ERRORS IN CHEMICAL ANALYSES

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Lecture 2

• ERRORS IN CHEMICAL ANALYSES

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• Measurements involve errors and uncertainties.
  Only a few of them are due to mistakes of
  experimenter. Mostly they are the result of
  faulty calibrations or standardizations or random
  variations and uncertainties in results.
• Frequent calibrations, standardizations, and
  analyses of known samples can be used to minimize
  the all but random errors and uncertainties.
• However, measurement errors are the inherit part
  of the quantized world. Thus, it is impossible to
  perform a chemical analysis that is totally free
  of errors or uncertainties.

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What should we do to estimate the quality of the
data we obtained?

• Experiments can be designed to disclose the
  existence of errors.
• Known standard samples can be analyzed and
  compared to know composition.
• Literature search may save you a great amount of
  time.
• Calibration of instruments usually enhances the
  quality of the data.
• Finally, statistics may serve you to estimate the
  quality of the data.

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• Because none of these techniques is perfect, we
  have to make judgments for the accuracy of our
  results.
• The first question to answer before beginning an
  analysis is What maximum error can I tolerate in
  the result.

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Some Important Terms

• The Mean and Median
• The mean
• The median is the middle result when replicate
  data are arranged according to increasing or
  decreasing

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Precision

• Decribes the reproducibility of measurements.
• Three terms are widely used to describe the
  precision of replicate data Standard deviation,
  variance, and coefficient of variation.
• These three are functions of how much an
  individual result xi differs from the mean, which
  is called the deviation from the mean, di.

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Accuracy

• Indicates the closeness of the measurement to the
  true or accepted value and is expressed by the
  error.
• Absolute Error
• Relative Error
• A more useful quantity than the absolute error.

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Accuracy vs Precision
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THE TYPES OF ERRORS IN EXPERIMENTAL DATA

• Random (or Intermediate Error) causes data to be
  scattered more or less symetrically around a mean
  value and they affect measurement precision.
• Systametic (or Determinate) Error casuses the
  mean of a data set to differ from the accepted
  value and affect accuracy of measurement
• Gross Error differs from both errors above. They
  occur occasionally, often large, may cause a
  result to be either high or low. They are usually
  product of human errors. Gross error leads to
  outliers.

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SYSTEMATIC ERRORS

• They have a definite value and assignable cause.
  They lead to bias in measurement results.
• Bias measures the systametic error associated
  with an analysis. It has a negative sign if it
  causes the results to be low and a positive sign
  otherwise.

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Sources of Systematic Errors

• Instrumental Errors
• Method Errors
• Personal Errors
• Systematic errors may be either constant or
  proportional.
• Constant error becomes more serious as the size
  of the quantity measured decreases.
• Detection of Systematic Errors
• Analysis of a standard sample
• Indipendent analysis
• Blank determination
• Variation in sample size

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CHAPTER 6

• RANDOM ERRORS IN CHEMICAL ANALYSIS

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The nature

• They can never be totally eliminated and often
  the major source of uncertainty in a
  determination

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Example
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Flipping coins
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(No Transcript)
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Statistical Treatment of Random Error

• We base statistical analyses on the assumption on
  random errors in analytical results follow a
  Gaussian distribution.
• Sample vs Population We try to get information
  about a population or universe from the
  observations made on a subset or sample.

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Sample Standard Deviation, s
The equation for s must be modified for small
samples of data, i.e. small N
Two differences cf. to equation for s
1. Use sample mean instead of population mean.
2. Use degrees of freedom, N - 1, instead of
N. Reason is that in working out the mean, the
sum of the differences from the mean must be
zero. If N - 1 values are known, the last value
is defined. Thus only N - 1 degrees of freedom.
For large values of N, used in calculating s, N
and N - 1 are effectively equal.
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Properties of Gaussian Curve

• The equation for a Gaussian curve has the form
• ? is population mean.
• ? population standard deviation
• sample mean
• s sample standard deviation

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SAMPLE
finite number of observations
total (infinite) number of observations
POPULATION
Properties of Gaussian curve defined in terms of
population. Then see where modifications needed
for small samples of data
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The population mean ? and the sample mean
Population Standard Deviation, ?

