The arithmetic mean is the
"standard" average, often
simply called the "mean"
The standard deviation (SD)
quantifies variability. If the
data follow a bell-shaped
Gaussian distribution, then
68% of the values lie within
one SD of the mean (on
either side) and 95% of the
values lie within two SD of the
mean. The SD is expressed
in the same units as your
data.
To apply a significance test, a hypothesis must be clearly stated and must have
a quantity with a calculated probability associated with it. This is the fundamental
difference between a hunch and a hypothesis test—a quantity and a probability.
The hypothesis will be accepted or rejected on the basis of a comparison of the
calculated quantity with a table of values relating to a normal distribution. As with
the confidence interval, the analyst selects an associated level of certainty,
typically 95%. The starting hypothesis takes the form of the null hypothesis
What is a null hypothesis? The null hypothesis is stated in such a way as to
say that there is no difference between the calculated quantity and the expected
quantity, save that attributable to normal random error. As regards to the outlier
in question, the null hypothesis for the chemist and the trainee states that the
11.0% value is not an outlier and that any difference between the calculated and
expected value can be attributed to normal random error.
The P value is a probability, with a value ranging from zero to one. If the
P value is small, you'll conclude that the difference is unlikely to be a
coincidence:
P<0.05 "significant”
P>0.05, "not significant"
t test
Another hypothesis test used in forensic chemistry is one that
compares the means of two data sets. In the supervisor–
trainee example, the two chemists are analyzing the same
unknown, but obtain different means. The t-test of means can
be used to determine whether the difference of the means is
significant. The t-value is the same as that used in for
determining confidence intervals. This makes sense; the goal
of the t-test of means is to determine whether the spread of
two sets of data overlap sufficiently for one to conclude
that they are or are not representative of the same population.
In the supervisor–trainee example, the null hypothesis could
be stated as “ The mean obtained by the trainee is not
significantly different than the mean obtained by the
supervisor at the 95% confidence level ” Stated another way,
the means are the same and any difference between them is
due to small random errors.
Table Bullets and fragments received by the FBI.
Specimen Description
Total weight, grains
Total weight,mg
CE 399 (Q1) Bullet from stretcher (lead core plus jacket)
158.6
10,277
CE 567 (Q2) Bullet fragment from seat cushion
(lead core plus brass jacket)
44.6
2,890
CE 569 (Q3) Bullet fragment from front seat (jacket)
21.0
1,361
CE 843 (Q4,5) Two lead fragments from President’s head[2]
1.65; 0.15
107; 9.7
CE 842 (Q9) Three lead fragments from Connally’s arm
0.5
32
CE 840 (Q14) Three lead fragments from rear carpet
0.9, 0.7, 0.7
58, 45, 45
CE 841 (Q15) Scraping from inside surface of windshield
None listed
Table : Individual determinations of antimony in the
FBI’s Run 4
Specimen
Weight of subfragment, mg
Sb, ppm
Q1
7.16
643
4.20
636
1.79
750
1.24
749
1.16*
749
15.55
705±60*
Table : Individual determinations of antimony in the
FBI’s Run 4
Specimen
Weight of subfragment, mg
Sb, ppm
Q9
1.92
690
2.07
662
1.34
677
5.33
676±14
Table : Individual determinations of antimony in the
FBI’s Run 4
Specimen
Weight of subfragment, mg
Sb, ppm
Q2
39.75
521
21.60
521
3.84
578
3.68
515
68.87
534±30
Table The FBI’s results for silver and antimony
in bullets and fragments (concentrations in ppm).
Specimen Q1
Q9
Q2
Q4,5
Q14
Ag
9.4±0.3
9.2±0.1
7.9±0.9
8.5±0.4
8.5±0.2
Sb Run 1
945±16
977±24
745±16
783±5
793±10
Sb Run 2
1002±13
1090±37
747±20
858±46
879±33
Sb Run 3
813±43
773±22
626±57
614±37
629±18
Sb Run 4
705±54
676±14
534±30
561±32
562±21