That highly depends on your definition of “average”. In the normal, even academic, speech “average” means arithmetic mean. The median is a location parameter of a distribution (together with the mean and the mode).
In a great deal of “normal” speech, I’m not convinced “average” has a well enough defined meaning to say that it means specifically the arithmetic mean or the median or any other particular average.
Sure, the median is a location parameter of a distribution. … Well, actually, I think this is imprecise. When you have a family of distributions that’s closed under translation, you can parameterize them by a location parameter (and if necessary some other parameters), and you might happen to take the location parameter to equal the median (which might or might not equal the mean and/or the mode). But the median is meaningful outside the context of distributions with location parameters.
Anyway, let’s stipulate that “the median is a location parameter of a distribution”. So what? You surely can’t be saying that that means the word “average” shouldn’t be used of it, since—as you even said—the mean is equally “a location parameter of a distribution”. Perhaps you mean that the median is a location parameter and nothing else; that it has no interest or importance beyond its use as a location parameter. But that’s obviously false.
(I agree that “the average” means the arithmetic mean much more often than it means any other specific measure of central tendency. I think it would be better if GWWC had said “median” rather than “average”, and more generally I think it would be better if almost all uses of “average”—especially those referring to some actual calculation—said explicitly what sort of average was meant. But I don’t think it is wrong to use the word “average” to mean the median, especially when you’re explicit about doing so.)
I agree that “the average” means the arithmetic mean much more often than it means any other specific measure of central tendency.
Do you, or are you just being agreeable?
I think the majority of usage is a vague confusion between median and mode. Maybe it means mean more than any other precise meaning, but most of the time it definitely doesn’t mean mean.
Yes, I really do think that, but I think you may have misunderstood what that is :-).
I agree (with you) that most of the time the word “average” doesn’t denote any specific kind of, er, average. And I agree that, in so far as people using it that way have any specific idea in mind, that idea probably has more of “median” and “mode” in it than of “arithmetic mean”. But I think when it does denote something specific it’s much more often the arithmetic mean than anything else. And that’s what I was saying.
“Specific” isn’t very specific. If you use it so narrowly as to only include examples where people have actually done a calculation, it means mean more than it means anything else, but not “much more often.” But I think that there are a lot of broader meanings of “specific” under which mean loses.
You surely can’t be saying that that means the word “average” shouldn’t be used of it
Why, but I am saying this!
There must be a misunderstanding. “That”, in what I wrote, is the fact (kinda) that the median is a location parameter. This fact is also true of the mean. Therefore, “that” cannot be justification for not calling the median “average” unless it is also justification for not calling the mean “average”. But your whole argument is that the mean should be called “average” and the median shouldn’t.
Are you saying the word “average” means “a location parameter of a distribution”?
No (and I don’t really understand how you could get that idea from anything I wrote). I am saying:
The word “average” is used in a variety of ways. I’d much prefer to see it used much less, and more specific terms used instead.
When it is used with a specific meaning, it is (as you say) usually used to mean the arithmetic mean but (I say) it can also legitimately be used to mean other measures of central tendency.
One should say which one.
(I didn’t say this, but:) I don’t claim to have a precise definition of the characteristics a thing should have for it to be reasonable to call it an average. In practice, since children are taught in school—at least where I come from—that “the” “three” averages are the mean, the median and the mode, it is probably best to avoid the term for other kinds of average unless there’s an especially strong reason.
If you are being explicit, “median” is a perfectly fine word.
As I already said, in these exact words: I think it would be better if GWWC had said “median” rather than “average”. But the question isn’t whether “median” is a good word to use—we are agreed that it is—but whether “average” is a legitimate word to use. I say it is; you have offered no actual grounds for disagreeing with that. Do you disagree, or are you just arguing for the sake of arguing?
Yes, I think so. We seem to have persistent difficulties in being clear to each other.
Your line of reasoning looks to me like this: We can call small felines (mean) pussycats (average). The small felines are mammals (location parameters). Ninjas (medians) are also mammals. Therefore we can can call ninjas pussycats.
Do you disagree
Yes, and I thought I was pretty explicit about that:
Why, but I am saying this!
I wasn’t taught in school that there are three averages. To me “average” is a colloquial term for the mean with the implied handwaviness of “something something middle, we don’t care to specify precisely”. I do not think that that the word “average” should be used in the meaning of “median” (or “mode”).
Your line of reasoning looks to me like this: [...]
No. I am not saying “We can call ninjas pussycats because they are mammals”. I am saying “The fact that ninjas are mammals is not a reason not to call them pussycats”. There are other reasons for not calling ninjas pussycats, and those (not the fact that ninjas are mammals) are why we shouldn’t call ninjas pussycats. Which is why I was puzzled that you wrote “In normal, even academic, speech, ‘pussycat’ means Felis catus. A ninja is a mammal (as is a dog or cat).” And you didn’t state any actual reasons for not calling ninjas pussycats.
I wasn’t taught in school that there are three averages.
Fair enough; I was and my 10-year-old daughter was. (For the avoidance of doubt, I am not saying ”… and therefore that is the correct usage” but ”… which tells us something about how the term is likely to be understood by generally informed readers without specialist knowledge of statistics”. Of course schools in different places may do different things.)
To me “average” is a colloquial term for the mean with the implied handwaviness [...]
Here’s the Shorter Oxford[1]. Its meaning I is an older but obscure one to do with shipping.
II transf.4 The determination of a medial estimate or arithmetic mean. [...] 5 The generally prevailing rate, degree, or amount; the ordinary standard; the arithmetic mean. [...]
(I promise the bits I have omitted don’t change the meaning or implications of what I quoted.) That word “medial”, as defined in the same dictionary, has the same double use: it can mean specifically “equal to the arithmetic mean” but can also mean “typical”, “central”, “kinda in the middle”, etc.
