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Measurements and Noise 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 Rounding • 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) 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 1.4562 g NaCl 58.54 g/mol = 0.0248752 mol 0. 0248752 mol 0.500 mL = 0.0497504 = 0.04975 M – (if the extra sig figs are left out...) 0.02487 mol 0.500 mL = 0.04974 M (-0.02% error) – (if the numbers were rounded to begin with) 1.5g 59 g/mol 0.5 mL = 0.051 M (2.5% error) 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? • what if the number were 0.999 ? • should you round up then drop a sig fig? • if a number barely rolls over into the next digit, add an extra digit to the sig figs (1.00) 1 Significant Figures • the standard deviation conveys 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) Error Sig Figs 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 or 20 (±1) mL is OK Read About the Following • 20.1 (±1.2) mL or 20 (±1) mL is OK • keep the extra sig fig of uncertainty to prevent rounding errors from increasing your uncertainty • guard digit the extra sig fig in uncertainty Measurement Statistics (read on your own) • the mean ( x) Use the spreadsheet – the center of the distribution function AVERAGE – the number that would result from the measurement if there were no errors – represents accuracy • the standard deviation (s) – is a measure of the error in a single measurement 2 – has the same units as the mean N x i N – represents precision 2 xi i 1 Use the spreadsheet N s i 1 function STDEV N 1 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 ( x ), and a standard deviation ( s ) • the sample statistics are usually the same as the population statistics when N>20 or 30 2 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 x x x x 2 s pooled What do you call this? 5000 averaged scans 2 i set1 Example: i set 2 12 averaged scans What do you call this? N1 N 2 2 s ( N1 1) s22 ( N 2 1) N1 N 2 2 2 1 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 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 • can eliminate with digital meters Hypothesis Testing Types of Errors (Summary) • this is the scientific method • random or indeterminate a measure of precision • systematic (bias) or determinate a measure of accuracy • PIBCAK • ID10T – – – – – 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) 3 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? Comparing An Experimental Mean and the True Value • you assay an NIST antacid tablet for CaCO 3 – you get 535 12 mg (N = 4) – NIST gets 550.0 mg • are they the same? is there significant bias? x t s N where is the true value, s is the std. dev. and N is the number of replicates • then compare tcalc with ttable from the Student’s t table • if |tcalc | ttable then reject the null hypothesis 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) as N (actually 20 or so), t z Comparing An Experimental Mean and the True Value • you assay an NIST antacid tablet for CaCO 3 – you get 535 12 mg (N = 4) – NIST gets 550.0 mg t • are they the same? t 535 550 2.5 12 / 4 x s N tcrit= ttable= 3.18 (@ 3 DOF and 95%) • |tcalc | ttable to reject the null hypothesis • 2.5 < 3.18, so the numbers are the same 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 z x N • if |zcalc | ztable then there is a significant bias (systematic error) in the method • the systematic error needs to be tracked down and eliminated 4 Comparing Experimental Means • aspirin tablets from two different batches are assayed for their aspirin content – batch #1: 328.1 2.6 mg/tablet – batch #2: 341.5 2.3 mg/tablet (N=4) (N=5) Comparing Experimental Means – batch #1: 328.1 2.6 mg/tablet – batch #2: 341.5 2.3 mg/tablet • are they the same? is there significant bias? s1 = 2.6 s2 = 2.3 • are they the same? x x x x 2 • since both samples were collected under the same conditions, use spooled to answer the question s pooled (N=4) (N=5) • are they the same? is there significant bias? t x1 x 2 s pooled N1N2 328.1 341.5 N1 N2 2.433 45 8.2098 45 • |tcalc | ttable to reject the null hypothesis • 8.21>2.36 (7 DOF and 95%), so the null hypothesis is rejected • there is a significant different between the two batches 2 i set1 i set 2 N1 N 2 2 s ( N1 1) s22 ( N 2 1) N1 N 2 2 2 1 spooled = 2.433 Summary Comparing Experimental Means – batch #1: 328.1 2.6 mg/tablet – batch #2: 341.5 2.3 mg/tablet (N=4) (N=5) • compare exp. mean with true – use the eqn as is – compare to tcrit from the table t x s N • comparing two exp. means (same method) – change xt to x 2 (bar) x x N1 N 2 t 1 2 – find spooled s pooled N1 N 2 These equation change slightly when you move from sample statistics to population statistics (s) Definitions