Errors in Analysis

1. Errors in Analysis EVERY measurement has errors every measurement has random errors some have systematic errors it is impossible to eliminate all sources of error it is important to be able to express the magnitude of the error (and minimize it)

2. Terms Accuracy ? the measure of how close a measurement is to the true value did you measure the true (literature) value? Precision ? the measure of how close replicate measurements are to each other how reproducibility is the method (or you)? you need to have both! high accuracy but low precision means you cannot be sure of what the true value is! high precision and low accuracy means you are consistent, but you cannot find the true value

3. Accuracy and Precision

4. Terms replicate measurements are always made in order to determine precision with replicate measurements, there is always a distribution of values mean ? a simple average of the measurements (m1+m2+m3+m4)/4 = average median ? the middle result of the measurements 12.2, 12.3, 12.4 median = 12.3

5. Replicates a replicate is the preparation and measurement of a sample that is nearly the same as another sample replicates provide a better measure of the precision of the method and measurement (instrument) Replicate measurements allow for the detection of outliers (measurements containing gross errors)

6. Mean and Median the mean and median are equal only in a symmetric distribution the mean is affected by outliers in the data set the median is not affected by outliers both are used to assess the accuracy of a measurement

7. Precision replicate measurements show precision precision is expressed by standard deviation variance

8. Absolute Error expresses the absolute magnitude of the error has the same units as the mean of the measurements example: 20.8 (� 0.5) g the true value probably occurs within � 0.5 g

9. Relative Error expresses the magnitude of the error relative to the size of the measurement has units of % or parts per thousand (using the per mille symbol � � think milli or 1/1000) example: 20.8 (� 0.5) g (absolute) (0.5/20.8)*100% = 2.4% (relative error) (0.5/20.8)*1000 � = 24 � (relative error) 0.5 g is the absolute error and 2.4% is the relative error make sure you can do these calculations

10. Error Sig Figs sig figs are meant to communicate uncertainty in a number a buret reading of 20.14 mL certain of the 20.1 mL uncertain of the 0.04 mL if a measurement is 20.14 (�1.2) mL the 0.04 mL is meaningless (so is the 0.2 mL, really) it is OK to use 2 sig figs of uncertainty 20.1 (�1.2) mL

11. Random Errors could be positive or negative effects the precision of the measurement since random errors tend to be positive as often as negative, then cancel out when the average is taken (and don�t affect accuracy) the effect is to cause measurements to scatter and a decrease in precision cannot be eliminated, only reduced sources: white noise in the signal

12. Systematic Errors (Bias) could be positive or negative (but not at the same time for the same error) effects the accuracy of the measurement can sometimes be eliminated different types: instrumental - did not calibrate instrument method errors - problems w/ chemicals, etc. strong acid/strong base titration with phenolphthalein personal - color blindness, bias in reading scale can eliminate with digital meters

13. Detecting Systematic Errors calibrate your instrument use standard reference materials their purity and composition are certified use another method which has been validated it�s like a second opinion

14. Gaussian Distribution ? is the mean or center ? is the standard deviation the probability of an event (or measurement) occurring is calculated by the area under the curve

16. Gaussian Distribution z is the normalized x-axis variable it is used when you want to discuss probability in a relative sense using ?

17. Measurement Statistics population (or universe) distribution created when N ? ? (infinity) yields a mean (?), and a standard deviation (?) since it is impossible to perform an infinite number of measurements, we are stuck with the sample distribution created when N is small yields a mean ( ), and a standard deviation ( s ) the sample statistics are usually the same as the population statistics when N>20 or 30

18. Measurement Statistics the mean (? or ) the center of the distribution the number that would result from the measurement if there were no errors represents accuracy the standard deviation (? or s) is a measure of the error in a single measurement has the same units as the mean represents precision

19. Measurement Statistics standard deviation (? or s)

20. Other Terms relative standard deviation (RSD) RSD ? std. dev./mean can be expressed as a % or ppt also called the coefficient of variation spread or range ? xmax - xmin

21. Pooled Standard Deviation if data from an instrument or method has been collected over a period of time, you have a track-record of the random error instead of having to make many replicates in one sitting (N>20), spooled can be used

22. Significant Figures scientists convey information by the numbers they report 4.21 mL means the 4.2 mL is certain and the 0.01 mL is uncertain it is important for you to convey the proper information when reporting numerical values

23. Rounding when doing a compound calculation, never round until the very end of the calculation it is a good idea to record 2 or three extra sig figs for future calculations notebook entry 1.23 g NaCl ? 58.44 g/mol = 0.021047 = 0.0210 mol 0.021047 mol ? 0.500 mL = 0.04209 = 0.0421 Molar (if the extra sig figs are left out ? 0.2% error) 0.0210 mol ? 0.500 mL = 0.0420 = 0.0420 Molar

24. Rounding(make sure you read this!) 4.2051 is rounded to 4.21 4.2049 is rounded to 4.20 4.2050 is rounded to 4.2? from 4.201, 4.202, 4.203, 4.204 round to 4.20 from 4.206, 4.207, 4.208, 4.209 round to 4.21 rounding 4.205 to 4.21 would provide a higher probability of rounding up (5 cases to 4 cases) since there are 4 cases of each round up to make an even number round down to make an even number 50% of the time you will round down (or up)

25. Rounding 24 ? 4.52 ? 100.0 = 1.0848 = 1.1 24 ? 4.02 ? 100.0 = 0.9648 = 0.96 should both answers have 2 sig figs?

26. Significant Figures standard deviations (error), absolute error, and relative error convey uncertainty round all types of error to 2 sig figs since uncertainty lies in the first digit, the second digit is even more uncertain the second digit is useful to prevent rounding errors the third, fourth, etc. digits are completely useless (in the final answer, but may be useful in propagation of error calculations)

