chap6 random
TRANSCRIPT
Random Errors in Chemical
Analysis
A. Nature of Random Errors• Uncontrollable variables are the source ofrandom errors• Contributors to random errors are not all– identifiable– individually detectable– quantifiable• The combined effect of random errors producethe fluctuation of replicate measurementsaround the mean• Random errors are the major source of
uncertainty
Distribution of Random ErrorsAssume four contributors to the random error of equal magnitude.Equal probability of occurrence of negative and positive deviation.Each can cause the final result to be high or low by ±U
1:4:6:4:1
Frequency of Occurrence and Probability
• The frequency of a deviation of a given
magnitude is ameasure of theprobability ofoccurrence of that
deviation
37.5%
25.0%25.0%
6.25%6.25%
• For ten equal sizeuncertainty
Gaussian Curve or Normal Error Curve• For a very
largenumber ofindividual
errors
Replicate Data on the Calibration of a 10mL Pipet
Generating a histogramFrequency within ranges
Distribution of the Experimental Errors Approachesa Gaussian Curve
A - Bar Graph or Histogram B – Gaussian Curve
Sources of Random Uncertainties
1. Visual Judgments2. Variations in the drainage time and in
the angle of the pipet as it drains3. Temperature fluctuations4. Vibration and drafts that cause small
variations in the balance readings.
Statistical Treatment of RandomErrors
• Distribution of the majority of analytical datadisplays characteristics of the normal
distribution• Therefore, Gaussian distribution is used toapproximate distribution of analytical data• Available standard statistical methods are
usedto evaluate analytical data assuming randomdistribution of errors
Terminology• Population: all possible observations/measurements/a universe of data• Types of population– Finite and real (lot of steel, a lot of Advil Tablets)– Hypothetical or conceptual (Calcium in blood,
lead inlake Ontario).• A sample of the population is analyzedSample: subset of the population• Results from the analysis are used to infer thecharacteristics of the population
Properties of Gaussian Curves
• The gaussian curve is
fully characterized bytwo parameters– the mean:μ– the standarddeviation:σ
• Population mean (μ) and
Standard Deviation(σ)
Universal Gaussian Curve
• Abscissa: deviation from the mean in
units of standard deviation
Properties of a normal error curve
• Mean occurs a the central point of maximum
frequency• Symmetrical distribution of positive and
negativedeviations• Exponential decrease in frequency as themagnitude of the deviations increases
The Sample Standard Deviation• Number of degrees offreedom: number ofindependent resultsneeded to compute thestandard deviation• As N approaches
infinity,s approaches σ andapproaches μ
ExampleThe following results were obtained in the
replicate analysis of a blood sample for its lead content: 0.752, 0.756, 0.752, 0.751, and 0.760 ppm Pb. Calculate the mean and the standard deviation of this set of data.
Sample Xi X2 0.752 0.565504 0.756 0.571536 0.752 0.565504 0.751 0.564001 0.760 0.5776sumXi = 3.771 2.844145
mean = 0.7542ppm Pb
S = 0.003768
Assignment
1. Consider the sets of replicate. Calculate: a. mean b. median c. spread d. standard deviation
a. 2.4, 2.1, 2.1, 2.3, 1.5b. 69.94, 69.92, 69.80c. 0.0902, 0.0884, 0.0886, 0.1000
Standard Error of the Mean
• The standarddeviation of the
mean= standard error
ofthe mean
Pooled Standard Deviation
Relative Standard Deviation
Significant Figures• All certain digits plus one uncertain digit• Rules– All initial zeros are not significant– All final zeros are not significant, unless theyfollow a decimal point– Zeros between nonzero digits are significant– All remaining digits are significant• Use scientific notation to exclude zerosthat are not significant
Significant figures in NumericalComputations