2010WICS-FD RNP1 SqrtSumSquaresExample 8-25-10
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Transcript of 2010WICS-FD RNP1 SqrtSumSquaresExample 8-25-10
Log Normal Distributions
Data A Data B Sum of A and B
Percentile Levels 0.97 0.97 0.97
Corresponding Data Values 112.1 95.7 217.6
Standard Deviations 66.9 69.6 95.3
Lognormal Random Error Data: 0.0 0.8 0.8
0.4 5.1 5.4
789.6 -0.8 788.8
4.4 2.3 6.7
-0.2 -0.9 -1.1
1.9 33.1 35.0
-0.6 5.3 4.7
7.6 -1.0 6.6
31.1 -0.4 30.7
2.0 7.4 9.4
0.1 -0.7 -0.5
55.9 1.4 57.3
-0.4 0.9 0.5
57.1 -0.9 56.2
9.7 0.6 10.3
9.9 16.9 26.8
0.0 0.0 0.0
0.4 7.8 8.2
-0.4 26.2 25.8
0.1 16.9 16.9
0.0 0.3 0.4
-0.1 2.1 2.0
-0.2 -0.6 -0.8
19.0 1.9 21.0
-0.9 5.2 4.3
1.2 -0.6 0.6
-0.8 0.5 -0.3
-0.9 0.0 -0.9
-0.9 0.9 0.0
1.6 4.1 5.6
2.5 3.5 6.1
27.0 2.4 29.4
-0.6 6.0 5.5
338.8 -0.8 338.0
-1.0 0.3 -0.7
14.4 1.8 16.1
7.2 -0.9 6.3
-0.9 1.1 0.3
-0.8 8.1 7.3
-0.7 -1.0 -1.7
1.1 -0.2 0.9
-1.0 0.5 -0.5
0.4 49.9 50.3
-0.6 69.2 68.6
0.9 7.8 8.7
-0.4 -0.7 -1.1
26.6 -0.9 25.6
5.1 60.8 65.9
15.8 47.7 63.6
-0.9 3.6 2.6
0.0 -0.8 -0.9
2.3 11.4 13.7
-0.1 -0.7 -0.8
20.4 -0.6 19.7
5.1 10.3 15.4
3.9 4.4 8.3
-0.3 35.1 34.8
71.6 -0.8 70.8
5.4 11.3 16.7
0.6 1.5 2.1
-0.7 -0.8 -1.5
6.2 0.6 6.8
33.6 0.0 33.6
1.9 0.6 2.5
-0.7 -0.4 -1.1
-0.1 0.3 0.2
0.1 -0.6 -0.5
6.2 -0.9 5.4
1.9 1.4 3.3
-1.0 415.1 414.1
14.1 4.7 18.7
6.2 55.8 62.0
1.3 -0.3 1.1
-0.8 -0.1 -0.9
2.6 12.0 14.5
29.2 77.1 106.3
0.6 4.2 4.8
51.0 2.1 53.1
4.2 37.1 41.3
-1.0 4.5 3.5
13.3 77.3 90.7
61.0 51.5 112.5
-0.1 0.0 -0.1
0.5 -0.7 -0.2
1.5 17.2 18.7
3.4 5.3 8.7
-0.8 128.2 127.3
-0.2 -0.9 -1.1
13.0 3.0 16.0
0.0 0.8 0.8
40.2 30.2 70.4
0.0 24.5 24.5
2.4 -0.9 1.5
1.1 -0.4 0.6
-0.6 0.0 -0.6
2.5 -0.2 2.3
743.1 1.0 744.1
6.0 27.6 33.6
0.6 26.9 27.5
-0.8 2.7 1.8
0.7 0.3 1.0
0.3 0.3 0.6
-0.6 1.4 0.8
11.2 2.6 13.7
0.5 2.5 3.0
-0.9 0.4 -0.5
-0.8 14.1 13.3
-0.4 -0.4 -0.8
19.1 5.2 24.3
4.0 6.0 10.0
6.0 3.2 9.3
-0.6 0.4 -0.3
-0.3 1.0 0.8
13.9 -0.1 13.8
15.9 0.5 16.4
2.2 178.1 180.3
-0.4 0.4 0.0
17.3 -0.9 16.4
18.5 2.4 20.9
1.1 1.2 2.3
4.7 25.8 30.5
2.0 2.0 4.0
254.3 6.1 260.4
5.0 4.0 8.9
23.7 183.6 207.2
5.2 2.3 7.5
5.2 -0.3 4.9
61.5 1.1 62.6
-0.7 6.4 5.8
0.1 16.4 16.5
402.9 -1.0 401.9
-0.9 -0.6 -1.5
2.5 1.2 3.7
2.1 112.6 114.6
3.1 1.6 4.6
0.2 0.3 0.5
0.3 2.3 2.6
-0.2 3.0 2.8
2.9 -1.0 2.0
81.6 -0.4 81.2
-0.1 -0.3 -0.4
118.9 -0.8 118.0
-0.6 1.2 0.6
3.2 16.1 19.4
-0.7 -0.8 -1.5
