Understanding the Mean: Exploring its Definition and Significance

[WiFO215]

I know what the definition of a mean is, but I'd like to know what it actually means. Let me give you an example. Suppose I have a regular dice, then the mean that I calculate using the normal process seems to say 3.5, although the dice will never realize that value. So what does it explain here?

What would the mean be if I had a dice with any 6 colors from VIBGYOR? In such a case, the mean isn't even defined properly is it? If I assigned each of the colors a number from 1-6, then I'd get a mean of 3.5 again which makes absolutely no sense. Odd to say that the mean in this case is turquoise. So what does it mean in this case?

 

[CRGreathouse]

I know what the definition of a mean is, but I'd like to know what it actually means. Let me give you an example. Suppose I have a regular dice, then the mean that I calculate using the normal process seems to say 3.5, although the dice will never realize that value. So what does it explain here?

One interpretation: as the number of rolls becomes large, the expected total approaches the number of rolls times the mean.

What would the mean be if I had a dice with any 6 colors from VIBGYOR? In such a case, the mean isn't even defined properly is it? If I assigned each of the colors a number from 1-6, then I'd get a mean of 3.5 again which makes absolutely no sense. Odd to say that the mean in this case is turquoise. So what does it mean in this case?

Colors aren't cardinal, so you can't give a mean. They also aren't ordinal, so you can't give a median. But you can give a mode, unhelpful as it may be in this case.

That's a natural hierarchy to me: as you go to less and less ordered sets, you can use fewer and fewer methods.

 

[bpet]

I know what the definition of a mean is...

To be absolutely clear, what definition are you using?

 

[WiFO215]

Good grief. I hardly even remember starting this topic. I don't even know why I started it. I posted it just before going to sleep. But let's continue anyway. I've never heard the terms ordinal and cardinal, so I might learn something new.

@bpet : I am using the definition $$\overline{x}$$ = $$\sum x_{i}p(x)$$ where the x_i are the realizations of a random variable X.

 

[HallsofIvy]

So the "mean" is the weighted average of the possible values, weighted by their probability.

If you are talking about a discrete problem (as you are since you use sum, not integral), you can think of each possible outcome as having "multiplicity" according to its probability. In that case, the mean is just the arithmetic average of all possible outcomes.

 

[bpet]

I don't even know why I started it...
I am using the definition $$\overline{x}$$ = $$\sum x_{i}p(x)$$ where the x_i are the realizations of a random variable X.

Ha, no it was a good question. You're on the right track - a random variable is an abstract object that maps an event space to real numbers and needn't have any physical meaning. For your coloured dice example the events (colours) could all be assigned to zero and the mean would be zero. Take care not to confuse the observed mean (sample mean) with the theoretical mean (expected value). Good luck with your studies!

 

[WiFO215]

For your coloured dice example the events (colours) could all be assigned to zero and the mean would be zero. Take care not to confuse the observed mean (sample mean) with the theoretical mean (expected value).

1. How/ why can you map all the colors to zero?
2. What is observed mean and sample mean? The only mean I know of is the one I posted above and it's analog for the continuous case.