Chemistry 155 Introduction to Instrumental Analytical Chemistry

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Chem 155 Unit 1 Page 16 of 313When you make real measurements of things you generally don’tknow the ‘true’ value of the thing that you are measuring. (Call thisthe true mean, μ, for now. For the purposes of this discussion let usassume that there is no systematic (accuracy) error (i.e. no bias).1.5.3 Confidence IntervalWHAT DO YOU DO TO ENSURE THAT YOUR ANSWER IS ASCLOSE AS POSSIBLE TO THE TRUTH?TAKE THE AVERAGEx AVGBut, you still don’t know the exact answer…SO WHAT DO YOU REALLY WANT TO SAY?I am highly confident that the true mean lies withinthis interval (e.g between 92 and 94 grams).In fact, there is only a 1 in 20 chance that I am wrong!How do you calculate what that interval is?You need to know:The average of the data set: xThe standard deviation: σ or sThe number of measurements (observations) made: NThis interval is called a confidence interval (CI). Which is better,abigger or a smaller CI?Smaller is better…How can you improve your CI?Make more measurements (N)Page 16 of 313
Chem 155 Unit 1 Page 17 of 313Confidence Interval (continued).If you know the standard deviation, σ, (less common case), then:The x% confidence interval for μ = x AVG ± zσ / N ½If you don’t know the standarddeviation, σ, (more commonly thecase), then:If you have only a roughestimate of x AVG , then you areless confident that it is close toμ, hence you divide by N ½ .The x% confidence interval for μ = x AVG ± ts / N ½In this case t is a function of NThis leaves only z and t – what are they? These numbers representthe multiple of one standard deviation (σ or s) that correspond to theconfidence interval. In the second case, s is only an estimate of σ, sothe error in s needs to be taken into account, so t is a function of the“number of degrees of freedom”. For our purposes, i.e. averagingmultiple identical measurements, the number of degrees of freedomis simply N-1.Page 17 of 313
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Chemistry 155 Introduction to Instrumental Analytical Chemistry

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