-0.3 -0.9 -1.1
1.5 15.0 16.5
13.7 76.9 90.6
7.8 1.8 9.6
14.5 2.6 17.1
2.4 22.3 24.7
-0.8 34.7 33.8
4.5 -0.9 3.6
-0.2 -0.3 -0.5
2.7 374.3 377.0
0.2 -0.8 -0.7
24.7 -0.3 24.4
17.3 5.1 22.4
-0.2 22.7 22.6
-0.5 -0.9 -1.4
0.9 1.9 2.7
2.7 38.4 41.1
-0.9 4.5 3.7
16.4 -0.6 15.8
5.7 1.8 7.5
141.2 -0.6 140.6
3.4 0.3 3.7
-0.9 57.0 56.1
0.3 4.2 4.5
9.6 26.6 36.2
111.9 27.8 139.6
13.3 2.3 15.6
1.3 -1.0 0.4
-0.3 -0.4 -0.7
-0.3 1.5 1.1
12.4 -0.5 12.0
-0.7 0.4 -0.3
5.4 0.3 5.7
3.8 12.9 16.7
1.2 20.9 22.1
-0.7 19.8 19.1
1.4 1.2 2.6
41.3 0.8 42.1
-0.6 -0.9 -1.5
-0.5 27.9 27.4
1.7 -0.3 1.4
4.0 0.1 4.0
4.8 1.7 6.5
-0.8 0.5 -0.3
1.3 0.1 1.3
4.7 1.0 5.7
2.6 -0.1 2.5
23.3 3.9 27.2
18.0 1.2 19.2
0.9 69.7 70.6
1.8 6.2 8.1
101.9 -0.3 101.6
-0.3 0.0 -0.3
55.8 1.5 57.3
82.0 9.6 91.6
7.4 69.9 77.3
5.4 5.5 10.9
0.4 49.2 49.5
-0.9 -0.3 -1.1
0.0 1.3 1.3
-0.5 3.8 3.4
1.6 -0.4 1.2
3.2 -0.5 2.7
28.9 -0.8 28.1
3.0 -0.9 2.1
3.5 -0.2 3.4
6.6 -0.7 6.0
1.5 3.6 5.0
2.7 43.7 46.4
0.9 6.6 7.4
2.6 -0.1 2.4
5.4 0.2 5.6
219.2 4.5 223.7
3.9 -0.8 3.1
-0.2 5.1 4.8
23.6 6.9 30.5
-0.8 -0.6 -1.4
1.7 0.1 1.8
-0.6 55.5 54.8
21.6 0.1 21.6
28.2 1.6 29.9
0.0 -0.8 -0.8
1.5 -1.0 0.5
2.0 -0.5 1.5
0.5 -0.3 0.2
19.7 -0.7 19.1
80.0 13.8 93.7
-0.8 31.1 30.4
74.7 1.3 76.0
3.6 15.9 19.5
-0.9 16.9 15.9
0.1 -0.3 -0.2
7.6 3.3 10.9
6.5 1.6 8.1
235.7 0.6 236.3
-0.8 1.6 0.8
-0.5 6.9 6.4
15.9 65.2 81.1
5.4 1.4 6.8
-0.8 3.1 2.3
0.7 -0.4 0.4
0.4 16.2 16.6
-0.9 6.7 5.8
0.1 64.1 64.2
0.4 3.1 3.5
0.2 6.1 6.4
0.1 41.0 41.1
-0.9 62.9 62.0
28.5 10.3 38.8
16.0 -0.7 15.3
-0.2 19.3 19.1
-1.0 -0.3 -1.3
-0.5 8.5 7.9
-0.3 1.4 1.1
1.1 -0.8 0.3
-0.1 16.2 16.1
-0.7 15.2 14.5
7.3 -0.8 6.6
-0.8 20.2 19.4
8.9 1.8 10.7
-0.2 53.5 53.4
1.5 8.7 10.2
6.8 1.6 8.4
5.1 274.0 279.1
14.4 58.9 73.3
-0.7 0.2 -0.6
30.7 22.7 53.5
7.5 0.8 8.2
0.0 -0.4 -0.4
2.3 5.3 7.6
21.5 145.3 166.8
-0.4 3.5 3.1
47.9 -0.3 47.6
4.3 -0.8 3.5
-0.8 49.4 48.6
-0.4 7.9 7.4
-0.1 1.1 1.0
562.7 4.2 566.9
5.7 -0.6 5.1
13.2 1026.6 1039.8
17.3 1.6 18.8
3.2 -0.3 2.9
29.0 -0.7 28.3
21.9 -0.3 21.6
-0.1 -1.0 -1.1
-0.9 5.3 4.4
156.3 -0.3 156.0
27.3 9.5 36.7
3.1 3.5 6.6
-0.8 4.4 3.6
5.7 7.0 12.8
51.2 0.8 52.0
6.3 7.6 13.9
24.2 840.3 864.5
-0.6 1.0 0.3
-0.4 16.4 16.0
3.9 -0.7 3.3
1.6 19.1 20.7
-0.2 -0.2 -0.5
0.0 0.4 0.4
10.7 2.4 13.1
1.7 2.5 4.2
5.7 95.4 101.1
12.1 -0.2 12.0
5.6 172.7 178.2
37.7 -0.9 36.8
5.6 -0.8 4.8
5.5 -0.1 5.3
13.9 1.6 15.5
2.0 1.4 3.4
51.9 12.5 64.5
