Propagation of Uncertainty

1
Propagation of Uncertainty A laboratory
example From the following data, calculate the
standard deviation in the molarity of NaOH
Final weighing -0.8348 ? 0.0001 g
Final reading 39.23 ? 0.02 mL Mass KHP Initial
weighing 0.0000 ? 0.0001 g Volume Initial
reading 0.27 ? 0.02 mL Mass
KHP 0.8348 ? ?
Vol NaOH 38.96 ? ? KHP NaOH NaKP
H2O
Note The largest contribution to the overall
uncertainty is the uncertainty in the volume
measurement
2
Statistics

• Uses of Statistics
• Determine the interval around the mean of a
  normally distributed set of replicate
  determinations within which the true mean is
  expected to be found with a certain probability
• Confidence interval or confidence limit
• Using the null hypothesis determine with some
  specified probability whether two means differ
• Is the difference between two means due to
  determinate or indeterminate error?
• Students t
• Determine whether two methods have a difference
  in precision
• Fisher F test
• How many replicates are required to assure a mean
  is within a certain interval around the true
  mean with a specified probability
• Students t
• Rejection of data
• Q test

3
Statistics

• Uses of Statistics
• Treatment of calibration data
• Determination of sensitivity of an analytical
  method
• Regression analysis including least squares
  regression
• Defining and establishing detection limits
• Minimum detectable quantity or concentration of
  analyte
• Students t
• Statistical terms
• Population all the objects in any set under
  investigation
• Real
• Potential
• Hypothetical
• Statistical sample a fraction of the population
• Many elements may comprise the statistical sample

4
Statistics

• Statistical terms
• Example
• Consider the problem of the determination of the
  concentration of Cl2 in a swimming pool
• The population is all the Cl2 in the swimming
  pool or all the 1.0 mL water replicates that can
  be taken from the swimming pool
• The statistical sample is the collection of
  replicates used for analysis
• The statistical sample contains a number of
  elements
• Chemists will call each element in the
  statistical sample an analytical sample
• Random sample a statistical sample extracted
  from the population in such a way that each
  component of the population has the same
  probability of being in the statistical sample
• The composition of chemical samples will often
  depend on how the replicates are extracted from
  the population
• Most statistical analyses depend on obtaining a
  random sample

5
Statistics

• Statistical terms
• Sample standard deviation from the mean
• Population standard deviation from the mean
• If N is small, is an estimate of m and one can
  only estimate s
• Given for a small set, only N-1 deviations
  from the mean are required to estimate s and,
  since deviations retain sign information, one of
  the deviations can be explicitly calculated
• Only N-1 deviations from the mean are required to
  give an independent measurement of s
• There is a negative bias in estimating s from a
  small set of data using N deviations from the
  mean
• See the pipet calibration data

6
Statistics
Table. Results of calculation fo s and s for
subsets of the pipet calibration data.
Conclusion A negative bias accompanies the
calculation of s for small sets of data. S give a
better estimate of s for small sets.
7
Statistics

• Properties of the normal error curve
• Examine Figure 3-4, FAC7, p 26
• Fig. A shows two frequency distributions where sB
  2sA both are plots of frequency vs.
  deviation from the mean, x-m
• Fig. B shows the two frequency distributions as
  plots of frequency vs. deviation from the mean
  divided by s or normalized to the standard
  deviation
• Both curves are superimposed
• The central point occurs at the mean or where the
  deviation from the mean is 0
• Zero deviation from the mean has the greatest
  probability or frequency
• The curves are symmetrical about the mean or zero
  deviation from the mean
• There is an exponential decrease in frequency as
  the deviations from the mean increase on
  either side of the maximum frequency

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Statistics
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Statistics

• Properties of the normal error curve
• Areas under the normal error curve between
  intervals defined in terms of the standard
  deviation give the probability that a single
  measurement may occur
• Within the interval -1s and 1s, area 68.3
• Within the interval -2s and 2s, area 95.5
• Within the interval -3s and 3s, area 99.7
• As N increases s becomes a better estimate of s
• Estimate s from s for a set of 20 or so
  replicates if the work is not too time consuming
• Pool the results of the analysis of several
  analyses for the same analyte in similar
  populations to obtain a good estimate of s

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Statistics
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Statistics
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Statistics
13
Statistics
Properties of the normal error curve Example
calculate a pooled estimate of s from the
following data Samples from the top and bottom
of the contents of a weighing bottle were
analyzed
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Statistics

• Confidence limits and confidence intervals
• Statistics allows the determination of a range
  about a measured mean within which the true
  mean may be found with some level of probability
• Confidence limit
• The calculation of the confidence interval
  depends on how well s and the number of
  elements in the statistical sample for which s
  and are determined
• If there is a good estimate of s, then the
  confidence limit for m is
• The value chosen for z determines the area under
  the normalized normal distribution curve
  between -z and z
• The value used for z gives the probability
  associated with the confidence interval
• If s is used to estimate s, i.e., there is a
  small set of data
• t depends on N and the level of probability as n
  ?, t z see Table 3-2

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Statistics
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Statistics
Confidence limits and confidence
intervals Example Calculate the confidence limit
for the data from the determination of the
analyte in the top of the weighing bottle
Examine example 4-3, FAC7, p 51