# [R] Re: A somewhat off the line question to a log normal distribution

Phineas Campbell pcampbell at econ.bbk.ac.uk
Thu Dec 2 13:42:03 CET 2004

```It has always been my understanding that an arbitrary lognormal distribution
has a sufficient quantity of finite moments to ensure a CLT holds, it is not
uniquely defined by these moments.  Thus although sums of IID lognormal
variates will converge to a normal distribution we cannot know, a priori,
what the parameters of this distribution will be.

If a distribution is strictly positive and has sufficient moments then the
sums of logs of the variables will converge to a normal distribution, thus
in estimating parameters that are closely related to the mean there would
appear to be little to loose by logging the data.

Phineas Campbell

-----Original Message-----
From: r-help-bounces at stat.math.ethz.ch
[mailto:r-help-bounces at stat.math.ethz.ch]On Behalf Of Vito Ricci
Sent: Thursday, December 02, 2004 10:08 AM
To: r-help at stat.math.ethz.ch
Subject: [R] Re: A somewhat off the line question to a log normal
distribution

Dear Siegfried,

I believe your boss is wrong saying that:
>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.

every statistician know that increasing the sample
size the sample distribution of the mean is proxy to a
gaussian distribution (Central Limit Theorem)
independently from the original distribution of data

See this:

The Central Limit Theorem is a statement about the
characteristics of the sampling distribution of means
of random samples from a given population. That is, it
describes the characteristics of the distribution of
values we would obtain if we were able to draw an
infinite number of random samples of a given size from
a given population and we calculated the mean of each
sample.

The Central Limit Theorem consists of three
statements:

 The mean of the sampling distribution of means is
equal to the mean of the population from which the
samples were drawn.

 The variance of the sampling distribution of means
is equal to the variance of the population from which
the samples were drawn divided by the size of the
samples.

-->  If the original population is distributed
normally (i.e. it is bell shaped), the sampling
distribution of means will also be normal. If the
original population is not normally distributed, the
sampling distribution of means will increasingly
approximate a normal distribution as sample size
increases. (i.e. when increasingly large samples are
drawn) <--

So results you got are just in this way!

I think your boss doesn't know well statistics!

Regards
Vito

You wrote:

Hello:

Oh yes I know it isn't so much related to R, but I
gather there are a
lot of statisticians reading the mailing list.

My boss repeatedly tried to explain me the following.

==
Lets assume you have got daily measurements of a
variable in natural
sciences. It turned out that 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.
==

I somehow get his explanation.

But I have measurements which are log-normal
distributed when evaluated
on a daily basis over the course of a year  but they
are close to a
Gaussian distribution when considered under the light
of monthly means
over the course of a year.

Is such a latter case feasible. And if not why.

Regards,
Siegfried Gonzi

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