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

8.0 3.8 11.8

4.6 -1.8 2.8

-11.4 18.4 7.0

18.5 -9.4 9.1

-4.4 -7.3 -11.7

13.6 2.2 15.7

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