1. STATISTICAL TESTS AND ERROR ANALYSIS

3. PRECISION AND ACCURACY
PRECISION – Reproducibility of the result
ACCURACY – Nearness to the “true” value

5. Uncertainty in every experiment (measurement)
How sure are you that the experimentally obtained value is close to the “true” value? How close is it?
Finding errors
Experimental error
Uncertainty in every experiment (measurement)

6. SYSTEMATIC / 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)

7. RANDOM / 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

8. 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

9. The mean or average, x
 the sum of the measured values (xi) divided by the number of measurements (n)
The standard deviation, s
 measures how closely the data are clustered about the mean (i.e. the precision of the data)
NOTE: The quantity “n-1” = degrees of freedom

10. Other ways of expressing the precision of the data:
Variance
Variance = s2
Relative standard deviation
Percent RSD / coefficient of variation

11. POPULATION DATA For 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

12. The Gaussian curve equation:
= Normalisation factor
It guarantees that the area under the curve is unity.
Probability of measuring a value in a certain range = area below the graph of that range
The Gaussian curve whose area is unity is called a normal error curve.
µ = 0 and σ = 1

13. The standard deviation measures the width of the Gaussian curve.
(The larger the value of σ, the broader the curve)
Range Percentage of measurements
µ ± 1σ
µ ± 2σ
µ ± 3σ
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

14. 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 4
751 5
760

15. Replicate
[Pb] / ppb 1
752 2
756 3 4
751 5
760
NB DON’T round a std dev. calc until the very end.

16. 754  4 ppb Pb
Also:
Variance = s2

17. Motor vehicle emissions Lead plumbing Pewter Lead-based paints
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).
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
Pb – where from?
Motor vehicle emissions
Lead plumbing
Pewter
Lead-based paints
Weathering of Pb minerals
Food stuffs < 2 mg/kg Pb
Next to highways mg/kg Pb
Near battery works mg/kg Pb
Metal processing sites mg/kg Pb

18. CONFIDENCE INTERVALS
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
The confidence interval is given by:
where t is the value of student’s t taken from the table.

19. A ‘t’ test is used to compare sets of measurements.
Usually 95% probability is good enough.

20. 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.
Find x = 1.63
s = 0.131
50% confidence:
t = for n-1 =
There is a 50% chance that the true mean lies between 1.58 and 1.68 ppm

21. x = 1.63 s = 0.131 90% confidence: t = for n-1 = 90% 50%
1.68
1.48
1.58
1.78
90%
50%
90% confidence:
t = for n-1 =
There is a 90% chance that the true mean lies between 1.48 and 1.78 ppm

22. Confidence intervals - experimental uncertainty

23. APPLYING STUDENT’S T: 1) COMPARISON OF MEANS
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!
Statistical tests are giving only probabilities.
They do not relieve us of the responsibility of interpreting
our results!

24. 2) COMPARISON OF REPLICATE MEASUREMENTS
For 2 sets of data with number of measurements n1 , n2 and means x1, x2 :
One sample, many measurements
Where Spooled = pooled std dev. from both sets of data
tcalc > ttable at 95% confidence level  difference between results is significant.
Degrees of freedom = (n1 + n2 – 2)

25. 3) COMPARISON OF INDIVIDUAL DIFFERENCES
Use two different analytical methods, A and B, to make single measurements on several different samples.
Perform t test on individual differences between results:
Many samples, one measurement
Where
Where d = the average difference between methods A and B
n = number of pairs of data
tcalc > ttable at 95% confidence level  difference between results is significant.

26. Are the two methods used comparable?
Example:
(di)
Are the two methods used comparable?

28. F TEST COMPARISON OF TWO STANDARD DEVIATIONS
Fcalc > Ftable at 95% confidence level
 the std dev.’s are considered to be different  the difference is significant.

30. Q TEST FOR BAD DATA The range is the total spread of the data.
The gap is the difference between the “bad” point and the nearest value.
Example:
Gap
Range
If Qcalc > Qtable  discarded questionable point

31. Arrange in increasing order: 0.1050M 0.1066M 0.1067M 0.1071M
EXAMPLE:
The following replicate analyses were obtained when standardising a solution: M, M, M and M. One value appears suspect. Determine if it can be ascribed to accidental error at the 90% confidence interval.
Arrange in increasing order:
0.1050M M M M
Q =
Gap
Range =
BUT these values are very close  rather do another analysis to confirm!!!

32. ANALYTICAL VARIANCE + SAMPLING VARIANCE
STATISTICS OF SAMPLING
A chemical analysis can only be as meaningful as the sample!
Sampling – process of collecting a representative sample for analysis
OVERALL VARIANCE =
ANALYTICAL VARIANCE + SAMPLING VARIANCE

33. Where does the sampling variance come from?
Consider a powder mixture containing nA particles of type A and nB particles type B.
Probability of drawing A: p =
Probability of drawing B: q = nA
nA+ nB nB
= 1 - p
If n particles are randomly drawn, the expected number of A particles will be np
and standard deviation of many drawings will be:

34. How much of the sample should be analysed?
Std dev.
Where p, q – fractions of each kind of particles present
Relative
Std Dev.
Relative
Variance
 nR2 = pq
The mass of sample (m) is proportional to number of particles (n) drawn, therefore:
Ks = mR2
Where R = RSD as a % and Ks (sampling constant) = mass of sample required to reduce the relative sampling standard deviation to 1%

35. How many samples/replicates to analyse?
Rearranging Student’s t equation:
Required number of
replicate analyses: e
µ = true population mean
x = measured mean
n = number of samples needed
ss2 = variance of the sampling operation
e = sought-for uncertainty
Since degrees of freedom is not known at this stage, the value of t for n → ∞ is used to estimate n.
The process is then repeated a few times until a constant value for n is found.

36. Example:
In analysing a lot with random sample variation, there is a sampling deviation of 5%. Assuming negligible error in the analytical procedure, how many samples must be analysed to give 90% confidence that the error in the mean is within 4% of the true value?
For 90% confidence:
t =
n = 6

38. SAMPLE STORAGE
Not only is the sampling and sample preparation important, but the sample storage is also critical.
The composition of the sample may change with time due to, for example, the following:
reaction with air
reaction with light
absorption of moisture
interaction with the container
Glass is a notorious ion exchanger which can alter the concentration of trace ions in solution.
Thus plastic (especially Teflon) containers are frequently used.
Ensure all containers are clean to prevent contamination.

39. EXAMPLE: (for you to do)
Consider a random mixture containing 4.00 g of Na2CO3 ( = g/ml) and g of K2CO3 ( = g/ml) with an approximated uniform spherical radius of mm.
How many particles of Na2CO3 are in the mixture? And K2CO3?
Na2CO3: g at g/ml
V = m 
= ml = 1.58 cm3
K2CO3: g at g/ml
V = m 

40. VNa2CO3 = 1.58 cm3
VK2CO3 = cm3
Particles: r = mm = cm
nNa2CO3 =
8.94x105 particles
nK2CO3 =
2.24x107 particles

41. What is the expected number of particles in 0.100 g of the mixture?
EXAMPLE:
Consider a random mixture containing 4.00 g of Na2CO3 ( = g/ml) and g of K2CO3 ( = g/ml) with an approximated uniform spherical radius of mm.
What is the expected number of particles in g of the mixture?
8.94x102 particles of Na2CO3 and
2.24x104 particles of K2CO3
in a 0.1 g sample

42. EXAMPLE:
Calculate the relative standard deviation in the number of particles for each type in the g sample of the mixture.