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Lecture 4

CHMB16Fall2012 Lecture 4 Notes.docx

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University of Toronto Scarborough
Kagan Kerman

CHMB16Fall2012 Lecture 4: Statistics I – Chapter 4 Evaluation of Analytical Data Example of s pooledsed to improve s 2" o o o since N<30, s pooledmuch more closely approximates the population standard deviation ( ) Statistical tests: o sometimes outliers resulting from Gross errors exist in a data set o when analyzing data the scientist must decide whether to use or discard suspect points o several statistical tests have been developed to decide this question o most commonly used is the Q-test 3" o these tests assume the data exhibits a Gaussian distribution  this assumption cannot be proven for small data sets (< 50 data points) Detection of Gross Errors: Q test o the quantity Q calc(the rejection quotient) is calculated as: o o where x = questionable result, x = nearest neighbor to the questionable result, w is the spread of q n the entire set o this value is then compared with a rejection value Q , foucritn the Q table 1 5" o o if Q calc Qcrithen the questionable result may be rejected at the indicated confidence level (i.e it is an outlier) Example of using Q-test in problem 6" Solution: Qcalc = (0.1035 – 0.1015)/ (0.1035-0.1012) Qcalc = 0.87 From Q table, at 90% confidence and where N = 4, Qcrit = 0.76 Since Qcalc > Qcrit, the data point can be discarded. Statistical tests: o an important question that often needs answering in science is whether a numerical difference between experimental values is real or the result of random error o easiest way to test which is true is to make a null hypothesis o an appropriate statistical test is then applied to determine if the null hypothesis fails o if it fails, the numerical difference is real, if it holds the numerical difference is the result of random error Grubb’s test for an outlier Gcalc = (questionable value – average) / s Student t test o the student t value used in determining confidence intervals may also be used to evaluate the null hypothesis 2 o there are 2 main situations where it may be used: to compare an experimentally determined mean with the true mean and to compare 2 experimentally determined means o generally, if the numbers are not within each other’s 95% confidence intervals, the null hypothesis is questionable and the differences are significant Comparing an Experimental mean with a known value : t test For Small Data sets (where s is not a good estimate of ) o select a critical value of t at the confidence level you require from the t-table  t crit o calculate t using: o o if tcalctcrithen the means are not considered identical at the conf
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