4- Data Analysis and Presentation
Statistics
CHAPTER 04: Opener
What is in this chapter?
1) Uncertainty in measurements: large N
distributions and related parameters and
concepts (Gaussian or normal distribution)
2) Approximations for smaller N (Student’s tand related concepts)
3) Other methods: G, Q (FYI: F)
4) Excel examples (spreadsheet)
What do we want from
measurements in chemical analysis?
• We want enough precision and accuracy to
give us certain (not uncertain) answers to
the specific questions we formulated at the
beginning of each chemical analysis. We
want small error, small uncertainty. Here
we answer the question how to measure it!
• READ: red blood cells count again!
Distribution of results from:
• Measurements of the same sample from
different aliquots
• Measurements of similar samples (expected
to be similar/same because of the same
process of generation)
• Measurements of samples on different
instruments
• ETC
FYI: Formalism and mathematical
description of distributions
• Counting of black and white objects in a sample,
or how many times will n black balls show up in a
sample  Binomial distribution
• For larger numbers of objects with low freqency
(probability) in the sample
Poisson distribution
and if the number of samples goes to infinity
Normal or Gaussian distribution
Normal or Gaussian distribution
• Unlimited, infinite number of measurements
• Large number of measurements
• Approximation: small number of
measurements
CHAPTER 04: Figure 4.1
4-1 IMPORTANT: Normal or Gaussian distribution
Data from many measurements of the same
object or many measurements of similar objects
show this type of distribution. This figure is the
frequency of light bulb lifetimes for a particular
brand. Over four hundred were tested
(sampled) and the mean bulb life is 845.2
hours. This is similar but not the same as
measurement of one bulb many times in similar
conditions! See also Fig 4.2
F ( x)  
1
2
e
(x μ)2

2σ 2
Find “sigma” and “mu” on
the Gaussian distribution
figure !!!!!!!!!!!!!!!!!!!!!!!!
CHAPTER 04: Equation 4.3
IMPORTANT
M

CHAPTER 04: Figure 4.3
Here is a normal or Gaussian distribution determined by two
parameters m, here =0, and , here a) 5, b) 10, c) 20. Wide
distributions such as ( c) are result of poor or low precision
.The distribution (a) has a narrow distribution of values, so it
is very precise.
Q: How to quantify
the width as
a measure of
precision?
A: “sigma” and
“s” standard
deviation
Another
example
with
data
Another way to get close to Gaussian distribution is to measure a lot of
data
Properties of the Gaussian or Normal Distribution or
Normal Error Curve
1. Maximum frequency (number of measurements with
same value) at zero
2. Positive and negative errors occur at the same
frequency (curve is symmetric)
3. Exponential decrease in frequency as the magnitude
of the error increases.
The interpretation of Normal distribution, Standard Deviation
and Probability: the area under the curve is proportional to the
probability you will find that value in your measurement. Clearly, we
can see form our examples that the probability of measuring value x
from a certain range of values is proportional to the area under the
normalization curve for that range.
Range
Gaussian Distribution
µ ± 1
68.3%
µ ± 2
95.5%
µ ± 3
99.7%
The uncertainty decreases in proportion to 1/(n)^.5, where n is the number of measurements.
The more times you measure a quantity, the more confident you can be that the
average value of your measurements is close to the true population mean, µ.
Standard deviation here is a parameter of Gaussian curve.
We can now say with certain confidence that the
value we are measuring will be inside certain
range with some well-defined probability.
This is what can help us in quantitative analysis!
BUT, can we effort measurements of large, almost infinite number
of samples? Or repeat measurement of one sample almost infinite
number of times???
We will introduce something that can be measured with
smaller number of samples, X, and s instead…….
As n gets smaller (<=5) µ  mean X and   s
This is the world we are in, not infinite number of
measurements !!!!!!!
