How to assume a distribution to be normal distribution or log-normal distribution?

Thank you very much for the quick reply. So I can just use the general AVERAGE and STDEV formula? Sorry for my ignorance. So I will get three values in Normal distrubtion right? Average, stdev and the one using NORMDIST?

Originally Posted by shg

Things that are constrained to be >= 0 (like the concentration of a chemical, which cannot be < 0), cannot be normally distributed

Theoretically correct, since the normal distribution goes from -infinity to +infinity.

(So even data with negative values cannot be truly normally distributed, since +/-infinity are unattainable. )

But in actual practice, I would disagree.

For example, consider the distribution of salaries from 100,000 to 1,000,000 with a mean of 550,000 and a std dev of 56,250.

If the salaries are (or should be) normally distributed, the probability of non-positive salaries (including zero) is 7.01E-23 = NORMDIST(0,550000,56250,1). Even the probability of salaries less than 100,000 is about 6.22E-16 = NORMDIST(99999.99,550000,56260,1).

Both probabilities are so small that we do not need to be concerned. We can still say the salaries are (or can be) normally distributed within the prescribed range.

Originally Posted by k4k4shi

Thank you very much for the quick reply. So I can just use the general AVERAGE and STDEV formula?

I think you are responding to my first response, which I subsequently "deleted". I deleted the posting because the details of your description are unclear, and my original response might have compounded the confusion.

But the substance of my original response is correct.

Yes, AVERAGE and STDEV are just arithmetic calculations. They do not depend on the type of distribution.

However, the usefulness and interpretation of those statistics, especially the std dev, does depend on assumptions about the type of distribution.

For example, we might want to say that 99.99% of a larger set of data should between mean - 4*stdDev and mean + 4*stdDev.

That is based on the assumption that the larger set of data is normally distributed, and that the data that you have was randomly selected ("sampled") from the larger set of data.

Note: I suggested that you use STDEV instead of STDEVP because your assignment asks you to "estimate" the std dev. I interpret that to mean: estimate the std dev of a theoretical larger set of data based on a presumedly random sample.

Originally Posted by k4k4shi

So I will get three values in Normal distrubtion right? Average, stdev and the one using NORMDIST?

As I recall, in my original response, NORMDIST was merely one example of how we might rely on the assumption of normal distribution when we apply the AVERAGE and STDEV.

We are not limited to just those 3 statistics.

Originally Posted by shg

Thank you for the world's longest definition of "negligable"

And thank you for presenting a distinction without a difference, in the first place.

I'm sorry that I presumed you thought you were making an important point, which I thought required clarification lest it be misunderstood by k4k4shi, who obviously does not have enough grounding in statistical application to realize just how negligible your original comment was.

Originally Posted by k4k4shi

How do I "assume" it to be in normal distribution or log normal distribution?

Originally Posted by shg

If your preferred assumption is that it is log-normal, then the only way I know to measure it is via regression.

When we say data is log-normally distributed ("log normal"), we simply mean that the log of the data is normally distributed. That does not require regression to "measure it", whatever that means.

However, it does require that the log of the data is computable. That is, the data must be greater than zero.

If the data is in B1:B100, we calculate the mean and std dev of the log data simply by array-entering the following formulas (press ctrl+shift+Enter instead of just Enter):

=AVERAGE(LOG(B1:B100))
=STDEV(LOG(B1:B100))

Again, we can do those calcuations without assuming the log data is normally distributed. And the fact that we can do those calculations does not imply the log data is normally distributed.

However, if we assume the data is log-normally distributed, we might expect, for example, that 99.99% of a larger set of log data -- the log of the "population" of data from which B1:B100 is a random sample -- lies between mean - 4*stdDev and mean + 4*stdDev.

@k4k4kshi: Also see my responses #6 and #9.

Originally Posted by k4k4shi

Exact question: Assume that the distribution of internal body concentration of chemical A among Japanese can be approximated to normal distribution. Estimate its average and std.

I apologize for my incessant posting, but it continues to bother me that the assignment says to assume a normal distribution when no such assumption is necessary to calculate an average and std dev.

I wonder if by "estimate its average", the assignment means: calculate a confidence interval around the mean based on some degree of confidence (typically 95%). In other words, calculate mean +/- delta, where delta is z*stdErr or t*stdErr.

The stdErr of the mean is: STDEV(B1:B100)/SQRT(COUNT(B1:B100)).

Alternatively: STDEVP(B1:B100)/SQRT(COUNT(B1:B100)-1).

For 95% confidence, z is -NORMSINV((1-95%)/2), which is about 1.96. t is TINV(1-95%,COUNT(B1:B100)-1), which is about 1.98. The larger the sample size, the closer Student's t is to z.

Although we can calculate the CI around the mean of random samples from a non-normal distribution, it tends to require large samples. So it is indeed "better" if we can assume the data is a random sample from a normal distribution.