Summary • compare the calculated t to the t value in the table (tcrit) for a given confidence interval and degrees of freedom (DOF) • blank the solution that contains everything except the analyte • matrix everything except the analyte • |tcalc | ttable to reject the null hypothesis • |tcalc | < ttable to accept the null hypothesis • difference? – The matrix is in every solution that contains the sample – The blank is one solution of the calibration curve made without the analyte 5 Figures of Merit • Precision standard deviation of a measurement - a measure of random error – example: white noise or static • Bias is measured value higher or lower than the actual? – example: analyte loss in processing • Sensitivity the slope of the calibration curve • Limit of Detection (LOD) how low can you go? (S is the signal, NOT the stdev (rep. by s or ) why the difference? Figures of Merit • Dynamic Range or Limit of Linearity (LOL) linear calibration curve from LOD to some concentration • Selectivity how easy is it to ignore contaminants? A selective method/instrument can detect a group of related analytes. Limit of Detection Example LOD: sigm = sigbl + 3 bl LOQ: sigm = sigbl + 10 bl • examine the error in the y-int of the graph and blank signal • take the larger of the two values (blank and error) – y-int: 0.0161470.0028 – blank: sigbl=0.02130.0018 • Specificity how selective is the method/instrument? The ultimate selectivity is one that is specific for a single analyte. Limit of Detection Example 0.16 Slope Std. Dev. in Slope (s m ) Y-intercept Std. Dev. in Y-int (s b ) 0.015178 0.000581 0.016147 0.0028 0.14 R2 0.991293 Error Analysis Standard error in y (s r ) N (count of standards) S xx ybar (avg absorbance) 0.005837 8 1.01E+02 6.56E-02 sigm = sigbl + 3 bl sigm = 0.0213 + 30.0028 = 0.0297 • this is the minimum signal, so substitute it into the linear eqn: y=mx+b (where y is the signal) x = (y-b)/m = (0.0297- 0.016147)/0.015178 = 0.8929 • this is the minimum signal for the LOD • Repeat for the LOQ (replace 3 with 10): 2.184 y = 0.0152x + 0.0161 R2 = 0.9914 0.12 0.1 0.08 0.06 0.04 0.02 0 0 2 4 6 8 10 12 Concentration (mg/L) Limit of Detection Example y-int: 0.0161470.0028 blank: sigbl=0.02130.0018 Calibration Curve 0.18 y = 0.0152x + 0.0161 R2 = 0.9913 0.16 0.14 0.12 Signal LOD: sigm = sigbl + 3 bl LOQ: sigm = sigbl + 10 bl Calibration Curve 0.18 Regression Analysis Signal – Qualitative (LOD): Sm = Sbl + 3bl – Quantitative (LOQ): Sm = Sbl + 10bl – We’ll do an example in a minute 0.1 0.08 0.06 LOD LOQ 0.04 0.02 0 0 2 4 6 8 10 12 Concentration (mg/L) 6 External Standards 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 • 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 External Standards Calibration Curve • IMPORTANT: the error in the unknown concentration is minimized if it is centered in the calibration curve 0.9 y = 0.0056x - 0.0076 R2 = 0.9998 0.8 unknown signal (measured) 0.6 0.5 Uncertainty in Calc. x 1 0.4 0.3 unknown concentration (calculated) 0.2 0.1 0 0 50 100 150 Observed Y Absorbance 0.7 0.8 0.6 Data Uncertainty 0.4 0.2 0 200 0 Concentration 50 100 150 200 Calculated x • disadvantage - must repeat the entire procedure for every unknown sample Multiple Standard Additions Multiple Standard Additions • advantage - the sample and standards have the same matrix, so there is not systematic error due to a matrix mismatch step 1: step 2: step 3: 7 Multiple Standard Additions (graphing method #1) 1) measure signal of all solutions 2) plot signal vs. volume of standard added 3) find the and use x-intercept (gives the equivalent volume of the standard that represents the unknown) Rephrase: gives the # of mL’s of the standard that, if diluted to the volume of the original unknown, would give the concentration of the unknown (this method is the one discussed in your textbook) Multiple Standard Additions (graphing method #1) 1) measure signal of all solutions 2) plot signal vs. conc of diluted standard 3) find the and use x-intercept (gives the equivalent conc. of the unknown in the diluted volume) Internal Standard • A compound that the analyte can be compared to in quantitation • example: – you need to inject a sample into an instrument, but it is difficult to get the injection volumes from one injection to another to be exactly the same Internal Standard • Example (continued): – if, after processing, 50% of the internal std. was lost, then 50% of the analyte was also lost – thus, the real conc. of analyte is twice the measured conc. – variations in the injection volume lead to variations in the analyte signal – add a known amount of an I.S. before the injections – if the injection volume is large, then the I.S. signal is large (vice versa for smaller injections) 8