BASIC CHEMOMETRIC EVALUATION
Arithmetic mean x :
∑=
=
n
i
ix
n
x
1
1
where: n …number of parallel analyses,
xi ….measured values.
Parameter estimation σ is standard deviation SD, s. The precision calculated for n ≥ 10 according
follow:
( )
( )∑=
−
−
=
n
i
i xx
n
s
1
2
1
1
For n ≤ 10, calculated according to Dean and Dixon with variation range R:
Rks nR ⋅=
minmax xxR −=
where: kn … Dean-Dixon’s coefficient for n measurements (values in table S.1),
xmax …the heights (maximum) value in their ascending order.
xmin ....the lowest (minimum) one.
Tab. S.1: Dean-Dixon’s coefficient kn
n kn n kn
2
3
4
5
6
0.8862
0.5908
0.4857
0.4299
0.3946
7
8
9
10
0.3698
0.3512
0.3367
0.3249
Relative standard deviation (RSD):
[ ]%100⋅=
x
s
sr
The confidence interval is interval in which is the value with predetermined probability 1 – α, if the
method is not affected by systematic error. Confidence level 1 – α is most often 0.95.
For n ≥ 10, the confidence interval L2,1 the following:
n
t
sxL v)(2/
1,2
α
±=
where: tα/2(v) is critical value of Student’s distribution for selected value α and number of degrees of
freedom v = n – 1 (in table S.2).
Tab. S.2: Critical values of Student’s distribution tα//2(v)
For n ≤ 10, the confidence interval L2,1 is according Dean and Dixon:
RKxL α
n ⋅±=1,2
where: Kn
α
... Dean-Dixon’s ccoefficient for pro for selected value α and number of measurements n
(values in table S.3).
Tab. S.3: Values of Dean-Dixon’s ccoefficient Kn
α
(1-α) (1-α)
n
0.95 0.99
n
0.95 0.99
2
3
4
5
6
6.40
1.30
0.72
0.51
0.40
31.80
3.01
1.32
0.84
0.63
7
8
9
10
0.33
0.29
0.26
0.23
0.51
0.43
0.37
0.33
α α
ν
0.1 0.05 0.039 0.01
ν
0.1 0.05 0.039 0.01
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
6.314
2.920
2.353
2.132
2.045
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
12.706
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
16.303
4.914
3.517
3.024
2.778
2.631
2.534
2.465
2.414
2.374
2.342
2.317
2.295
2.277
2.262
63.657
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
16
17
18
19
20
25
30
40
50
60
80
100
200
∞
1.746
1.740
1.734
1.729
1.725
1.708
1.697
1.684
1.676
1.671
1.664
1.660
1.652
1.645
2.120
2.110
2.101
2.093
2.086
2.060
2.042
2.021
2.008
2.000
1.990
1.984
1.972
1.960
2.248
2.237
2.226
2.217
2.209
2.178
2.159
2.134
2.120
2.210
2.099
2.092
2.078
2.066
2.921
2.898
2.878
2.861
2.845
2.787
2.750
2.704
2.678
2.668
2.639
2.626
2.606
2.576
2.2. The outlier test
The Grubbs (T-test) or Dean-Dixon’s test are used in practice. The tests eliminate the outlier values
from results.
The testing is performed for the highest (xn) and lowest (x1) value of results ordered in ascending order
by size.
Grubbs test:
n
n
n
s
xx
T
−
= .
ns
xx
T 1
1
−
=
( )∑=
−=
n
i
in xx
n
s
1
21
Calculated values Tn and T1 are confirmed with tabulated critical values Tn
α
. If is Tn ≥ Tn
α
, or T1 ≥ Tn
α
.,
the highest, or lowest results outlier. In this case, we must eliminate this value from results and the next
evaluation carries out without it. The tabulated critical values Tn
α
are in table S.4.
Tab. S.4: Critical tabulated values of Tn
α
α α
n
0.01 0.025 0.05 0.1
n
0.01 0.025 0.05 0.1
3
4
5
6
7
8
9
10
11
1.414
1.723
1.955
2.130
2.265
2.374
2.464
2.540
2.606
1.414
1.710
1.917
2.067
2.182
2.273
2.349
2.414
2.470
1.412
1.689
1.869
1.996
2.093
2.172
2.237
2.294
2.343
1.406
1.545
1.791
1.894
1.974
2.041
2.097
2.146
2.190
12
13
14
15
16
17
18
19
20
2.663
2.714
2.759
2.800
2.837
2.871
2.903
2.932
2.959
2.519
2.562
2.602
2.638
2.670
2.701
2.728
2.754
2.778
2.387
2.426
2.461
2.493
2.523
2.551
2.577
2.600
2.623
2.229
2.264
2.297
2.326
2.354
2.380
2.404
2.426
2.447
For numbers of measurements, n = 3–10, the outlier test for extreme values can be perform according
Dean-Dixon’s test. All three values must be to each other different!:
,1
R
xx
Q nn
n
−−
=
R
xx
Q 12
1
−
=
Calculated values Qn and Q1 are confirmed with tabulated critical values Qn
α
. If is Qn ≥ Qn
α
, or Q1 ≥
Qn
α
, the highest, or lowest results outlier. In this case, we must eliminate this value from results and the
next evaluation carries out without it. The tabulated critical values Qn
α
are in table S.5.
Tab. S.5: Critical tabulated values of Qn
α
α α
n
0.01 0.025 0.05 0.1
n
0.01 0.025 0.05 0.1
3
4
5
6
0.886
0.679
0.557
0.482
0.941
0.765
0.642
0.560
0.972
0.846
0.729
0.644
0.988
0.889
0.760
0.698
7
8
9
10
0.434
0.399
0.370
0.349
0.507
0.468
0.437
0.412
0.586
0.543
0.510
0.483
0.637
0.590
0.555
0.527