As a further indication of how the word is used casually by a mathematically literate writer, here’s an extract from Darrell Huff’s famous “How to lie with statistics”:
When you are told that something is an average you still don’t know very much about it unless you can find out which of the common kinds of average it is—mean, median or mode.
This sort of usage really isn’t uncommon, and it’s why I think saying “average” when you mean the median (or even, for nice unimodal distributions, the mode) is reasonable—at least if, as GWWC did, you say somewhere what sort of average you are using.
[1] For the avoidance of doubt: Not because I think dictionaries determine meanings, but because good dictionaries record actual usages; the SOED is a very good dictionary.
If you are being explicit, “median” is a perfectly fine word. Shorter than “average”, too.
The problem is that if you write for a general audience many people might not know what median means. I think in the average person the words that are chosen make the person think of the right concept. I think that the “average person” think’s that the “average citizen of the world” has a “average income”.
If I speak of what the “average person on the street” thinks then the word average doesn’t equal “sum/amount”. The word is understand to point to to a quality that’s not defined by a fixed mathematical formula. It takes math training for people to associate the word with the fixed.
In our statistics for bioinformatics class the general idea that they taught us was that it’s usually a bad idea to use the straight arithmetic mean as is, as using it means one measurement errors can throw of your whole data set. Data-cleaning is usually needed to get useful statistics.
When I see the word average I don’t associate it with a specific formula but with ‘we want to know a statistics that represents “the middle” of a data set’. A middle that’s appropriately calculated for the context in question.
In think that’s the sense that most people who are not well educated in math use. They don’t focus on a specific formula.
The notion that the GWWC calculator is firmly aimed at mathematical illiterates has the slight problem that they put a note right there which says “We use equivalised income” with a Wikipedia link.
So you are saying they bothered to explain “equivalised” but didn’t bother to explain “average”?
I think it’s quite obvious which what’s meant with “average”, if you talk about the fact that many people are poor. On the other hand it’s less obvious what’s meant with income.
This is true. For skewed distributions (and the income distribution is quite skewed) mean and median are different and if you have to choose you pick the one that serves your purposes better. For the comparisons of population well-being the median is, indeed, the preferred metric.
In the normal, even academic, speech “average” means arithmetic mean.
In mathematical speech it doesn’t.
As far as normal speech I’m not sure. If you ask a group of people how much the average wealth of the people in a bar changes when Bill Gates walks into a city of 10,000 citzens, I’m not sure that a majority will tell you that the average wealth shoot up a great deal.
But saying “the global average” suggests a meaning of “average” such that there is only one of. So it cannot generically mean “measure of central tendency” there.
In colloquial language, an average is the sum of a list of numbers divided by the number of numbers in the list. In mathematics and statistics, this would be called the arithmetic mean. In statistics, mean, median, and mode are all known as measures of central tendency.
Basically it says that “average” is not a mathematical or statistical term but means “arithmetic mean” which is the proper expression to use in math/stat context.
The median is just as good an average as the mean.
That highly depends on your definition of “average”. In the normal, even academic, speech “average” means arithmetic mean. The median is a location parameter of a distribution (together with the mean and the mode).
In a great deal of “normal” speech, I’m not convinced “average” has a well enough defined meaning to say that it means specifically the arithmetic mean or the median or any other particular average.
Sure, the median is a location parameter of a distribution. … Well, actually, I think this is imprecise. When you have a family of distributions that’s closed under translation, you can parameterize them by a location parameter (and if necessary some other parameters), and you might happen to take the location parameter to equal the median (which might or might not equal the mean and/or the mode). But the median is meaningful outside the context of distributions with location parameters.
Anyway, let’s stipulate that “the median is a location parameter of a distribution”. So what? You surely can’t be saying that that means the word “average” shouldn’t be used of it, since—as you even said—the mean is equally “a location parameter of a distribution”. Perhaps you mean that the median is a location parameter and nothing else; that it has no interest or importance beyond its use as a location parameter. But that’s obviously false.
(I agree that “the average” means the arithmetic mean much more often than it means any other specific measure of central tendency. I think it would be better if GWWC had said “median” rather than “average”, and more generally I think it would be better if almost all uses of “average”—especially those referring to some actual calculation—said explicitly what sort of average was meant. But I don’t think it is wrong to use the word “average” to mean the median, especially when you’re explicit about doing so.)
Do you, or are you just being agreeable?
I think the majority of usage is a vague confusion between median and mode. Maybe it means mean more than any other precise meaning, but most of the time it definitely doesn’t mean mean.
Yes, I really do think that, but I think you may have misunderstood what that is :-).
I agree (with you) that most of the time the word “average” doesn’t denote any specific kind of, er, average. And I agree that, in so far as people using it that way have any specific idea in mind, that idea probably has more of “median” and “mode” in it than of “arithmetic mean”. But I think when it does denote something specific it’s much more often the arithmetic mean than anything else. And that’s what I was saying.
“Specific” isn’t very specific. If you use it so narrowly as to only include examples where people have actually done a calculation, it means mean more than it means anything else, but not “much more often.” But I think that there are a lot of broader meanings of “specific” under which mean loses.
Maybe, but they happen not to be the meaning of “specific” I was using when I wrote the words in question.
(Of course you needn’t care about that. The author is dead, etc.)
It’s incomplete. You don’t need to have a distribution to have a median—all you need is a set of numbers.
Why, but I am saying this!
Are you saying the word “average” means “a location parameter of a distribution”? I wonder how many people can you find to agree with that.
If you are being explicit, “median” is a perfectly fine word. Shorter than “average”, too.
There must be a misunderstanding. “That”, in what I wrote, is the fact (kinda) that the median is a location parameter. This fact is also true of the mean. