~ 1/1000 therefore 4 significant figs in answer

  10. Multiplication and Division • The relative uncertainty of the answer must fall between 0.2 and 2.0 times the largest relative uncertainty in the data used in the calculation. • Unless otherwise specified, the absolute uncertainty in an experimental measurement is taken to be +/- the last digit

  11. Multiplication and Division • With known errors - add squares of relativeuncertainties • r/R = [(a/A)2 + (b/B)2 +(c/C)2]1/2

  12. Logs • Only figures in the mantissa (after the decimal point) are significant figures • Use as many places in mantissa as there are significant figures in the corresponding number • pH = 2.45 has 2 sig figs

  13. Definitions • Arithmetic mean, (average) • Median -middle value • for N=even number, use average of central pair

  14. Accuracy • Deviation from true answer • Difficult to know • Best way is to use Reference standards • National Bureau of Standards • Traceable Standards

  15. Precision • Describes reproducibility of results • What is used to calculate the confidence limit • Can use deviation from mean • or relative deviation • 0.1/5 x 1000 = 20ppt (parts per thousand) • 0.1/5 x 100% = 2%

  16. Precision of Analytical Methods • Absolute standard deviation s or sd • Relative standard deviation (RSD) • Standard deviation of the mean sm • Sm = s/N½ • Coefficient of variation (CV) s/x x 100% • Variance s2

  17. Standard Curve Not necessarily linear. Linear is mathematically easier to deal with.

  18. Correlation coefficients • Show how good a fit you have. • R or R2 • For perfect correlation, R = 1, R2 = 1

  19. LINEST • Calculates slope and intercept • Calculates the uncertainty in the slope and the intercept • Calculates R2 • Calculates s.d. of the population of y values • See page pp 68-72, Harris.

  20. Use these values to determine the number of sig figs for the slope and intercept

  21. Dealing with Random Errors

  22. Indeterminate Error • Repeating a coarse measurement gives the same result • eg weighing 50 g object to nearest g - only error would be determinate - such as there being a fault in the balance • If same object was weighed to several decimal places -get random errors

  23. How many eggs in a dozen? • How wide is your desk? • Will everyone get the same answer? • What does this depend on?

  24. With a few measurements, the mean won’t reflect the true mean as well as if you take a lot of measurements

  25. Random errors • With many measurements, more will be close to the mean • Various little errors add in different ways • Some cancel - sometimes will all be one way • A plot of frequency versus value gives a bell curve or Gaussian curve or normal error curve • Errors in a chemical analysis will fit this curve

  26. Equation for Gaussian Curve

  27. If z is abscissa (x axis) • Same curve is always obtained as z expresses the deviation from the mean in units of standard deviation

  28. Statistics • Statistics apply to an infinite number of results • Often we only do an analysis 2 or 3 times and want to use the results to estimate the mean and the precision

  29. 68.3%: ±1 , 95.4%: ±2 , 99.7%: ±3 68

  30. Standard deviation • 68.3% of area is within ± 1 of mean • 95.5% of area is within ± 2 of mean • 99.7% of area is within ± 3 of mean • For any analysis, chances are 95.5 in 100 that error is ± 2 • Can say answer is within  ± 2 with 95.5% confidence

  31. For a large data set • Get a good estimate of the mean,  • Know this formula -but use a calculator • 2 = variance • Useful because additive

  32. Small set of data • Average (x )   • An extra uncertainty • The standard deviation calculated will differ for each small set of data used • It will be smaller than the value calculated over the larger set • Could call that a negative bias

  33. s • For  use N in denominator • For s use N-1 in denominator (we have one less degree of freedom - don’t know ) • At end, round s to 2 sig figs or less if there are not enough sig figs in data

  34. Confidence Interval • We are doing an analysis to find the true mean  - it is unknown • What we measure is x but it may not be the same as  • Set a confidence limit eg 4.5 ± 0.3 g • The mean of the measurements was 4.5 g • The true mean is in the interval 4.2-4.8 with some specified degree of confidence

  35. Confidence limit • A measure of the reliability (Re) • The reliability of a mean (x ) increases as more measurements are taken • Re = k(n)1/2 • Reliability increases with square root of number of measurements • Quickly reach a condition of limiting return

  36. Reliability • Would you want a car that is 95% reliable? • How often would that break down?

  37. Confidence Interval • For 100 % confidence - need a huge interval • Often use 95 % • The confidence level chosen can change with the reason for the analysis

  38. Confidence Interval when s ~ s • µ ± xi = 1.96  for 95 % confidence • z = (xi - µ)/ =1.96 • Appropriate z values are given as a table • This applies to a single measurement • The confidence limit decreases as (N)1/2 as more measurements are taken

  39. Confidence Interval • In the lab this year I will make you go home before you can get enough data for s to =  • Therefore we will have to do a different kind of calculation to estimate the precision.

  40. Student’s t-test The Student's t-Test was formulated by W. Gossett in the early 1900's. His employer (brewery) had regulations concerning trade secrets that prevented him from publishing his discovery, but in light of the importance of the t distribution, Gossett was allowed to publish under the pseudonym "Student". The t-Test is typically used to compare the means of two populations

  41. t-test • t depends on desired confidence limit • degrees of freedom (N-1)

  42. For practical purposes • Assume  = s if you have made 20 measurements • Sometimes  can be evaluated for a particular technique rather than for each sample • Usually too time consuming to do 20 replicate measurements on each sample

  43. CONFIDENCE