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Lecture 4

# CHMB16Fall2012 Lecture 4 Notes.docx

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University of Toronto Scarborough

Chemistry

CHMB16H3

Kagan Kerman

Fall

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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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