# Defining the normal distribution

I am preparing some materials for an introduction to sample size calculation for life sciences graduate students with no or very limited previous statistical experience. Right at the start I will be defining the basic distributions (normal, binomial, Poisson and possibly exponential). Mostly, this is trivial (“Poission distribution describes probability of a given number of events occurring in a specified interval of time or space”, “binomial distribution describes number of results of a particular type when a process with two possible outcomes is samped a certain number of times”, etc) but I’ve noticed that I normally describe the general shape of the distribution and give some examples of the types of real-world processes that result in a normal distribution (heights of a group, that sort of thing) rather than define it in terms of the general class of processes that the distribution describes. Does anyone have any experience of what works best (in terms of student comprehension) in this situation?

The most obvious way I can see is to do it in terms of the central limit theorem – something like ‘the normal distribution describes the distribution of the average of groups of measurements of most random variables. The distribution of these averages approaches the normal distribution more closely as the number of groups increases.’ Any thoughts on this as a working definition? I will then give some real-world examples, but I want a stand-alone definition first. I don’t think I need to go into things like finite variance given the audience (and similarly I think they will interpret ‘average’ as ‘arithmetic mean’ by default) – I want to keep the language as accessible as possible.

(I'm aware that real-world data of this type is not necessarily normally-distributed in practice.)

• You could say more precisely " ... mean of many variables where each variable only has a low weight in the mean". Then you can use examples like heigth (normal, many influences none of them dominating) and weight (usually not normal, probably many influences, but some of them dominating). Apr 21, 2017 at 17:23

You could say something like: The normal distribution is used when data consists of real number measurements, especially when we can think conceptually of the data as mean of many variables where each variable has only a low weight in the mean. Then you can use examples like height (normal, likely very many influences none of them dominating), errors of measurement when using some measuring instrument like a length stick (often close to normal, especially for very precise measurement. With more lousy measurement, the probability of large errors often are larger than accommodated by the normal distribution, see Why should we use t errors instead of normal errors?) and weight (not well approximated by a normal, probably some few variables may be dominating).

I would say that a distribution is normal if it can be obtained from the Standard Normal from the transformation inverse to standardization:

If we know a random variable X to be normal $$N(\mu, \sigma)$$, then :

    $$Z=(x- \mu)/ \sigma$$


Is standard normal, i.e., $$N(0,1)$$

Conversely, I believe, any normal Y can be obtained from the standard by the inverse transform :

           $$Y=Z\sigma +\mu$$

• That neatly reduces the question to describing and explaining the standard Normal distribution--but it stops short of actually giving an answer!
– whuber
Jul 12, 2019 at 21:27
• @whuber: Can you think of a normal distribution that cannot be obtained this way? What else can we use that is viable : a mgf? Seems clunky. EDIT: I guess we can test for skewness/symmetry, approximations to the 1-2-3 68/95/99 % rule.
– MSIS
Jul 12, 2019 at 21:41
• The point is that you don't supply any definition or characterization of the standard Normal distribution. It's a little like describing a new kind of fruit, say a "jackberry," by stating that it is any fruit that regardless of its size looks like a standard jackberry. Although that might be correct, it's useless.
– whuber
Jul 15, 2019 at 13:50