1.4 0.8 2.2
-0.9 -0.1 -1.0
1.8 0.9 2.7
0.4 17.0 17.4
7.5 -0.4 7.1
336.8 -0.9 335.9
-0.7 1.7 1.0
4.0 -0.6 3.4
12.9 1.2 14.1
-0.2 1.0 0.7
7.2 -0.1 7.0
30.7 -1.0 29.7
271.7 -0.4 271.3
0.6 -0.4 0.1
31.4 10.4 41.8
0.0 12.3 12.3
1.5 7.6 9.1
6.5 -0.6 5.9
8.3 3.3 11.6
43.4 15.3 58.7
0.0 2.9 2.8
0.0 -0.5 -0.5
5.9 3.8 9.6
1.0 1.9 2.9
-0.9 7.2 6.3
-0.8 -0.1 -0.9
0.9 0.9 1.8
-0.8 2.8 2.1
1.6 -1.0 0.7
-0.4 3.2 2.8
-1.0 88.7 87.7
65.9 4.0 69.9
0.0 -0.8 -0.8
-1.0 -1.0 -1.9
1.7 -0.8 0.8
10.1 2.7 12.8
1.8 -0.7 1.1
-0.9 16.5 15.6
0.8 -0.9 -0.2
8.8 57.2 66.0
10.1 6.0 16.0
2.3 -0.9 1.3
7.0 15.2 22.2
-0.6 -0.9 -1.5
10.4 9.0 19.4
1.1 -0.7 0.3
-0.5 10.8 10.3
22.9 0.6 23.5
-0.9 -0.2 -1.2
-0.4 -0.7 -1.1
-0.8 5.0 4.2
0.6 -0.7 -0.1
6.5 4.5 11.0
0.9 52.6 53.4
0.5 4.2 4.7
-0.7 0.0 -0.7
5.3 8.5 13.9
-0.4 -0.1 -0.5
-1.0 -0.6 -1.6
8.5 0.1 8.7
51.8 4.1 55.9
0.2 26.3 26.5
-0.9 3.6 2.8
12.4 2.4 14.8
84.0 4.2 88.2
1.2 -0.4 0.8
-0.5 3.4 2.9
5.9 211.6 217.4
0.8 -0.4 0.4
1.8 1.4 3.2
-1.0 7.5 6.5
-0.4 11.1 10.7
18.8 9.7 28.5
0.9 6.3 7.2
0.9 -0.4 0.6
11.5 0.6 12.1
0.5 33.8 34.3
2.3 -0.8 1.5
3.8 22.3 26.1
14.8 21.6 36.4
-0.8 7.2 6.4
12.1 0.4 12.4
0.2 1.4 1.6
8.0 -1.0 7.1
4.1 8.1 12.2
10.1 -0.9 9.1
5.7 2.0 7.7
-0.8 8.9 8.1
-0.1 -0.9 -1.0
2.0 1.4 3.4
0.5 2.1 2.6
7.4 -0.1 7.3
-0.7 6.2 5.5
3.5 18.3 21.9
34.1 6.8 40.8
-0.5 16.0 15.4
0.5 -0.6 -0.1
-0.5 -0.9 -1.3
-0.8 -0.3 -1.1
-0.4 52.9 52.5
-0.7 -0.3 -1.1
9.8 6.2 16.0
89.8 1.3 91.2
0.9 4.2 5.1
1.2 0.0 1.2
10.9 -0.9 10.0
23.6 -0.8 22.9
8.8 90.3 99.0
0.7 2.4 3.0
-0.6 -0.9 -1.5
4.3 7.1 11.4
128.3 -0.6 127.7
57.6 -0.7 56.9
16.3 -1.0 15.3
-0.7 1.9 1.2
-0.1 13.9 13.8
10.1 -0.6 9.4
8.2 -0.3 7.8
10.6 0.5 11.1
1.3 7.4 8.7
20.5 9.2 29.7
-0.5 4.1 3.6
-0.8 107.9 107.1
0.1 10.4 10.4
6.6 1.5 8.2
1.0 -0.8 0.1
-0.1 33.4 33.3
0.4 20.5 21.0
-0.8 2.2 1.4
2.5 4.8 7.4
-0.7 75.8 75.0
-0.8 14.9 14.1
8.6 -0.8 7.8
4.4 7.3 11.6
0.9 16.2 17.0
-0.4 50.2 49.8
1.4 0.7 2.1
0.8 9.5 10.3
2.7 -0.9 1.8
21.7 3.1 24.8
37.4 4.4 41.8
53.8 1.4 55.2
146.3 -0.1 146.3
-0.8 2.0 1.2
-0.7 0.0 -0.7
5.3 4.1 9.3
-0.9 0.7 -0.2
-0.4 14.0 13.6
0.4 4.4 4.8
-0.8 5.8 4.9
0.6 1.7 2.3
-0.8 1.4 0.6
3.4 56.1 59.5
6.2 -0.2 6.0
-0.9 0.8 -0.1
22.5 3.8 26.2
-0.3 1.2 0.8
14.0 169.5 183.5
4.8 3.6 8.4
104.6 14.8 119.4
0.3 21.8 22.1
33.4 147.9 181.3
8.3 4.4 12.8
1.0 -0.9 0.1
10.3 34.1 44.4
0.0 -0.1 -0.2
-0.6 6.8 6.2
-0.3 -0.4 -0.8
-0.4 0.8 0.4
12.4 24.1 36.5
-0.7 15.2 14.5
