Statistical Treatment of Data 1 of

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• Statistical Treatment of Data
• Statistics will give us information on whether
  we can be confident about a result to a given
  level.
• Remember, an experiment is repeated due to the
  existence of errors.

Thus, there are no absolutes when reporting a
"determined value."
We can only reasonably repeat an experiment a
small number of times, not an infinite number of
times.
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• Question So, what information can we get from
  finite sampling?
• Answer Sample Parameters (vs. Population
  Parameters)
• Arithmetic Mean - (Average)
• n number of values
• xi individual values
• Standard Deviation - s is a measure of random
  error

variance s2 (c.f. Measurements and Errors)
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• So if s is small, the values are close to the ,
  and thus confidence of a measurement is high.
• Also, on the topic of sample parameters
• The value about which there are an equal number
  of points above and below, is referred to as the
  median.
• Range

(highest value - lowest)
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• Now, from where do we get the Gaussian
  Distribution or Normal Distribution?
• Example from Davies and Goldsmith,
• "Statistical Methods in Research and Production,"
• "Analysis of Carbon in a Powder"

If an infinite number of samples could be taken,
obtain a smooth curve.
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• The smooth curve is known as the
• Normal Distribution Curve or Gaussian Curve.
• It can be described by the equation
• s determines the breadth of the curve.
• s1lts2lts3lts4

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• If we express the abscissa in terms of s, life is
  easier.
• In all of the curves above and those we will work
  with, the total area under the curve will be
  unity (Probability of finding any value 1)

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• The area under the curve, or any portion, is
  directly related to the probability of finding a
  value between the defined limits.
• µ 1s
• µ 2s
• µ 3s
• (Note s (or s) is unique for a given data set)
• So, the probability of measuring z in a certain
  range is proportional to Area of that range.
• This allows us to assign some confidence to
  determined values

thus, Confidence Limits can be used for data.
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• Evaluation of Data
• Student's t - the degree of confidence that the
  true mean µ, is likely to fall within a
  particular interval of measured x otherwise
  known as Confidence Intervals.
• values of t are in Harris
• Let's do a Confidence Interval calculation
• data 10.19, 9.89, 9.98, 10.35, 10.41
• n 5 2.236 10.164
• s (10.19 - 10.16)2 (9.89 - 10.16)2 (9.98
  - 10.16)2 (10.35 - 10.16)2 (10.41 -
  10.16)2/(5-1)1/2
• s 0.226
• t at 95 (n 5) is 2.776
• µ 10.164 (2.776)(0.226)/(5)1/2
• µ 10.16 0.28 _at_ 95 confidence
• and at 99 µ 10.16 0.47

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• t-Test If two procedures are used to analyze for
  a particular quantity ( and ) and we
  want to see if the two means are different, use
• "t Test"
• pooled s

If tcalc gt ttab, the two results are
significantly different at the confidence level
in question.
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• t-Test Example Analysis of a solution of Cl-
  using two different methods
• Trial 1 2 3 4 5
• Method 1 (M) 0.2876 0.2871 0.2878 0.2880 0.2875
• Method 2 (M) 0.2843 0.2848 0.2839
• sp
• tcalc
• For n (degrees of freedom n1n2-2) 6 at 95
  confidence level,

0.2876 0.2843
3.9 x 10-4
(0.2876 - 0.2843)/(3.9 x 10-4)((5 x 3)/(5
3))1/2 12
ttab 2.447
Because tcalc gt ttab, the two methods are
significantly different at the 95 level.
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• Bad Data -- Get rid of it? The Q-test
• Question Is the datum (one point) inconsistent
  enough to neglect it in future calculations?
• Answer
• Qcalc gap/range
• (nearest point - bad point)/(high -
  low)
• If Qcalc gt Qtab, reject point. (at 90
  confidence limit)
• Example
• 5.96, 5.83, 5.89, 5.68, 5.85
• for n 5, Qtab(90) 0.64.

(Never apply any statistical test with removal to
fewer than 4 points (3 must remain))
The best rejection test is known as the Q- test
Qtab gt Qcalc, Keep the point.
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• Calibration of Instrument Responses with Analyte
  Properties Linear Least-Squares Regression
  Method
• Calibration Curves

• Make standard samples of known analyte amounts
• Make the amounts in each standard different
• Measure response of each standard sample
• Compensate for any Background responses
  (non-zero y-intercept)
• Plot response versus amount in each standard

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• Thus, for any response in the future (an
  unknown), we can obtain the property of that
  sample IFF we have a mathematical relationship.
• Linear Least-Squares Regression Method allows
  fabrication of a line through data points by
  minimizing the vertical (ordinate or y values)
  values of deviation between the points and the
  calculated line.

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• Linear Least-Squares Method

• based on minimizing the sum of the squares of the
  ordinate deviations (y-residual values)
• minimization of the residuals leads to equations
  for the best slope and intercept
• also, a litmus test number, the Correlation
  Coefficient - r can be calculated

r should be gt0.99 for a truly linear correlation
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• A Few Bits and Pieces
• You may have heard about the Limit of Detection
  and Sensitivity when it comes to Calibration
  Plots.
• SIGNAL Limit of Detection (SLOD) smallest
  amount of analyte giving a response significantly
  different from a blank or background response
• Sensitivity of Response Curve (calibration plot)
  the slope of the response curve
• Also, what is that thing called the Matrix?

ANALYTE Limit of Detection

• All of the other things in the sample
• Causes background responses
• Can alter the response of the analyte
• Can interfere with the analyte response