All our chemical analysis calculations starts here
from these “approximations” of Gaussian or
Normal distributions: mean and standard
deviation
Mean value and Standard deviation
–
x =
 xi
i
n
Examples-spreadsheet
 (xi–x–)2
s=
i
n-1
Also interesting are :
median (same number of points above and below ,
range ( or span, from the maximum to the minimum
CHAPTER 04: Equation 4.1
CHAPTER 04: Equation 4.2
Example
For the following data set, calculate the mean and standard deviation.
Replicate measurements from the Calibration of a 10-mL pipette
Trial
Volume delivered
1
9.990
2
9.993
3
9.973
4
9.980
5
9.982
– ) = (9.990 + 9.993 + 9.973 + 9.980 + 9.982) = 9.984
Mean (x
5
Standard Deviation (s) =
(9.990–9.984)2 + (9.993–9.984)2 + (9.973–9.984)2 + (9.980–9.984)2 + (9.982–9.984)2
5–1
s = 8 x 10–3
CHAPTER 04: Unnumbered Table 4.1
THE TRICK: Student's t (conversion
to a small number of measurements, by
fitting )
Student's t Table .
normal
Degree of freedom = n-1
t
x
Shown above are the curves for the t
distribution and a normal distribution
Confidence level(%)
Degrees of freedom
90%
95%
1
6.314
12.706
2
2.920
4.303
3
2.353
3.182
4
2.132
2.776
5
2.015
2.571
6
1.943
2.447
7
1.895
2.365
8
1.860
2.306
9
1.833
2.262
10
1.812
2.228
15
1.753
2.131
20
1.725
2.086
25
1.708
2.068
30
1.697
2.042
40
1.684
2.021
60
1.671
2.000
120
1.658
1.980

1.645
1.960
Student's t table, see
Table 4-2 book and
handouts
CHAPTER 04: Equation 4.4
CHAPTER 04: Figure 4.2
Link: Can we also use parameters similar to normal distribution to
characterize certainties and uncertainties of our measurements?
2=25
2=100
2=400
The square of the standard deviation is called the variance (s2) or 2
s
The standard deviation (or error) of the mean =
n
Typically use small # of trials, so we never measure µ or 
The standard deviation, s, measures how closely the data are clustered about the mean. The
smaller the standard deviation, the more closely the data are clustered about the mean.
The degrees of freedom of a system are given by the quantity n–1.
THE TRICK: Student's t (conversion to a
small number of measurements, by fitting )
normal
t
Shown above are the curves for the t distribution and a normal distribution.
x
The confidence interval is an expression stating that the true mean, µ, is
likely to lie within a certain distance from the measured mean, x-bar.
Confidence interval:
ts
–
µ=x ±
n
where s is the measured standard deviation, n is the number of observations, and t is the
Student's t Table .
Degree of freedom = n-1
4.2 Confidence interval
• Calculating CI
CI for a range of values will show the
probability at certain level (say 90%) that
you have the true value in that range.
Note : true value .
Confidence level(%)
Degrees of freedom
90%
95%
1
6.314
12.706
2
2.920
4.303
3
2.353
3.182
4
2.132
2.776
5
2.015
2.571
6
1.943
2.447
7
1.895
2.365
8
1.860
2.306
9
1.833
2.262
10
1.812
2.228
15
1.753
2.131
20
1.725
2.086
25
1.708
2.068
30
1.697
2.042
40
1.684
2.021
60
1.671
2.000
120
1.658
1.980

1.645
1.960
Student's t table, see
Table 4-2 book and
handouts
Example
A chemist obtained the following data for the alcohol content in a sample
of blood: percent ethanol = 0.084, 0.089, and 0.079. Calculate the
confidence interval 95% confidence limit for the mean.
0.084 + 0.089 + 0.079
–
x =
= 0.084
3
s=
(0.000)2 + (0.005)2 + (0.005)2
= 0.005
3–1
From Table 4–2, t at the 95% confidence interval with two
degrees of freedom is 4.303.