0.8 39.1 39.9
0.4 2.6 2.9
2.5 -0.2 2.2
0.8 8.4 9.2
1.9 -0.2 1.7
1.3 -0.8 0.5
5.8 -1.0 4.8
-0.4 0.8 0.4
-1.0 1.5 0.5
-0.2 0.0 -0.2
17.7 0.4 18.1
15.2 19.9 35.0
4.1 4.9 9.0
Percentile/Sigma
Levels from
Formula
SQRT Sum
Squares
Formula Error
Percentile Levels
147.4 32% Corresponding Data Values
96.5 1% Standard Deviations
Normal Random Error Data:
This worksheet shows the fact that percentiles of distributions do not
necessarily follow the square-root of the sum of squares formula
when data is not normally distributed-- even given uncorrelated data
sets.
In Columns B-G is data and summary statsitics for two sets of
randomly generated data: "Data A" and "Data B". The data are
taken from a lognormal distribution. Data values corresponding to
the 97th percentile are shown in row 6 for both data sets. Column E
contains a third set of data and statistics representing the simple
sum of Data A and Data B values.
Column F computes the 97th percentile level and standard deviation
of the sum from the approximation from the square-root sum of
squares formula. The percentiles and standard deviations should
closely match those in Column E if the formula is a good
approximation. The formula would be exact for the standard
deviation if the correlation between Data A and Data B were
identically zero. There is a small coincidental correlation that is
reflected in the differences.
Column G simply summarizes the difference between the computed
statistics and the statistics estimated from the formula. Generally
the error in standard deviation should be small, because the formula
is correct (absent correlation) for standard deviation, but only correct
for percentiles if the distributions are normal (i.e., Gaussian).
The whole procedure is repeated for normally distributed errors in
columns J through O. Generally, the formula approximation is within
a few percent of accuracy for both percentile and standard deviation
in the normal distribution-- but this is not the case for percentiles
calculated in the lognormal case. The data can be changed by
recalculating the worksheet to see that in general, the formul
approximation of percentiles in lognormal distributions can be
quite large. In contrast, the square root sum of squares formula is
correct for computing standard deviations irrespective of the
distribution type (only depends on correlation).
