# Interpretation of mean in right statistical context

It is very basic question on this platform. I always found difficulty in interpreting mean to layman person. After reading few answers on SE, I found mean definition as measure of central tendency. But why it is called "central tendency"? Does tendency have specific meaning in statistical context and what does does this word represent?

My second question is how mean can be interpreted when random variable can only take two possible outcomes? Because, in such a cases it does not matter how long we continue our experiment, we will never get sample value equal to its mean value. For example, a random variable, $X$, can take two possible value either -1 or 1 with probability 0.5 each. In this case, the mean is 0. But the truth is that how long we continue to sample we will never get 0, so what 0 (mean) indicate in this case and what is relevance of it ?

I found on certain sites that interpret mean as measure of central tendency where the data seems to cluster around. But in case of uniform distribution there is no clustering of data around any specific value. So how this definition of mean is right?

My simple question is how mean[arithmetic mean] can be interpreted in simple right statistical context?

I'm not quite sure why "central tendency" was used, but I preferred "measure of center" when teaching my students. It was a way to quantify the center of the data. "Central tendency" seemed to be a more complicated term for no real reason when explaining statistics to laymen.

If a random variable can take on two values, say 0 and 1, the mean will be somewhere between 0 and 1 (theoretically inclusive). While the mean will never be a value we can observe, the mean is then the proportion of events when the random variable takes on the value 1. If you define your binary outcomes to be 0 and 1, your "mean of 0" problem is solved. If you define your outcomes to be -1 and 1, this becomes more troublesome and must be interpreted within the context of the problem.

This should extend to multiple outcomes, conditional on the fact that the random variable's outcomes lie on an interval scale. (i.e. The mean of one roll of dice is 3.5.)

• They use this "central tendency" language in high schools these days. I also think it's confusing. Imagine Beta distribution with $\alpha=\beta=0.5$. Its mean doesn't fit into intuitive understanding of "central tendency", then you have to explain what do you mean by central tendency and so on – Aksakal Feb 10 '16 at 20:49
• They also teach calculus before statistics in most high schools. :) I think it's just silly - the term "measure of central tendency" is more complicated than "measure of center." We shouldn't treat introductory statistics courses as if its students are bound for rigorous statistical theory courses; the goal should be to articulate statistics at a practical level while providing a solid understanding of statistics. – Matt Brems Feb 10 '16 at 20:59
• In my days statistics was not taught in school, but calculus was. I was surprised and delighted to see they teach stats in high school these days (elective), but calculus is elective too. So, you can finish a school not knowing either of them. – Aksakal Feb 10 '16 at 21:06
• "in high schools" could mean anywhere in the world: which country/countries are being referred to (in an international forum)? – Nick Cox Feb 10 '16 at 21:27
• I meant to say "American high schools." – Matt Brems Feb 10 '16 at 21:50

I'm not sure where exactly mean comes from but the way I understood it was from physics. Draw a uniform distribution [0,1] on a piece of card board, then cut it out. Then put it on the pencil so that it balances perfectly. The pencil must be perpendicular to X axis. The mean of this distribution is where the pencil is. It'll be exactly at 0.5. Obviously you can't really cut out Gaussian distribution, because its X goes to infinity, but you can do a thought experiment, and see where the edge ends up. That would be the mean again.

In physics this is analogous to a center of mass. I think that's why the probability density function (PDF) is called a density function. It's similar to density of physical bodies.

• Can you please explain how this answer to question asked? – Neeraj Feb 11 '16 at 14:57
• Mean to the random variable is like a center of mass to a physical body, where probability density is like a mass density of a body. – Aksakal Feb 11 '16 at 15:51