• When N reaches 20 to 30 measurements, the
  difference is negligible.
• is a statistic that estimates the population
  parameter ?.

The Population Standard Deviation (?)
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s measure of precision of a population of
data, given by
Where m population mean N is very large.
The equation for a Gaussian curve is defined in
terms of m and s, as follows
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z term

• The normal error curve can be obtained by
  defining a new variable, z. The quantity z
  represents the deviation from the population mean
  relative to the standard deviation.
• z is the deviation of a data point from the mean
  relative to one standard deviation. When x-??, z
  is 1 when x-?2?, z is 2, when x-?3?, z is 3
  and so on.
• ?2 is called variance.

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Two Gaussian curves with two different standard
deviations, sA and sB (2sA)
General Gaussian curve plotted in units of z,
where z (x - m)/s i.e. deviation from the
mean of a datum in units of standard deviation.
Plot can be used for data with given value of
mean, and any standard deviation.
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Area under a Gaussian Curve
From equation above, and illustrated by the
previous curves, 68.3 of the data lie within ??
of the mean (?), i.e. 68.3 of the area under
the curve lies between ?? of ?.
Similarly, 95.5 of the area lies between ???,
and 99.7 between ???.
There are 68.3 chances in 100 that for a single
datum the random error in the measurement will
not exceed ??. The chances are 95.5 in 100
that the error will not exceed ???.
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The Sample Standard Deviation A measure of
Precision

• Sample standard deviation is given by
• (N-1) is number of degrees of freedom. When N-1
  is used instead of N, s is said to be an unbiased
  estimator of the population standard deviation,
  ?.
• Faliure to use N-1 in calculating the standard
  deviation for small samples results in values of
  s that are, on average, smaller than the true
  standard deviation ?
• s2 is important in statistical calculations and
  is an estimation of ?2.

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Alternative Expression for s
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Standard Error of the Mean

• It shows the scatter of the mean as the
  measurement number, N, changes. For example, as N
  icreases deviation from the mean decreases.

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Percent Relative Standard Deviation and The
Coefficient of Variation (CV)
Relative Standard Deviation (RSD) Standard
deviations are used more frequently in relative
than absolute terms.

• When reative standard deviation is multiplied by
  100, it is called coefficient of variation (CV).
  CV is given by

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STANDARD DEVIATON OF CALCULATED RESULTS

• Standard Deviation of a Sum or Difference

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• Standard Deviation of a Product or Quotient

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CHAPTER 7STATISTICAL DATA TREATMENT AND
EVALUATION

• Again! Why do we need statistics?
• To sharpen our judgement on the quality of the
  measurement.
• Some of the common applications of statistical
  tests to the treatment of analytical results
• Defining Confidence Interval (CI)- the numerical
  interval around the mean with a certain
  probability.
• Determining the number of replicate measurements
  required to ensure that an experimental mean
  falls within a certain range with a given level
  of probability.

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• Estimating the probability that
• an experimental mean and a true value
• or two experimental means are different
• it means whether the difference is real or simply
  result of random error.
• Determing whether the precision of two sets of
  data-set differs at a given probability level.
• Comparing the means of more than two samples to
  determine whether differences in the means are
  real or the result of random error. (known as
  analysis of variance)
• Deciding to reject an data point as an outlier.

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CONFIDENCE INTERVALS

• An interval surronding an experimentally
  determined mean within which the population mean
  ? is expected to lie with acertain degree of
  probability.

