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CHEM2017
ANALYTICAL CHEMISTRY
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STATISTICAL TESTS STATISTICAL TESTS AND ERROR AND ERROR ANALYSISANALYSIS
PRECISION AND ACCURACYPRECISION AND ACCURACY
PRECISION – Reproducibility of the result
ACCURACY – Nearness to the “true” value
TESTING ACCURACY
TESTING PRECISION
SYSTEMATIC / DETERMINATE ERRORSYSTEMATIC / DETERMINATE ERROR
• Reproducible under the same conditions in the same experiment
• Can be detected and corrected for
• It is always positive or always negative
To detect a systematic error:
• Use Standard Reference Materials
• Run a blank sample
• Use different analytical methods
• Participate in “round robin” experiments (different labs and people running the same analysis)
RANDOM / INDETERMINATE ERRORRANDOM / INDETERMINATE ERROR
• Uncontrolled variables in the measurement
• Can be positive or negative
• Cannot be corrected for
• Random errors are independent of each other
Random errors can be reduced by:
• Better experiments (equipment, methodology, training of analyst)
• Large number of replicate samples
Random errors show Gaussian distribution for a large number of replicates
Can be described using statistical parameters
For a large number of experimental replicates the results approach an ideal smooth curve called the GAUSSIAN or NORMAL DISTRIBUTION CURVE
Characterised by:
The mean value – x
gives the center of the distribution
The standard deviation – s
measures the width of the distribution
The mean or average, x
the sum of the measured values (xi) divided by the number of measurements (n)
n
x
x
n
1ii_
The standard deviation, s
measures how closely the data are clustered about the mean (i.e. the precision of the data)
2
ii
1n
xx
s
NOTE: The quantity “n-1” = degrees of freedom
• Variance
• Relative standard deviation
• Percent RSD / coefficient of variation
x
sRSD
Other ways of expressing the precision of the data:
Variance = s2
100x
s%RSD
POPULATION DATAPOPULATION DATAFor an infinite set of data,
n → ∞ : x → µ and s → σ
population mean population std. dev.
The experiment that produces a small standard deviation is more precise .
Remember, greater precision does not imply greater accuracy.
Experimental results are commonly expressed in the form:
mean standard deviation
sx
_
The more times you measure, the more confident you are that your average value is approaching the “true” value.
The uncertainty decreases in proportion to n1/
EXAMPLE
Replicate results were obtained for the analysis of lead in blood. Calculate the mean and the standard deviation of this set of data.
Replicate [Pb] / ppb
1 752
2 756
3 752
4 751
5 760
Replicate [Pb] / ppb
1 752
2 756
3 752
4 751
5 760
n
xx i_
2i
1n
xxs
NB DON’T round a std dev. calc until the very end.
Also:
x
sRSD
100x
s%RSD
0.00500 754
3.77
0.500% 100754
3.77
Variance = s2 14.2 3.77 2
754x
3.77s 754 4 ppb Pb
The first decimal place of the standard deviation is the last significant figure of the average or mean.
Lead is readily absorbed through the gastro intestinal tract. In blood, 95% of the lead is in the red blood cells and 5% in the plasma. About 70-90% of the lead assimilated goes into the bones, then liver and kidneys. Lead readily replaces calcium in bones.
The symptoms of lead poisoning depend upon many factors, including the magnitude and duration of lead exposure (dose), chemical form (organic is more toxic than inorganic), the age of the individual (children and the unborn are more susceptible) and the overall state of health (Ca, Fe or Zn deficiency enhances the uptake of lead).
Pb – where from?• Motor vehicle emissions• Lead plumbing• Pewter• Lead-based paints• Weathering of Pb minerals
European Community Environmental Quality Directive – 50 g/L in drinking water
World Health Organisation – recommended tolerable intake of Pb per day for an adult – 430 g
Food stuffs < 2 mg/kg Pb
Next to highways 20-950 mg/kg Pb
Near battery works 34-600 mg/kg Pb
Metal processing sites 45-2714 mg/kg Pb
CONFIDENCE INTERVALSCONFIDENCE INTERVALS
n
tsxμ_
The confidence interval is given by:
where t is the value of student’s t taken from the table.
The confidence interval is the expression stating that the true mean, µ, is likely to lie within a certain distance from the measured mean, x.
– Student’s t test
A ‘t’ test is used to compare sets of measurements.
Usually 95% probability is good enough.
Example:
The mercury content in fish samples were determined as follows: 1.80, 1.58, 1.64, 1.49 ppm Hg. Calculate the 50% and 90% confidence intervals for the mercury content.
n
tsx_
μ
50% confidence:
t = 0.765 for n-1 = 3
4
0.1310.7651.63 μ
05.01.63 μ
There is a 50% chance that the true mean lies between 1.58 and 1.68 ppm
Find x = 1.63
s = 0.131
n
tsx_μ
90% confidence:
t = 2.353 for n-1 = 3
4
0.1312.3531.63 μ
15.01.63 μ
There is a 90% chance that the true mean lies between 1.48 and 1.78 ppm
x = 1.63
s = 0.131
1.63
1.68
1.48
1.58
1.78
90%
50%
Confidence intervals - experimental uncertainty
1) COMPARISON OF MEANS
ns
xvalueknowntcalc
Statistical tests are giving only probabilities. They do not relieve us of the responsibility of interpreting
our results!
Comparison of a measured result with a ‘known’ (standard) value
tcalc > ttable at 95% confidence level
results are considered to be different the difference is significant!
APPLYING STUDENT’S T:APPLYING STUDENT’S T:
For 2 sets of data with number of measurements n1 , n2 and means x1, x2 :
Where Spooled = pooled std dev. from both sets of data
2nn
1)(ns1)(nss
21
2221
21
pooled
21
21
pooled
21calc nn
nn
s
xxt
2) COMPARISON OF REPLICATE MEASUREMENTS
tcalc > ttable at 95% confidence level difference between results is significant.
Degrees of freedom = (n1 + n2 – 2)
Compare two sets of data when one sample has been measured many times in each data set.
3) COMPARISON OF INDIVIDUAL DIFFERENCES
e.g. use two different analytical methods, A and B, to make single measurements on several different samples.
ns
dt
dcalc
tcalc > ttable at 95% confidence level difference between results is significant.
1n
)d(ds
2i
d
Where
d = the average difference between methods A and B
n = number of pairs of data
Perform t test on individual differences between results:
Compare two sets of data when many samples have been measure only once in each data set.
Example:
(di)
Are the two methods used comparable?
1n
)d(ds
2i
d
16
04.002.011.011.022.002.0s
222222
d
12.0sd
ns
dt
dcalc
60.12
0.06tcalc
2.1tcalc
ttable = 2.571 for 95% confidence
tcalc < ttable
difference between results is NOT significant.
22
21
calcs
sF
Fcalc > Ftable at 95% confidence level
the std dev.’s are considered to be different the difference is significant.
F TEST
COMPARISON OF TWO STANDARD DEVIATIONS
Q TEST FOR BAD DATAQ TEST FOR BAD DATA
range
gapQcalc
The range is the total spread of the data.
The gap is the difference between the “bad” point and the nearest value.
Example:
12.2 12.4 12.5 12.6 12.9
Gap
Range
If Qcalc > Qtable discarded questionable point
EXAMPLE:
The following replicate analyses were obtained when standardising a solution: 0.1067M, 0.1071M, 0.1066M and 0.1050M. One value appears suspect. Determine if it can be ascribed to accidental error at the 90% confidence interval.
Arrange in increasing order:
Q = GapRange
