What is the median value of an odd-numbered set of letters?

[snorkack]

I wonder what parts of statistics have specific terms existing for them - I see a relevant notion which would be relevant, but not sure if there is a term for it.

If variable values can be ordered then it possesses a median.
If the values can also be added then they also possesses an average.
A data set to illustrate it:
There are a bit over 7 milliard people in world.
Total wealth owned in world is 241 000 milliard US $. It is a summable variable.

Since it is summable, it possesses average. Which comes at about 34 000 US$ per head.

50 % of world population possesses about 1700 milliard US$, which is about 0,7 % of the total. This the average wealth of the second half is about US$ 500. The average wealth of the richer half is about US$ 68 000.
The value of which 50 % people are richer and 50 % are poorer is the median.

But now to illustrate my question.
1 % of people own 46 % of all wealth - 110 000 milliard US$. This makes about 1 500 000 US$ per member of that 1 %.

It would be interesting to know:
1) the actual number of richest people who own, not 46 % of all wealth, but exactly 50 % of all wealth. Which is clearly a bit bigger than 1 %, but exactly what?
2) the actual value of wealth such that people richer than that possesses 50 % of the wealth - obviously less than 1 500 000 US$, but again how much?

And my question on statistics is:
are there any established terms to call the answers to 1) and 2)? How to talk of quantiles of the sum of the values, as opposed to quantiles of the number of observations?

 

[snorkack]

Since it is summable, it possesses average. Which comes at about 34 000 US$ per head.

Or US$ 51 600 per adult.

50 % of world population possesses about 1700 milliard US$, which is about 0,7 % of the total. This the average wealth of the second half is about US$ 500. The average wealth of the richer half is about US$ 68 000.
The value of which 50 % people are richer and 50 % are poorer is the median.

Which is US$ 4000

But now to illustrate my question.
1 % of people own 46 % of all wealth - 110 000 milliard US$. This makes about 1 500 000 US$ per member of that 1 %.

And they each own more than US$ 753 000

 

[Stephen Tashi]

I wonder what parts of statistics have specific terms existing for them - I see a relevant notion which would be relevant, but not sure if there is a term for it.

If variable values can be ordered then it possesses a median.
If the values can also be added then they also possesses an average.

Assuming data is given as a set of real numbers, your question is whether there is any standard terminology that denotes how many properties of the data set are known. I don't know of such a terminology. (Given how statistics seems to love terminology, this is rather surprising!)

 

[snorkack]

Assuming data is given as a set of real numbers, your question is whether there is any standard terminology that denotes how many properties of the data set are known.

No, my question was about the specific properties I described.

 

[Simon Bridge]

Sorry for the resurrect, but I'm not sure any of the responses above actually answer the specific questions you ask, so I thought I'd have a go starting you off.

You start out very general, then narrow it down to a specific instance. I'll start out by following you, I'll try to establish some semantics, and then slightly rework your example to help answer your actual questions.

I wonder what parts of statistics have specific terms existing for them...

... very many - those properties which have a specific term to label them would be called "named" properties.
Generally a property will have a name if it is useful for it to be named: i.e. it gets talked about.

Fine points in statistics get nit-picked over ad-infinitem - so statisticians do love their terminologies. If you cannot seem to find a specific name for something, then it probably does not get talked about much, at least, not in the way you are thinking of.

But I wouldn't worry so much about what something is called so much as what it tells you.

- I see a relevant notion which would be relevant, but not sure if there is a term for it.

... then the term you are asking for would be "relevant" wouldn't it? :D

It is difficult asking about something you don't have the terminology for isn't it?

If variable values can be ordered then it possesses a median.
If the values can also be added then they also possesses an average.

Can you come up with a data-set which can be ordered but not added (so it has a median but not an average) ... or, conversely, one that can be added but not ordered?

I have a feeling that you are using non-standard terminology. i.e.
A "median" is a kind of average ... so, technically, you cannot have a median without an average. From your example, though, by "average" you seem to be referring to the "arithmetic mean" ... is that correct?

Total wealth owned in world is 241 000 milliard US $. It is a summable variable.

I have not been able to find the term "summable variable" in statistics works.
I have been able to find several different kinds of "statistical summability" though.
i.e.
http://www.sciencedirect.com/science/article/pii/S0895717708002239
http://www.journalofinequalitiesandapplications.com/content/2012/1/172

If you just mean that it is something that can be added up: then what is "total wealth" being added to: it's just one data point after all?

By context, you seem to mean that it can be divided - since you go on to find the mean wealth.

Making sure we are on the same page:
Although the mean and median are thought of a "middle values", this is not strictly the case. As you have observed, a few people may hold most of the wealth. Although half the wealth may be above the mean, half the people may not be.

Which seems to agree with:

The value of which 50 % people are richer and 50 % are poorer is the median.

... so I'm confident we understand each other.

It would be interesting to know:
1) the actual number of richest people who own, not 46 % of all wealth, but exactly 50 % of all wealth. Which is clearly a bit bigger than 1 %, but exactly what?
2) the actual value of wealth such that people richer than that possesses 50 % of the wealth - obviously less than 1 500 000 US$, but again how much?

And my question on statistics is:
are there any established terms to call the answers to 1) and 2)? How to talk of quantiles of the sum of the values, as opposed to quantiles of the number of observations?

1. if $$W=\sum_{n=1}^{P}x_n$$ ...is the total wealth of a population of $P$ people, where $x_n$ is the wealth of the $n$th person ordered by wealth.

The median wealth is: $M=x_{(P/2)}$ the [(P/2)th value of x] - 50% of the pop are above this

The mean wealth is: $W_{ave}=W/P$ - 50% of the wealth is above this value.