A. Finding the Confidence Interval When ? is
Known or s is a good estimate of ?
The confidence level is the probability that true
mean lies within a certain interval. It is often
expressed as a percentage. The probability that
results is outside the confidence interval is
often called the significance level.
CI for ? x ? ?z
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Confidence Levels for Various Values of zTable
7.1
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• CI for ? x ? ?z
• We almost all the time make more than one
  measurement. Thus, in the equation above we use
  experimental mean, , of N measurements as a
  better estimate of ?. Consequently, we replace x
  in the equation above with and ? with
  standard error of mean, .
• CI for

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B. Finding the Confidence Interval When ? is
Unknown

• We use statistical parameter t (Students t),
  which is exactly the same way as z except that s
  is substituted for ?.
• For the mean of N measurements,
• CI for

Remember! N is equal to N-1 for a small set of
data.
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DETECTION OF GROSS ERROR

• The Q Test
• xq is questionable result, xn is its nearest
  neighbor, and w is the spread of data set. Qcrit
  values are given in the tables. If Q is greater
  than Qcrit, the questionable result can be
  rejected with the indicated degree of confidence.

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Comparision of Two Experimental Means

• To compare the results of two different methods,
  analysts, conditions e.g. there are two sets of
  tests N1 and N2. If t calculated is greater than
  t in tabulated one at the 95 confidence level,
  the two results are considered to be different.

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Comparison Precision (F-test )

• In many cases it is important to compare standard
  deviations of two sets of data.This could be
  one-tailed i.e. one direction (is Method A better
  than method B) or two-tailed (do methods A and B
  differ in their precision). The F-test considers
  the ratio of the two samples variances i.e. the
  ratio of the squares of the standard deviations.
  The quantity calculated (F) is given by (make
  sure F is always gt or 1)
• Ideally the variance ratio should be close to 1.
  This means that the population variances are
  almost equal. Differences from 1 occur because of
  random variations but if the difference is too
  great it can no longer be attributed to this
  cause.
• So if the calculated value of F exceeds a certain
  critical value (obtained from Tables) then the
  two sets of data are different. This critical
  value of F depends on the size of both the
  sample, the significance level and the type of
  test performed.

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Example 1

• pH values 6.59, 6.69, 6.73, 6.83, 6.90
• Calculated 6.75.
• s 0.12
• N (N -1 4) For a 95 confidence level, t
  2.78 so
• 6.75 ? 2.78 x0.12/(5)1/2 6.75 ? 0.15
• Means There is a 95 chance that the true pH
  value to fall between 6.90 and 6.60.

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Example 2

• A significance test tests whether the difference
  between two results significant or can be
  accounted for merely by random variations. Such
  tests are widely used in the evaluation of
  experimental results.
• ? 0.123
• x 0.116
• s 0.0032
• t ( 0.116 - 0.123 /0.0032)x41/2 4.38
• By checking the tables given, at 95 confidence
  level t - value is 3.18 (N -1 3). So, 4.38 gt t
  - value methods are not comparable indicates
  possible bias.

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Example 3
36.98 (), 35.56, 31.61, 34.69, 45.88, 36.37,
36.73,36.44, 37.16, 36.84 Rearrange in
increasing order 31.61, 34.69, 35.56, 36.37,
36.44, 36.73, 36.84, 36.98, 37.16, 45.88 If Q gt
Qcritic, the data must be rejected (see Table
for tabulated values of Qcritic)
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CHAPTER 8

• SAMPLING, STANDARDIZATION, AND CALIBRATION

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Analytical Samples and Methods

• Types of Samples and Methods
• Sample Size
• gt0.1 g Macro Analysis
• 0.01 to 0.1 g Semimicro Analysis
• 0.0001 to 0.01 g Micro Analysis
• lt10-4 g Ultramicro Analysis
• Constituent Types
• 1 to 100 Major
• 0.01 (100 ppm) to 1 Minor
• 1 ppb to 100 ppm Trace
• lt1 ppm Ultratrace

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Sampling and Sample Handling

• Automated Sample Handling
• Discrete Methods-Automated systems Sample
  injection to chromatographic systems etc.
• Continuous Flow Methods
• Flow Injection Analysis (FIA)
• Lab-on-a-Chip

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What is Lab-on-a-Chip?