(4.303)(0.005)
3
= 0.084 ± 0.012
So the 95% confidence interval = 0.084 ±
What is the CI at the 90% CL for this example?
Representation and the meaning of the confidence interval
the error bars include the target mean (10,000) more often for the 90% CL than for the
50% CL
Important information for real process!!!
Representation and the meaning of the confidence interval
A control chart was prepared to detect problems if something is out of
specification. As can be seen when  3 away at the 95% CL then there is a
problem and the process should be examined.
Student's t values can aid us in the interpretation of results and
help compare different analysis methods.
4-3 Comparison of Means ,
hypothesis
• Case 1:
• Case 2
• Case 3
Underlying question is are the mean values
from two different measurements
significantly different?
Hypothesis about the TRUE VALUES and/or ESTABLISHED VALUES
We will say that two results do not differ from each other unless there is a >
95% chance that our conclusion is correct
Student's t values can aid us in the interpretation of
results and help compare different analysis methods.
The statement about the comparison of values is the same statement as the concept of a
"null hypothesis in the language of statistics ".
The null hypothesis assumes that the two values being compared, are in fact, the
same.
Thus, we can use the t test (for example) as a measurement of whether the null
hypothesis is valid or not.
There are three specific cases that we can utilize the t test to question the null
hypothesis.
A Answers on
analytical
chemistry
questions:
Are the results
certain and do
they indicate
significant
differences that
could give
different answers
?
How to
establish
quantitative
criteria?
Example
Case #1: Comparing a Measured Result to a
"Known Value"
A new procedure for the rapid analysis of sulfur in kerosene was tested
by analysis of a sample which was known from its method of preparation
to ccontain 0.123% S. The results obtained were: %S = 0.112, 0.118, 0.115,
and 0.119. Is this new method a valid procedure for determining sulfur
in kerosene?
Looks
good,
but…..
One of the ways to answer this question is to test the new procedure on the
known sulfur sample and if it produces a data value that falls within the
95% confidence interval, then the method should be acceptable.
–
x = 0.116
s = 0.0033
(3.182)(0.0033)
95% confidence interval = 0.116 ±
4
–
± x = 0.116 ± 0.005
±–
x = 0.111 to 0.121
which does not contain the "known value
0.123%S"
Because the new method has a <5% probability of being correct, we can
conclude that this method will not be a valid procedure for determining
sulfur in kerosene.
µ= –
x ± ts / n
– - µ ) ( n / s)
± t = (x
…but this is the correct
method to avoid problems.
The statistical " t " value is found and compared to the table "t" value.
If t found > t table , we assume a difference at that CL (i.e. 50%, 95%,
99.9%).
Is the method acceptable at 95% CL?
dof = (n - 1) = 3 & @ 95% the tt = 3.182 (from student's t table)
– - µ) ( n /s)
± tf = (x
± tf = (0.116 - 0.123) * [( 4)/0.0033] = 4. 24
t found > t table ?
4.24 > 3.18,
so there is a difference, (thus the same conclusion.)
**If you have m than use it instead of mean
Another example:
Case 1: Let's assume µ is known, and a new method is being developed.
If @ 95% tfound > ttable then there is a difference.
We have a new method of Cu determination and we have a NIST Standard for
Cu.
The NIST value of Cu = 11.87 ppm. We do 5 trials
and get –
x =10.80 ppm , s = 0.7 ppm.
Is the method acceptable at 95% CL?
dof = (N - 1) = 4 & @ 95% the tt = 2.776
– - µ) (n /s)
± tf = (x
= (10.80 - 11.87) * [(5)/0.7] = 3.4
tfound > ttable  3.4 > 2.8, so there is a difference present
Case #2: Comparing Replicate Measurements
(Comparing two sets of data that were independently done using the "t" test. Note: The
question is; " Are the two means of two different data sets significantly different?"
