Accident Reconstruction |
Ethan Parker PE
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Statistically Based Analysis
(Monte Carlo Simulation)
A statistical analysis can provide a more in-depth understanding of the probability of events in an accident. This is particularly true when information, such as drag factors, vehicle weights, or some other critical data, is known to lie somewhere within a range but its exact value is unknown. In many accidents, a number of key pieces of information are known only within a range of values. Without statistical analysis reconstructionists are left with no other choice but to generate unrealistic answers calculated from the extremes of the ranges of the values.
Statistical analysis provides the "mathematical common sense" that can narrow this range from what is theoretically possible to what is actually probable.
A good way to visualize how this works is to imagine having 6 dice that represent 6 different parameters in the reconstruction of a crash. By rolling all the dice at the same time you simultaneously generate a value for each of these parameters. Adding the values on the top face of each die gives the answer. Common sense, based on your experience playing board games with more than one die, tells you that the sum is more likely somewhere in the middle of the possible range between 6 and 36, rather than at 6 or 36.
Now lets make a mathematical model to show that our common sense is right! The following table is taken from a spreadsheet modeling this scenario. The first six columns generate random integers from 1 to 6, while the seventh column sums their values. Notice that the values in columns 1 through 6 are equally likely to be any value from 1 to 6, but their sums are not equally likely to be any value from 6 to 36. This is because the probability of rolling all 6s or all 1s is less than rolling the more numerous combinations of numbers that would result in totals that lie near the middle of the range. As a result the total of the sums equaling each value from 6 to 36 tend to fall into a normal distribution. This can be seen in the data in the ninth and tenth columns towards the bottom of the table and is illustrated graphically in the chart at the bottom of the page.
Since this data tends to follow a normal distribution, we can use
its mean and standard deviation to calculate a 51% inclusion.
Because of this we can say that you are more likely than not to roll
sums between 18 and 24. As you can see this type of analysis
gives the reconstructionist a much more meaningful answer than
"somewhere between 6 and 36".
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Mathematical
roll of 6 dice |
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n=1000 |
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Mean |
SD |
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20.95 |
4.14 |
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Die
1 |
Die
2 |
Die
3 |
Die
4 |
Die
5 |
Die
6 |
Sum |
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2 |
2 |
2 |
2 |
5 |
1 |
14 |
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18.05 |
min (51%
inclusion) |
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6 |
6 |
5 |
1 |
2 |
4 |
24 |
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23.85 |
max (51%
inclusion) |
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6 |
6 |
3 |
6 |
3 |
6 |
30 |
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5 |
2 |
1 |
4 |
2 |
5 |
19 |
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More
likely than not between |
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6 |
1 |
3 |
2 |
2 |
4 |
18 |
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18 |
and |
24 |
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6 |
5 |
4 |
5 |
6 |
3 |
29 |
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1 |
3 |
2 |
1 |
4 |
1 |
12 |
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Distribution |
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5 |
5 |
5 |
3 |
3 |
6 |
27 |
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Value |
No.
of sums |
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6 |
4 |
2 |
5 |
1 |
5 |
23 |
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6 |
0 |
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5 |
1 |
3 |
3 |
3 |
4 |
19 |
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7 |
0 |
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6 |
4 |
1 |
5 |
5 |
5 |
26 |
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8 |
0 |
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3 |
5 |
6 |
5 |
3 |
4 |
26 |
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9 |
1 |
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3 |
4 |
3 |
2 |
1 |
3 |
16 |
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10 |
1 |
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6 |
1 |
5 |
5 |
2 |
1 |
20 |
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11 |
6 |
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6 |
1 |
4 |
1 |
5 |
2 |
19 |
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12 |
7 |
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1 |
6 |
3 |
1 |
6 |
2 |
19 |
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13 |
23 |
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3 |
2 |
3 |
2 |
3 |
3 |
16 |
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14 |
19 |
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2 |
4 |
5 |
4 |
1 |
4 |
20 |
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15 |
29 |
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2 |
6 |
2 |
6 |
5 |
2 |
23 |
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16 |
55 |
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5 |
2 |
5 |
3 |
3 |
3 |
21 |
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17 |
68 |
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3 |
5 |
1 |
3 |
2 |
3 |
17 |
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18 |
83 |
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6 |
1 |
3 |
4 |
2 |
3 |
19 |
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19 |
83 |
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1 |
4 |
1 |
1 |
6 |
3 |
16 |
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20 |
91 |
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3 |
4 |
5 |
3 |
5 |
2 |
22 |
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21 |
93 |
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6 |
4 |
5 |
4 |
4 |
2 |
25 |
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22 |
84 |
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