• The lab-on-a chip is a microfluidic structure
  first built by Mike Ramsey 15 years ago to
  demonstrate the separation of chemicals in very
  small volumes.

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Why Lab-on-a-Chip?

• High-Throughput
• Cost
• Easy of use
• Minute amount of sample
• Integrated analysis
• DNA, protein, and other types of separations.
• For example, a lab-on-a-chip for immunological
  assays probably would integrate sample input,
  dilution, reaction, and separation, whereas one
  designed to map restriction enzyme fragments
  might have an enzymatic digestion chamber
  followed by a relatively long separation column.
• Microfabricated electrophoresis device at Oak
  Ridge National Laboratory. This "Lab-on-a-Chip"
  electrophoresis device allows mixtures of DNA or
  proteins to be separated at 1 of the time
  required by conventional capillary
  electrophoresis while using much less sample.

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STANDARDIZATION AND CALIBRATION

• Calibration determines the relationship between
  the analytical response and analyte
  concentration.
• Comparision with Standards
• External Standard Calibration
• An external standard is prepared separately from
  the sample.
• Calibration is accomplished by obtaining the
  response signal (absorbance, peak hiegth, peak
  area) as a function of the know analyte
  concentration.
• The Least-Squares Method

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The Least-Squares Method

• As is typical and usually desirable, the plot
  approximates a straight line. However, due to the
  indeterminate errors in the measurement process,
  not all the data fall exactly on the line.
• Thus, we must try to draw the best straight
  line among the data points. Regression analysis
  provides the means for objectively obtaining such
  a line and also for specifying the uncertainities
  associated.

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More...
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Assumption of the Least-Squares Method

• Two assumptions are made
• A linear relationship exists between the measured
  response y and the standard analyte concentration
  x.
• The mathematical relationship describing this
  assumtion is called the regression model
  represented as ymxb

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Assumption of the Least-Squares Method

• Any deviation of the individual points from the
  straight line arises from the error in the
  measurement. This means that there is no error in
  the x values of the points (concentrations).
• When there is significant uncertainty in the x
  data, wheighted least-squares analysis migth be
  necesseary.

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Minimizing Errors in Analytical Procedures

• Separations-sample clean up may minimize the
  error (fitration, precipitation, dialysis,
  solvent extraion, chromatographic separations
  etc.)
• Saturation, Matrix Modification, and Masking
• Saturation is the addition of the interference to
  the metrix that it becomes indipendent of the
  original concentration of interfering species.
• Masking is the selectively surpresing the
  interference by addition of a reactant reacts
  selectively with interfering species.
• Dilution and Matrix Macthing

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Minimizing Errors in Analytical Procedures

• Internal Standard Methods
• A known amount of a reference species is added to
  all samples, standards, and blanks. The response
  signal is not the analyte signal but the ratio of
  the analyte signal to the reference species
  signal.
• Standard Addition
• This method is used when it is difficult or
  impossible to dublicate the sample matrix.
• A know amount of standard solution of analyte is
  added to one portion of the sample. The responses
  before and after the addition are measured and
  used to obtain the analyte concentration.

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FIGURES OF MERIT FOR ANALYTICAL METHODS

• Sensitivity and Detection Limit(DL) or Limit of
  Detection (LOD)
• Calibration sensitivity is the change in the
  response signal per unit change in analyte
  concentration.
• The detection limit is the smallest concentration
  that can be reported with a certain level of
  confidence.
• sb is standard devian of the blank measurement.
• m is the calibration sensitivity.
• k is a factor, which is chosen 2 or 3.
• Linear Dynamic Range
• The concentration range that can be determined
  with a linear calibration range.