You want to know
$$N: W/2=W-\sum_{n=1}^{N<P} x_n$$ ... the number of rich people who, collectively, own half the world's wealth. (solve that relation for N)

I have a tickle in the back of my mind that there is a name for this value, but it eludes me.
I have a feeling the expression evaluates into something more commonly known.

2. this would be the x_Nth wealth value.
... which would be the average right?

But when you say "interesting" - consider: what would these values tell you? What is it that makes the value interesting besides idle curiosity? Could that information be represented by a more commonly discussed property?

i.e. if you represent the data as a set, you can use the language of sets to talk about them. (re: "quantiles" part of your question.)

Whatever - the above should get you started.

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More nitpicky:
"billion" is more usual than "milliard" outside of Europe - but milliard is technically more correct with less chance of confusion. Inside Europe it is still pretty rare... your use here suggests you are taking pains to be careful with your terminology which is why I do a lot of checking above.

The dollar-sign would normally go ahead of the number and certainly ahead of the country abbreviation.

Thus "$241000 milliard US" gets shortened to:

$US 241T = $241T US = 241T USD ... where the T="(short scale) trillion" or "x10¹²" ... it's neat because it matches the metric "Tera" for the same power.

241 Terabucks has a ring to it...

 

[snorkack]

Can you come up with a data-set which can be ordered but not added (so it has a median but not an average) ... or, conversely, one that can be added but not ordered?

Cannot imagine the second. The first... how about alphabet? You cannot say that A+D=B+C... but you could identify a median.

I have a feeling that you are using non-standard terminology. i.e.
A "median" is a kind of average ... so, technically, you cannot have a median without an average. From your example, though, by "average" you seem to be referring to the "arithmetic mean" ... is that correct?

Yes. I suspect that when "average" is mentioned without specifying which, it tends to be "arithmetic mean", rather than "median", "geometric mean" or some other.

If you just mean that it is something that can be added up: then what is "total wealth" being added to: it's just one data point after all?

By context, you seem to mean that it can be divided - since you go on to find the mean wealth.

Yes - it can be added and also multiplied and divided. In contrast to something like alphabet letters, which can be "ordered" but not added or divided.

Making sure we are on the same page:
Although the mean and median are thought of a "middle values", this is not strictly the case. As you have observed, a few people may hold most of the wealth. Although half the wealth may be above the mean, half the people may not be.

As I pointed out, half of the wealth may not be above mean either.

1. if $$W=\sum_{n=1}^{P}x_n$$ ...is the total wealth of a population of $P$ people, where $x_n$ is the wealth of the $n$th person ordered by wealth.

The median wealth is: $M=x_{(P/2)}$ the [(P/2)th value of x] - 50% of the pop are above this

The mean wealth is: $W_{ave}=W/P$ - 50% of the wealth is above this value.

You want to know
$$N: W/2=W-\sum_{n=1}^{N<P} x_n$$ ... the number of rich people who, collectively, own half the world's wealth. (solve that relation for N)

I have a tickle in the back of my mind that there is a name for this value, but it eludes me.
I have a feeling the expression evaluates into something more commonly known.

2. this would be the x_Nth wealth value.
... which would be the average right?

Certainly not the arithmetic mean!

But when you say "interesting" - consider: what would these values tell you? What is it that makes the value interesting besides idle curiosity? Could that information be represented by a more commonly discussed property?

i.e. if you represent the data as a set, you can use the language of sets to talk about them. (re: "quantiles" part of your question.)

Look at it this way:
who has the purchasing power in the world?
It turns out that Credit Suisse wealth report looked at the distribution of wealth between adults (a bit fewer than 5 milliards).

Then the average wealth of adults is US$ 50 000.
(If D in USD goes after US, why not $?)
The poorer half of mankind is 2,5 milliard people owning median US$ 4000 or less. The total wealth of that half is US$ 1700 milliards, so their average is US$ 680.

Now, regarding the arithmetic mean: it is the arithmetic mean between small number of people owning very large sums (10 % of mankind owns 83 % of all wealth) and large number of people owning small sums (the remaining 90 % owns just 17 %).

So we are still finding "averages" from a long but very low tail whose combined size is still small. Which is why it would seem interesting to find the value and number of set members which represents one half (or a different fraction) of the total distribution.

 

[Simon Bridge]

Just so I understand you:
There are 26 letters[*] in the (English) alphabet - there is a traditional ordering to them, so let's say that I accept that the alphabet is an ordered set.
What then is the median of this set?

As I pointed out, half of the wealth may not be above mean either.

The mean wealth does not have to be half way between the max and min wealth values no.

Certainly not the arithmetic mean!

... prove it :)
[edit]by which I mean: if you go about proving this, you will probably discover the answers to your questions.

I think you need to sort out your concepts before you can get a proper answer to your question.
I appreciate that you are trying to find the proper words for things you only have your own made-up terms for.
Suggest you try to write out what you mean as a mathematical expression.

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[*] 52 if I include upper and lower case, 62 if I include numerals.
Still even numbers so what's the median?
You can get an odd number of letters if you count the "space" as a letter - but then, why not other formatting marks like punctuation?

Naturally any odd-number collection of letters can be said to have a median - but only in the loose non-mathematical sense that any ordering of an odd number of elements has to have an element in the middle. That's not what "median" means - the median is the numerical value of the middle of the set. There need be no member of the set that has that value.

We can assign the letters a numerical value from 1-26, then the median is 13.5 (giving each letter equal weight - consider scrabble values though). However, we can certainly add these values up: i.e. value[A]+value=3=value[C].