CHM 302

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CHM 302

• Chapter 2
• Error Analysis

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• In all analytical measurements there is the
• possibility of making mistakes

• These mistakes are referred to as sources of
• error.
• Results of analytical measurements are usually
• the arithmetic mean of a series of replicate
• measurements

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Precision and accuracy

• Precision
• Precision describes the reproducibility of
  results
• -the agreement between numerical values
• for two or more replicate measurements
• or measurements that have been made in
• exactly the same manner
• Precision easily determined by simply repeating
• the measurement

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• Three terms figures of merit are used
• to describe precision

• - standard deviation, variance, and
• coefficient of variation
• Accuracy
• Accuracy describes the correctness of an
• experimental measurement
• Is a relative term

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• Accuracy is expressed in terms of either
• absolute error or relative error

• Absolute error Ea

• Relative error Ea/Xt
• Note that both absolute and relative errors
• bear a sign

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Random Errors

• Random errors produce values that are
• sometimes too high or too low
• The presence of random errors is reflected in
• the precision of the data
• In chemical analyses replicate measurements
• are usually distributed in an approximately
• Gaussian or normal form

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• Gaussian characteristics
• The most frequently observed result is
• the mean of the set of data
• 2. The results cluster symmetrically around
• this mean value
• 3. Small divergences (from the mean) are
• more common than large ones
• 4. In the absence of systematic errors, the
• mean of a large set of data approaches the
• true value

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• Random errors are usually treated with
• statistics
• Precision is usually indicated by the
• standard deviation, s

• The relative standard deviation
• is given by

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• This is usually expressed as a percentage
• called coefficient of variation, CV
• CV RSD x 100

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Confidence Limits

• The true or population mean (?) of a
• measurement must always be unknown
• In the absence of systematic errors, limits
• can be set within which the population mean
• can be expected to lie.
• These limits are called confidence limits
• Statistical measure calculated from the sample
• standard deviation

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• In considering CLs we need
• ? ? standard deviation of single
• measurement, and
• ?m ? standard deviation of mean

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• Note ?m ?/vN
• The most common confidence interval is
• stated at 95 confidence limit

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• When ? is unknown
• - usually the case for chemical analyses
• In this case, use is made of the statistical
• parameter, t, called the students t

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Systematic Errors

• Systematic (determinate) errors are
• reflected in the accuracy of the measurement
• Systematic errors
• - have a definite value
• - have an assignable cause
• - are of the same sign magnitude for
• replicate measurements

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• Three types of systematic errors
• Instrumental errors
• - electronic drift
• - calibration errors
• - decreases in voltage of batteries
• 2. Personal errors introduced by judgments of
• the experimenter
• 3. Method errors introduced from non-ideal
• chemical and physical behavior of reagents

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• Method errors can be detected and
• corrected by validation
• - using the method for analysis of
• standard materials
• Standards must have known content
• Instrument errors are handled by calibration
• with suitable standards
• Calibration requires the use of a blank a more
• general term is control

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Pooling Data
To pool data one requires (i) the mean, X (ii)
the standard deviation, s (iii) The number of
determinations, N for all sets of data to be
pooled
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Propagation of Uncertainty

• The process of combining errors from
• different measurements is called propagation
• of uncertainty
• Addition of subtraction x pq-r
• sx uncertainty in x

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• For multiplication and division x p?q/r

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Discordant data

• Sometimes referred to as outliers
• The Dixon or Q-test is a statistical measure
• of which data point(s) may be considered an
• outlier