Chemistry 155 Introduction to Instrumental Analytical Chemistry

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Chem 155 Unit 1 Page 14 of 3131.4 Vocabulary: Basic Statistics ReviewPrecisionRandom ErrorAccuracySystematic Error(bias)HistogramProbabilityDistributionIf repeated measurements of the same thing areall very close to one another, then ameasurement is precise. Note that precisemeasurements may not be accurate (see below).Random error is a measure of precision.Random differences between sequentialmeasurements reflect the random error.If a measurement of something is correct, i.e.close to the true value, then that measurement isaccurate.Systematic error is the difference between themean of a population of measurements and thetrue value.A graph of the number or frequency ofoccurences of a certain measurement versus themeasurement value.A theoretical curve of the probability of a certainmeasured value occurring versus the measuredvalue.A histogram of a set of data will often look like a Gaussian probability distribution.AverageThe sum of the measured .xvalues divided by the numbernnof measurements.MedianVariance (σ 2 )Standard Deviation(σ)Relative StandardDeviationPropagaion ofErrorx MEANHalf of the measurements fall above the medianvalue, and half fall below.A measurement ofprecision. The sum of thesquares of the randommeasurement errors.n 1A widely accepted‘standard’ measurement ofprecision. The square root nσof the variance.n 1The standard deviation divided by the mean, andoften expressed as : %RSD=σ/x MEAN 100%.σ 2 n x MEAN x n2When the mean of a set of measurement (x) hasa random error (σ x ), it is reported as x±σ x . If wewish to report the result of a calculation y=f(x)based on x, we propagate the error through thecalculation using a mathematical method.nx MEAN x n2Page 14 of 313
Chem 155 Unit 1 Page 15 of 3131.5 Statistics Review1.5.1 Precision and AccuracyHistogram of normally distributed events80MeanNumber of times it was observed604020Mean -onestandarddeviationMean +onestandarddeviation00 1 2 3 4 5 6Value observed1.5.2 Basic FormulaeHistogram of 1024 eventsMean : μlimN → ∞N∑ x NNPopulationStandardDeviation:σ xlimN → ∞x N − μ∑ ( )2NNAverage:x avg∑ x NNNSampleStandardDeviation:s xx N − x∑ ( avg )2NN−1Bias or absolute systematic error = x avg − μRelative standard deviation =sx avgPage 15 of 313
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Chemistry 155 Introduction to Instrumental Analytical Chemistry

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