# [R] A somewhat off the line question to a log normal distrib

(Ted Harding) Ted.Harding at nessie.mcc.ac.uk
Thu Dec 2 17:52:46 CET 2004

```On 02-Dec-04 Thomas Lumley wrote:
> On Thu, 2 Dec 2004 Ted.Harding at nessie.mcc.ac.uk wrote:
>
>> Hmm, perhaps you should think again! If X and Y have log-normal
>> distributions (mathematically exactly), then (X+Y)/2 does not
>> (mathematically) have a log-normal distribution -- still less
>> the arithmetic mean of some 30 such variables. So one wonders
>> what the basis of his "explanation" was.
>
> The original poster's boss did not (as far as we know) claim
> that the measurements were either independent or identically
> distributed.  While the problem would be simpler if they were,
> there is no guarantee that the answer would be remotely relevant.

Not quite sure of your point here, Thomas. I certainly wasn't
writing on the basis that the boss had claimed that they were
either independent or identically disitributed, and the paragraph
you quote was in reposnse to:

"The aformentioned daily measurements follow a log-normal
distribution when considered over the course of a year.
Okay. He also tried to explain me that the monthly means
(based on the daily measurements) must follow a log-normal
distribution too then over the course of a year."

which I interpreted as arguing that "if daily data log-normal,
then monthly means must consequently be log-normal", i.e. that
the mean of log-normals is log-normal; and I was simply pointing
out that this is a false implication (which would be the case
even if the data are neither independent nor identically distributed,
except in the extreme case where they are all copies of the one
log-normal variable).

Granted I later used i.i.d log-normals as examples; but then
pointed out that the mean of log-normals could remain sufficiently
skew that a log-normal could still be a useful distribution to

Of course the boss may in reality have argued that the distribution
of monthly means was, as a matter of fact, skew, and therefore
log-normal would be approriate; but Siegfried Gonzi stated that,
according to his observations, these were more like Gaussian.
So I came to the conclusion that my first interprtation was
appropriate. Maybe Siegfried could clarify?

Best wishes,
Ted.

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Date: 02-Dec-04                                       Time: 16:52:46
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