Question on Normal Distributon.

[sudeepdoon]

Hi David,

I was going through sample questions and came across the following:

Which of the following statements about a normal distribution is least accurate?

A) The mean, median, and mode are equal.
B) The mean and variance completely define a normal distribution.
C) A normal distribution has excess kurtosis of three.
D) Approximately 68% of the observations lie within +/- 1 standard deviation of the mean.

The first question that came to my mind was : Is this a Valid question, because how can you comment on "A" and "B" if the distribution is normal distribution and is not specified if it is a standard normal..

and is "kurtosis" and "excess kurtosis" the same.

If yes then we cannot comment on "C"also till when we dont assume standard normal

same is for the fourth...

so if I try to answer the question I dont get the answer.. But when I look at the answer by the author... It says :

The correct answer was C) A normal distribution has excess kurtosis of three.
Even though normal curves have different sizes, they all have identical shape characteristics. The kurtosis for all
normal distributions is three; an excess kurtosis of three would indicate a leptokurtic distribution. The other
choices are true.

Please comment !

 

[David Harper CFA FRM]

Hi Sudeep,

It's correct. The standard normal is merely a transformed normal, any normal is symmetrical (skew = 0) with "normal" kurtosis. The symmetry imples mean = median = mode. (A is true for normal, or for that matter, the student's t)

B is true: this is the elegance of normal, we write a variable is approximately normal as ~ N(mean, variance) where standard normal is but a special case ~N(0,1). B is true b/c we don't need third moment (skew) nor fourth moment (tail property)

C is a little skeaky: any normal has kurtosis = 3 or, equivalently, excess kurtosis = 0. Excess kurtosis = kurtosis - 3.

D is true because it does not depend on the units, it is true of the normal (s.d. = 1) or any other normal (s.d.. = x)

Hope this helps, David