This could be used to decide if two materials are the same or if two independently done
analyses are essentially the same or if the precision for the two analysts performing the
analytical method is the same. or two sets of data consisting of n1 and n2 measurements with
averages x1 and x2 ), we can calculate a value of t by using the following formula
| –x 1 – –x2|
t=
sspooled
n1n2
n1+n2
s12(n1–1) + s22(n2–1)
n1 + n2 –2
where sspooled =
Cont.
• The value of t is compared with the value of t in
Table 4–2 for (n1 + n2 – 2) degrees of freedom. If
the calculated value of t is greater than the t value
at the 95% confidence level in Table 4–2, the two
results are considered to be different.
•
• The CRITERIUM
•
If tfound > ttable  there is a difference!!
The Ti content (wt%) of two different ore samples
was measured several times
by the same method. Are the mean values
significantly different at the 95% confidence level?
n
X
s
Sample 1 5
0.0134
4E-4
Sample 2 5
0.0140
3.4E-4
sspooled =
s12(n1–1) + s22(n2–1)
n1 + n2 –2
sspooled =
(4.0 x 10–4)2(5–1) + (3.4 x 10–4)2(5–1)
5 + 5 –2
sspooled = 3.7 x 10–4
|–
x1 – –
x 2|
t=
sspooled
t=
n1n2
n1+n2
|0.0134 – 0.0140|
3.7 x 10–4
(5)(5)
10
t = 2.564
t from Table 4–2 at 95% confidence level and 8 degrees of freedom is 2.306
Since our calculated value (2.564) is larger than the tabulated value (2.306), we can say that the mean
values for the two samples are significantly different.
If t found > t table then a difference exists.
Case #3: Comparing Individual Differences
(We are using t test with multiple samples and are comparing the differences
obtained by the two methods using different samples without the duplication of
samples. For example; it might be reference method vs. new method. This would
monitor the versatility of the method for a range of concentrations.
This case applies when we use two different methods to make single
measurements on several different samples.
–
–
d
t=
n
sd
 (di – d)2
sd =
i
where d is the difference of results between the two methods
n–1
Sample
Composition
by method 1
(old)
Composition
by method 2
(new)
Delta -d
A
0.0134
0.0135
0.0001
B
0.0144
0.0156
0.0012
C
0.0126
0.0137
0.0011
D
0.0125
0.0137
0.0012
E
0.0137
0.0136
0.0001
–
 (di – d)2
i
sd =
n–1
sd = (16.6 X10 -7 / 4)
sd = 6.4 x 10–4
–
d
t=
sd
n
t = (0.00070/0.00064) x ( 5)
t = 2.45
t from Table 4–2 at 95% confidence level and 4 degrees of
freedom is 2.776
Since our calculated value (2.45) is smaller than the tabulated
value (2.776), we can say that the results for the two samples are
not significantly different.
If t found > t table then a difference exists.
2.45 > 2.776 is not true, So no difference exists!
Known true
value and CI:
CI
tabulated t
values show
C
rejection of
hypothesis
A
A acceptance
What to do with outliners? Points far
from the rest. Keep them or not?
G-method
FWI: also Q method
(both should give similar estimates)
CHAPTER 04: Unnumbered Figure 4.4
CHAPTER 04: Equation 4.13
CHAPTER 04: Table 4.5
Q 4-6) Test for Bad Data (Q-test and Dixon’s outliners)
Sometimes one datum appears to be inconsistent with the remaining data. When this happens, you are
faced with the decision of whether to retain or discard the questionable data point.The Q Test allows you to
make that decision:
gap = difference between the questionable point and
the nearest point
gap
Q=
range
Q table for the rejection of
data values
range= spread of data
# of
observations
90%
95%
99%
3
0.941
0.97
0.994
4
0.765
0.829
0.926
5
0.642
0.71
0.821
6
0.56
0.625
0.74
7
0.507
0.568
0.68
8
0.468
0.526
0.634
9
0.437
0.493
0.598
10
0.412
0.466
0.568
If Q ( observed or calculated) > Q(tabulated),
the questionable data point should be discarded.