27. Evaluating Data how certain are you that your number is the true number? would you be more certain if you gave a range of numbers instead of a single value? how often (out of 100 times) would you be wrong? use a confidence limit (-------- -------) ? (--------------------- -------- ? ---------)

28. Confidence Intervals since the std. dev. of the measurement provides the precision, use multiples of it to provide a confidence interval

30. Confidence Intervals what is convincing? 50% confidence? 80% ? 90% ? 95% ? 100% ? 100% confidence covers - ? to + ? and is not practical 90% to 95% are typically used NOTE: stating a confidence interval assumes that all systematic errors have been removed

31. Confidence Intervals is s a good estimate of ? ? (Is N>20 ?) if so, then assume s = ? , x = ? CL = ? � z ? or (where z is from table)

32. Confidence Intervals if s is not a good estimate of ? ( N < 20) need to use Student�s t values if N is small, then the precision of the measurement is not well defined the confidence interval must expand in order to represent the same certainty t is bigger than z think of t as z ? (a fudge factor) as N? ? (actually 20 or so), t ? z

35. Confidence Interval Range in one of the CAPA sets, you will be asked to provide a range or span of the confidence interval

36. Hypothesis Testing this is the scientific method make a hypothesis make some measurements do the results support the hypothesis? if yes, then use the hypothesis again if no, then abandon the hypothesis you will use this when you need to compare your results to something the true value or theoretical value to another measurement (another method)

37. Hypothesis Testing (t-testing) Null Hypothesis ? two numbers (means) are the same is my mean the same as ? within error? is the mean from my sample the same as the mean from your sample? is the mean from instrument #1 the same as the mean from instrument #2? is the standard deviation from instrument #1 the same as the standard deviation from instrument #2?

38. Comparing An Experimental Mean and the True Value you assay an NBS antacid tablet for CaCO3 you get 535 ? 12 mg (N = 4) NBS gets 550.0 mg are they the same? is there significant bias?

39. Comparing An Experimental Mean and the True Value you assay an NBS antacid tablet for CaCO3 you get 535 ? 12 mg (N = 4) NBS gets 550.0 mg are they the same?

40. Comparing An Experimental Mean and the True Value if |tcalc | ? ttable then there is a significant bias (systematic error) in the method the systematic error needs to be tracked down and eliminated NOTE: just because the null hypothesis is correct, does not mean that your method is acceptable (std error may be too large)

41. Comparing An Experimental Mean and the True Value what if s is a good estimate of ?? then, t becomes z and you look up ztable from the table

42. Comparing Experimental Means aspirin tablets from two different batches are assayed for their aspirin content batch #1: 328.1 ? 2.6 mg/tablet (N=4) batch #2: 341.5 ? 2.3 mg/tablet (N=5) are they the same? since both samples were collected under the same conditions, use spooled to answer the question

43. Comparing Experimental Means batch #1: 328.1 ? 2.6 mg/tablet (N=4) batch #2: 341.5 ? 2.3 mg/tablet (N=5) are they the same? is there significant bias?

44. Comparing Experimental Means batch #1: 328.1 ? 2.6 mg/tablet (N=4) batch #2: 341.5 ? 2.3 mg/tablet (N=5) are they the same? is there significant bias?

45. Summary compare exp. mean with true use the eqn as is compare to tcrit from the table

46. Summary compare the calculated t to the t value in the table (tcrit) for a given confidence interval and degrees of freedom (DOF) |tcalc | ? ttable to reject the null hypothesis |tcalc | < ttable to accept the null hypothesis

47. Quantitation the absorbance of an analyte is proportional to its concentration (A ? C) the absorbance of an analyte is proportional to the amount of sol�n it passes through the its absorption properties of the chemical A = a2 C (Beer�s Law) where C = concentration a2 = a constant; slope of the graph

48. Methods of Quantitation method of external standards - commonly associated with calibration curves method of multiple standard additions - used when the matrix of the sample is complicated the method of Internal Standard � an exotic compound is added to the original sample before processing to gauge the efficiency of the analysis method and to quantitate the unknown this method will not be used in this class

49. External Standards produce as series of solutions with known analyte concentration produce a calibration curve (signal vs. conc.) measure the signal of the unknown use the regression statistics to find unknown concentration, uncertainty advantage - curve can be used for several unknowns disadvantage - since std�s are made w/ pure solvent, matrix is not the same as unknown

51. External Standards IMPORTANT: the error in the unknown concentration is minimized if it is centered in the calibration curve

52. Multiple Standard Additions advantage - the sample and standards have the same matrix, so there is not systematic error due to a matrix mismatch disadvantage - must repeat the entire procedure for every unknown sample

53. Multiple Standard Additions

54. Multiple Standard Additions

56. Regression Analysis We will go over the following slides in lab when we construct the regression spreadsheet

57. Linear Regression we will not focus on the math of linear regression in this course focus your attention on the spreadsheet functions described in the book

58. Spreadsheet Functions linear regression assume the data set fits y = mx + b m is the slope, b is the y-intercept y is the dependent variable x is the independent variable slope(y values, x values) intercept(y values, x values) rsq(y values, x values) (R=correlation coefficient) calculates the R2 value, which indicates the quality of the fit ( 0? R2 ? 1, R2 =1 means a perfect fit) steyx(y values, x values) (see a later slide) varp(x values) (see a later slide)

62. Comparing STEYX to sr start with sy and b substitute and pull out a 1/[n(n-2)]

63. Errors in Slope and Y-int

65. Uncertainty Formula