Re: Airbus Safety

Date:         13 Oct 98 02:48:14 
From:         standaert@mail.chem.tamu.edu (Bob Standaert)
Organization: Department of Chemistry, Texas A&M University
References:   1 2
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It is hard to conclude much about the relative safety of the A320 and
737 on the basis of statistics alone. We are (most fortunately)
comparing frequencies of low probability events, and the number of
events and flights is too small to produce a meaningful distinction.
What follows is an illustration of the point in my typically longwinded
style... I apologize for the length and hope that there is enough
substance to make up for it, at least in part.

Interpreting the observed frequency of improbable events is something
that must be undertaken with caution. I'll add extra caution by saying I
am not a statistician, only someone who has occasion to use statistics
and tries to be careful with them. Let's suppose we do, in fact, know
the "true" probability P that an event will occur. We can use the
Poisson distribution to calculate the probability F that we will observe
X events in N tries:

     F = [(NP)^X]*[exp(-NP)]/X!

What we will find is that many values of X are reasonably possible.
Taking Karl's criterion of the hull loss as the event of interest, let's
work from the assumption that both airliners have an identical frequency
of events, P = 1.2/Mflight. For the sake of argument, let's also lump
all of the 737's together. The probability F of observing X losses in N
flights is given in the table below. The "SUM" columns refer to the
cumulative probability; for instance, with the 737, the chance that we
would observe <70 events is 4.6%, the chance we would observe more than
100 is 4.6% (100 - 95.4), and there is a roughly 90% chance that we
would observe between 70 and 100 events. For the A320, the corresponding
range is 3-10.

|   737 (N = 70.6M)    |     |    A320 (N = 5.2 M)   |
|______________________|     |_______________________|
|                      |     |                       |
  X     F(%)     SUM            X     F(%)     SUM
----   ------   ------        ----   ------   ------
 <70              4.6           0      0.2      0.2
  70     1.2      5.8           1      1.2      1.4
  71     1.5      7.2           2      3.8      5.2
  72     1.7      9.0           3      7.9     13.1
  73     2.0     11.0           4     12.3     25.4
  74     2.3     13.2           5     15.4     40.8
  75     2.6     15.8         * 6 *** 16.0 *** 56.8 *
  76     2.9     18.7           7     14.3     71.0
  77     3.2     21.8           8     11.1     82.1
  78     3.4     25.3           9      7.7     89.9
  79     3.7     29.0          10      4.8     94.7
  80     3.9     32.8          11      2.7     97.4
  81     4.1     36.9          12      1.4     98.8
  82     4.2     41.1          13      0.7     99.5
  83     4.3     45.4          14      0.3     99.8
  84     4.3     49.8          15      0.1     99.9
* 85 *** 4.3 *** 54.1**        16      0.0    100.0
  86     4.3     58.3          17      0.0    100.0
  87     4.1     62.5          18      0.0    100.0
  88     4.0     66.5          19      0.0    100.0
  89     3.8     70.3          20      0.0    100.0
  90     3.6     73.9
  91     3.3     77.2
  92     3.1     80.3
  93     2.8     83.0
  94     2.5     85.6
  95     2.2     87.8
  96     2.0     89.8
  97     1.7     91.5
  98     1.5     93.0
  99     1.3     94.3
 100     1.1     95.4
>100            100.0

In both cases, the most probable outcome (starred) is that we will
observe the average number of events (P*N), but there are many other
outcomes that are nearly as likely, or at least reasonably likely. As
noted above, 90% of the probability falls between 70-100 events for the
737, and 3-10 for the A320, but there is still a 10% probability that we
would observe event numbers outside these ranges due to chance alone!

I started with the assumption that both birds have identical "true"
event frequencies. We can go one step further and ask the following
question: what "true" event frequencies are consistent with the observed
number of events? Let's take as our criterion that the observed number
of events must be within the middle 90% of the probability distribution
for the "true" frequency. With a little hacking around on a spreadsheet,
I came up with the following ranges (events/Mflight):

     737         A320
  ---------   ---------
  1.00-1.44   0.63-2.02

With the 737, we have a much better idea of where the "true" frequency
is because we have many more data. The A320 could be significantly
better, significantly worse, or (most likely) about the same. There just
isn't enough experience yet to pin it down so well.

I find it much more informative to discuss (or at least read while
others discuss) the specifics of individual events and what they reveal.

Regards,

Bob

--I don't speak for Texas A&M, Texas A&M doesn't speak for me, and we're
both happier for it.