Example
For the 90% CL, Can the value of 216 be rejected from the following set
of results?
Data: 192, 216, 202, 195 and 204
Q=
gap
range
gap = 216 – 204 = 12
range = 216 – 192 = 24
Q=
12
= 0.50
24
Q(tabulated) = 0.64
Q(observed) is less than Q(tabulated) (0.50 < 0.64) so the data
point cannot be rejected.
Note:
You must deal with the result of the Q-test. The simplest way is to
throw away the datum!
It is unethical to KEEP the datum!!!
4-4 FYI:
F Test
The F test provides a simple method for comparing the precision of two sets of identical
measurements.
s12
F= 2
s2
where s1 is the standard deviation of method 1 and s2 is the standard deviation of method 2
The F test may be used to provide insights into either of two questions:
(1)
Is method 1 more precise than method 2, or
(2)
Is there a significant difference in the precisions of two methods
For the first case, the variance of the supposedly more precise procedure is denoted s2.
For the second case, the larger variance is always denoted s1.
Critical Values for F at the Five Percent Level
Degrees of Freedom
(Denominator)
Degrees of Freedom (Numerator)
2
3
4
5
6
12
20
2
19.00
19.16
19.25
19.30
19.33
19.41
19.45
19.50
3
9.55
9.28
9.12
9.01
8.94
8.74
8.66
8.53
4
6.94
6.59
6.39
6.26
6.16
5.91
5.80
5.63
5
5.79
5.41
5.19
5.05
4.95
4.68
4.56
4.36
6
5.14
4.76
4.53
4.39
4.28
4.00
3.87
3.67
12
3.89
3.49
3.26
3.11
3.00
2.69
2.54
2.30
20
3.49
3.10
2.87
2.71
2.60
2.28
2.12
1.84

3.00
2.60
2.37
2.21
2.10
1.75
1.57

1.00
Example
The standard deviations of six data obtained in one experiment was 0.12%, while the standard
deviations of five data obtained in a second experiment was 0.06%. Are the standard deviations statistically
the same for these two experiments?
This example deals with option #2, so the larger standard deviation is placed in the numerator:
(0.12)2
F=
= 4.00
2
(0.06)
Note : If F calculated > F table then difference exists.
F(tabulated) = 6.26, so the standard deviations are statistically insignificant.
i.e. no difference
Additional material
cont
Systematic error
• What if all measurements are off the true
values?
SFYI: SYSTEMATIC ERROR
T(1) Back to Types and Origin(s) of Uncertainties Errors
We must address errors
when designing and evaluating any analytical method or performing an analysis determination.
Systematic errors or determinate errors. When they are detected, we must remove them, or
reduce or have them under control. Signature of the determinate error is that all are on one side of the
true value.
Examples of systematic error
IInstrument errors: a thermometer constantly reads two degrees too high. We can use a correction
factor. A balance is out of calibration, so we must have it calibrated.
Method errors: Species or reagents are not stable or possible contamination. Relationship about
analyte and signal invalid (standard curve not linear). Limitations on equipment, tolerances,
measurement errors of glassware, etc). Failing to calibrate you glassware or instrumentation. Lamp
light source not properly aligned.
Personal errors: color blindness, prejudice (You think the measurement is OK or is bad). We make
these a lot of the time! Not reading instruments correctly, not following instructions!
Suggested ways of eliminating systematic error: equipment calibration, self-discipline, care, analysis
of known reference standards, variation of sample size, blank determinations, independent lab testing
of method or sample.
Random error is always present and always
“symmetrical”!!! Random errors or indeterminate error
They cannot be reduced-unless you change the instrument
or method; so they are always present and are distributed
around some mean (true) value. Thus data readings will
fluctuate between low and high values and have a mean
value. We often use statistics to characterize these errors.
A measurement
Examples: "Parallax error reading a buret“ or could be
instrumental noise such as electrical
voltage noise of recorder, detector, etc.