Normal Distribution

Hi all,

(Schweser, 2014, exam 3, afternoon session, p.160)

Q 29) Which of the following statements about the normal distribution is least accurate?

A. has a mean of zero and standard deviation of one.

B. is completely described by its mean and standard deviation.

C. is bell-shaped, with tails extending without limit to the left and to the right.

Answer: A

I thought A and B would be true and C is false? (Due to the without limit part)

A standard normal distribution has a mean of zero and standard deviation of 1. So, A is not true for all normal distributions.

A normal distribution is completely described by the mean and standard deviation (variance, naturally).

The without limit part is correct. Theoretically, the tails of the distribution extend indefinitely, and the probability under the curve gets infinitesimally small in these regions.


So . . . _ non theoretically_ they don’t?

If by non-theoretically, you mean practically, then yes?

I don’t have any idea what I mean by nontheoretically. tickersu qualified his answer with “theoretically”, so I was just tryint to ascertain what happens without that qualification.

(Personally, I despise the adverb “theoretically”. Generally, the perpetrator of this word has no idea to what theory they’re referring, so the word is meaningless, and the conclusions even more so.)

what happens without that qualification is that you need another page (theoretically infinity pages :wink: ) to illustrate the normal distribution, ie. no text book has ever replicated it with accuracy :slight_smile:

Well, based on the fact that you’re not going to observe these infinite tail limits in practice, it is more of a theoretical idea, isn’t it? My qualification was to point out that you’re not going observe these bounds in a practical scenario, but the bounds of this theoretical normal distribution are +/- infinity.

Also, feel free to correct my conclusion if it is meaningless/incorrect.

Thanks guys for the points here!

Do you know the upper and lower limits of what the mean and S.D. could represent under a normal distribution?

When you say the tails extend to infinity, the variance could be -infinity to +infinity?

The mean can be any real number; the SD must be greater than 0. The variance can definitely be very large, but infinite? I would have to think on that one and maybe even dig out an old stats textbook.

For a standard distribution, with a given mean and standard deviation it means that the probablility that an observed value is +/- infininty is greater than zero.

For example, P(X>n) > 0 for any n in the set of real numbers.