Normal Distributions
Data A Data B Sum of A and B
Percentile/Sigma
Levels from
Formula
SQRT Sum
Squares
Formula
Error
0.97 0.97 0.97
19.3 17.8 25.7 26.2 2%
10.1 10.0 14.3 14.2 1%
13.1 -22.5 -9.5
6.2 8.1 14.3
-6.5 3.1 -3.4
-8.0 -8.4 -16.4
-6.1 -14.5 -20.6
6.8 2.2 8.9
-13.6 -14.9 -28.5
1.0 2.7 3.7
-6.7 9.5 2.8
-12.8 -17.9 -30.6
-12.2 -25.8 -38.0
6.4 -13.4 -7.0
9.9 -14.5 -4.7
-2.5 1.1 -1.4
-10.7 -12.1 -22.8
0.3 -1.2 -0.8
-6.7 -8.7 -15.4
-7.8 -14.1 -21.9
-3.6 -11.6 -15.2
10.5 -0.4 10.1
-2.1 2.8 0.7
16.0 -11.8 4.2
-3.5 -2.6 -6.1
-7.4 -6.5 -13.9
11.4 -11.4 0.0
-1.0 -25.4 -26.3
10.2 -11.0 -0.8
5.1 -4.2 0.9
-0.7 1.7 1.0
10.8 3.7 14.4
8.1 -10.7 -2.7
10.8 0.6 11.4
3.8 1.4 5.1
-1.4 2.5 1.0
6.9 0.4 7.3
4.5 -6.0 -1.5
5.1 -25.6 -20.5
3.1 -6.2 -3.1
20.2 11.9 32.2
-16.3 -0.4 -16.7
7.8 17.4 25.3
6.8 -5.8 1.0
2.2 10.6 12.8
-16.2 10.3 -5.8
-25.2 -10.2 -35.4
-5.9 0.2 -5.7
-6.6 4.0 -2.6
-1.4 12.8 11.4
-9.0 3.2 -5.8
6.8 0.5 7.3
3.0 8.2 11.2
-0.2 -13.8 -13.9
-11.2 0.6 -10.6
-4.5 3.2 -1.3
1.9 5.2 7.0
9.9 -18.1 -8.2
-9.3 13.4 4.2
-3.1 -12.2 -15.3
11.9 -11.3 0.6
-6.8 -10.1 -16.9
-0.5 12.3 11.9
10.9 -5.4 5.4
-0.2 15.2 15.0
-18.2 21.5 3.2
8.5 5.5 14.0
-5.9 0.8 -5.1
-10.5 13.5 3.1
6.7 -9.1 -2.4
-9.3 2.8 -6.4
2.8 7.9 10.7
3.3 0.4 3.8
-10.9 1.7 -9.2
-7.1 2.9 -4.1
-3.6 -10.0 -13.7
5.3 3.8 9.1
-6.4 0.0 -6.4
3.5 -15.8 -12.4
7.6 -4.6 3.0
-2.7 9.2 6.5
18.3 2.3 20.6
5.9 -4.9 1.0
11.1 2.4 13.5
-23.2 0.4 -22.8
17.9 3.2 21.1
-11.2 6.0 -5.2
-2.9 2.8 0.0
11.3 -6.4 4.9
7.1 4.4 11.5
-1.9 13.4 11.5
-7.3 10.0 2.6
19.3 -0.3 19.0
6.2 -2.9 3.4
5.9 -2.7 3.1
4.5 13.0 17.6
3.4 2.2 5.6
-1.2 0.4 -0.8
0.4 6.5 6.9
-14.8 -1.0 -15.8
-18.5 3.8 -14.8
-4.1 -10.5 -14.6
-23.2 -7.8 -31.0
-10.0 7.6 -2.3
-11.8 -0.2 -12.0
4.5 11.7 16.2
-2.3 -6.6 -8.9
-4.3 -18.9 -23.2
-2.6 -1.8 -4.4
-6.9 -12.3 -19.2
1.3 -2.4 -1.0
-10.8 0.9 -10.0
-5.5 -14.8 -20.3
-2.2 -6.7 -8.9
3.4 -9.1 -5.7
-5.3 9.2 3.8
5.8 2.0 7.8
-13.1 5.2 -7.9
-12.1 -6.5 -18.6
0.1 7.6 7.6
6.6 -7.7 -1.1
-1.0 -0.5 -1.5
2.2 2.7 4.9
19.0 -17.2 1.7
3.4 -3.0 0.4
-4.5 -16.6 -21.2
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