Lecture 3
Statistics: Significance Tests
Reading: Ch. 4
Outline
1. Comparison of means with Student’s t (4-3)
2. Comparison of standard deviations with the
F test(4-4)
3. Bad data testing with the Q test (4-6).
Null hypothesis: our method is not a subject to systematic
error
Comparison of means with Student’s t (4-3)
1. Comparison of an experimental mean with a known value.
2. Comparison of the means of two samples.
3. Paired t-test: comparison of two analytical methods
Comparison of an experimental mean with a known
value.
μ = x±
ts
(x − μ) n
⇒t =
s
n
Example: Determining of mercury by cold vapor atomic absorption.
Standard reference material contains 38.9% mercury
Mercury content:
Experiment 1
Experiment 2
Experiment 3
38.9%
37.4%
37.1%
t =
( x − μ ) n (37.8 − 38.9) 3
=
= 1.98
s
0.964
From the table t=4.3 (95% and 2 degrees of freedom)
The null hypothesis is retained because the observed t value is less than critical
value.
J.C.Miller and J.N.Miller, Statistics for Analytical Chemistry, 3rd edition, 1993.
Comparison of the means of two samples (comparing
two sets of replicate measurements).
A. If two samples have standard deviations which are not significantly different, a pooled
estimate standard deviation can be calculated:
SET 1
SET 2
∑ (x − x ) + ∑ (x − x )
2
s pooled =
i
i
1
i
2
i
n1 + n2 − 2
2
=
s (n1 − 1) + s (n2 − 1)
n1 + n2 − 2
2
1
2
2
x1 − x2
s pooled
tcalculated =
n1n2
n1 + n2
Example: Determining of boron in water samples (micrograms / mL).
Spectrofotometry
Fluorimetric method
n = 10
n = 10
mean = 28.0
mean = 26.25
standard deviation = 0.3
standard deviation = 0.23
spooled = 0.267
tcalculated = 14.7
t = 2.1 (95%, 18 degrees of freedom)
The null hypothesis is rejected because the calculated (observed) t value is more than critical
value.
B. If two samples have standard deviations which are significantly different, t can be calculated
as follows:
⎧
⎫
⎪ 2
2 ⎪
⎪ s1 / n1 + s22 / n2
⎪
x1 − x2
degrees of freedom = ⎨
⎬−2
2
2
2
2
tcalculated =
/
/
s
n
s
n
⎪ 1 1 + 2 2 ⎪
s12 / n1 + s22 / n2
⎪ n +1
n2 + 1 ⎪⎭
⎩ 1
(
(
) (
)
)
the result is rounded to the nearest whole number
J.C.Miller and J.N.Miller, Statistics for Analytical Chemistry, 3rd edition, 1993.
Paired t-test: comparison of two analytical methods.
Do the two methods give values, which differ significantly?
Two methods, several samples, single measurement on each sample.
Important assumption: errors, either random or systematic, are independent on concentration.
d i = xi1 − xi2
sd =
∑ (d
i
− d )2
i
n −1
tcalculated =
d
sd
n
Example: concentration of Pb determined by two different methods for each the four test
samples
Sample
1
2
3
4
Method 1(wet oxidation)
71
61
50
60
tcalculated = 0.70
Method 2(direct oxidation)
76
68
48
57
t = 3.18 (95%, degrees of freedom = 3)
The null hypothesis is retained because the observed t value is less than critical
value.
J.C.Miller and J.N.Miller, Statistics for Analytical Chemistry, 3rd edition, 1993.
Comparison of standard deviations with the F test (4-4)
Use F test to compare two standard deviations (comparing two sets of replicate measurements).
Fcalculated
s12
= 2
s2
s1 ≥ s2 ⇒ Fcalculated ≥ 1
If Fcalculated > F the difference is significant
Comparison of standard deviations with the F test (4-4)
Use F test to compare two standard deviations (comparing two sets of replicate measurements).
Example: Determining of boron in water samples (micrograms / mL).
Spectrofotometry
Fluorimetric method
Fcalculated
s12
= 2
s2
Fcalculated = 1.7
n = 10
n = 10
mean = 28.0
mean = 26.25
s1 ≥ s2 ⇒ Fcalculated ≥ 1
F = 3.18
Fcalculated < F
the difference is not significant
standard deviation = 0.3
standard deviation = 0.23
Bad data testing with the Q test (4-6): Outliers
Use Q test to reject “bad” data points.
Qcalculated =
suspect value − nearest value
largest value − smallest value
If calculated value for Q exceeds the critical
value the suspect value is rejected
Example:
Determining of nitrite in ground water (mg / L).
0.403
0.410
0.401
0.380 (?)
suspect
Qcalculated = |0.380-0.401|/|0.410-0.380| = 0.7
Q = 0.76 (from the table)
The experimental value should be retained
Q
(95% confidence)
Number of
observations
0.831
4
0.717
5
0.621
6
0.570
7
0.524
8
0.492
9
0.464
10
J.C.Miller and J.N.Miller, Statistics for Analytical Chemistry, 3rd edition, 1993.