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Purpose: Check for excessively large or small deviations
standard deviations too bunched (are concentrated in the
central region)indicates undergraduation ( too close to crude
rates) or heterogeneity
standard deviations too spread out (errors not centered):
Roughly half the standardised deviations should fall in the
2
range (− 3 , 2 ).
3
A lot of values in the tails (big absolute deviations) indicates
overgraduation or the existence of duplicates (lives not
independent).
25/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Standardised Deviations Test Procedure I
Calculate the standard deviation zx for each age group
Categorize the standardised deviations into ranges –for
example, ∞ to −3,3 to 2;2 to 1, 1 to 0, 0 to +1, +1 to
+2, +2 to +3, +3 to +∞. (there should be no standardised
deviations lying outside range (3,3))
Note: when the number of age groups is small, use smaller
no. of intervals, such that the expected no. of standardised
deviations in each interval is not less than ﬁve.
Compare
the observed no. of the zx that fall in each interval (actual);
with
the expected no. of the zx that fall in each interval assuming
zx ∼ N (0, 1) (expected)
26/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Standardised Deviations Test Procedure II Use the test statistic
X=
all intervals (actualexpected)2
expected X has a χ2−1 distribution where n is the number of intervals
n
used
Note other tests for normality could be used (such as a QQ plot
(quantilequantile)) 27/72 Actuarial Statistics – Module 8: Method of Graduation
Statistical tests
Standardised Deviations Test Example For a graduation, we have calculated the standardised deviations as
follows
x
30
31
32
33
34
35
36
37
zx
2.29 2.71 0.29 0.55 2.14 1.18 0.06 1.55
x
38 3940
41
42
43
44
45
zx 1.09 2.05 1.08 1.52 0.04 0.39 1.48 1.20
x
46
47
48
49
zx
0.21 2.24 0.60 1.99
Analyse the distribution of the standardised deviations for this
graduation. 28/72 Actuarial Statistics – Module 8: Method...
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 Statistics, Normal Distribution, Statistical tests, actuarial statistics, Parametric Formula

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