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Calculate the mean (x) and the relative standard deviation. (RSD). sample # ppm Fe. 1. 5.01. 2. 4.98. 3. 4.99. 4. 5.00.
Typology: Lecture notes
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vs. Accuracy , which indicates how close the measured analyte concentration is to the true analyte concentration in the sample.
or relative standard deviation , RSD
reference materials)
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Neither A, but not P A & P P, but not A
Results close together, but far from true value
Results close together and close to true value
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probability of obtaining the true value.
replicate measurements
Analytical Terms – Cont.
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Data handling and error analysis:
The results are back from the lab. Now what do we do?
Accuracy = closeness of result to the accepted or true value; measured with:
Precision refers to the reproducibility of results; measured with:
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Bias is a quantitative term describing the difference between the measured quantity (or its mean) and its true or known value.
Bias = measured value – known value
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where x = individual measurements; = mean of all measurements and n = total number of measurements
X
s RSD = X
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x = 5.00 4
Example : Determining the precision of a set of measurements
The following data were obtained for the determination of iron in a vitamin tablet using flame atomic absorption spectrometry. Calculate the mean (x) and the relative standard deviation (RSD).
sample # ppm Fe 1 5. 2 4. 3 4. 4 5. 5 5.
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The true or accepted value is not always available.
However, remember that the higher the precision, the greater is the probability of obtaining an accurate result
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Scenario:
The presence of dissolved copper in drinking
water is typically due to corrosion of household
plumbing systems. Its levels in drinking water is
regulated by the EPA because of its health effects,
such as gastrointestinal distress and liver or kidney
damage
The maximum contaminant level for
Cu is 1.3 ppm (or 1300 ppb)
(http://www.epa.gov/safewater/contaminants/index.html#mcls).
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The maximum contaminant level for Cu is 1.3 ppm. The following data were obtained from the analysis of water samples collected over a 5-day period.
Sample ID
Amount of Cu in mg/L Day 1 1. Day 2 1. Day 3 1. Day 4 1. Day 5 1.
Based on the 5-day mean copper l evels, is this water safe for drinking?
Mean ppm Cu = 1.
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Scenario: Let’s say the analysis of the same sample of water for Cu was repeated 100 times by atomic absorption spectroscopy. Let’s also limit errors to random error.
Question: How would the results look like?
Results will tend to cluster around the mean value for Cu levels
Gaussian distribution (Figure 4-1, p. 69) Increase repetitions, smoother curve
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Gaussian Distribution
Figure 4-1, p. 69. Bar graph and Gaussian curve for lifetimes of hypothetical light bulbs
Bell-shaped Normal distribution of variation in expt’l data
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Gaussian Distribution (Cont.)
In real life, we repeat experiments 3-4 times , not 100 times!
Small set of results
Estimate statistical parameters
Large set of results
Describes
Allows estimation of statistical behavior from a small number of repetitions (analysis) 21
Characteristics of the Gaussian Curve
(Smooth orange line)
Sum of measured values, xi, divided by the number of measurements, n
1. Mean, x (also called average )
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1
n i i
∑
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Gaussian distribution. Image available at http://introcs.cs.princeton.edu/java/11gaussian/
Gaussian curve and standard deviation – Cont.
Practical significance
Method A Method B
Comparing two methods (and two ) to measure % Fe in ore
s 0.8 1.
x 32.4 31.
Interpretation of results About 68 % of measurements from Method A will fall between 31.6 - 33.2 (vs. 30.7 - 32.9 from Method B)
95.5 % of measurements from Method A will fall between 30.8 - 34.0 (vs. 29.6 - 34.0 from Method B) 26
Parameters for Finite vs. Infinite Set of Data
Finite set Infinite set
Sample mean, x Standard deviation, s
Population mean, μ Population std. dev., σ
NOTE: μ and σ cannot be measured, but …
… as the number of measurements increases values of x and s approach μ and σ
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Degrees of freedom, n – 1
Used in the calc. of s
Squared std. dev.
Relative standard deviation, RSD (or the coefficient of variation ) Std. dev. expressed as percentage of the mean
Confidence interval
An expression which states that the true mean, μ, is likely to lie within a certain distance from the measured mean,
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n – 1 =
mean (^) 0.15 (0.148) stdev 0.03 (0.028) deg. freedom 5.00 sqrt (n) = 2. t at 90 % CI 2. t at 99 % CI 4.
CI (90% ) = 0.148 +/- (2.0150.028)/sqrt(6) = 0.148 +/- 0. = 0.15 +/- 0. CI (99% ) = 0.148 +/- (4.0320.028)/2. = 0.148 +/- 0. = 0.15 +/- 0.
95 % chance that the true mean lies within the range 0.12 % to 0.18 % (of additive) 99 % chance that the true mean lies within the range 0.10 % to 0.20 % (of additive) 32
Confidence Intervals -Cont.
A closer look at the meaning of confidence interval
50 % C.I: There is a 50 % chance that true mean, μ , lies between 12.4 to 12.7 % carbs
90 % C.I: There is a 90 % chance that true mean, μ , lies between 12.1 to 12.9 % carbs
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Allows for comparison of two sets of measurements (e.g. using 2 different methods) to decide whether or not they are the same
Recall the following results for measuring % Fe in ore using two different methods
Method A Method B
s 0.8 1.
x 32.4 31.
Are the results from the 2 methods different? (i.e. Are the two means different? )
Perform a t-test
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where
s 12 (n 1 –1) + s 22 (n 2 -1) n 1 + n 2 - 2
spooled = √
NOTE: spooled is a pooled standard deviation from both sets of data
Step 2: Compare tcalc with tabulated t (Table 4-2) for n 1 + n 2 – 2 degrees of freedom.
If tcalc is greater than ttable at the 95 % confidence level, the two results are considered to be different
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Case 2 – Cont.
Exercise: Determine if the two means from the measurements below are significantly different. Assume that each method consisted of 5 replicate measurements.
Method A Method B
s 0.8 1.
x 32.4 31.
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Can be performed with Excel (Case 2 only) – read Section 4-5, p. 82, “ t tests with a spreadsheet”
3. Comparing individual differences (Paired t Test)
Two different methods are used to make single measurements on several different samples (NOTE: There are no replicate measurements)
Answers the question: Is method A systematically different from method B?
Not as commonly used in chem labs as Case 2
For more information on Case 3, read p. 78
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Grubb’s Test for an Outlier
Outlier = a data point that is far from other points
Given twelve results for determining the mass % Zn in galvanized nail:
Outlier?
Q. Should the reading 7.8 be discarded? We can’t simply discard bad data But we can perform a Grubbs test to determine if an outlier can be discarded (^40)
Mass % Zn Mass % Zn 10.2 10. 10.8 9. 11.6 11. 9.9 9. 9.4 10. 7.8 11.
