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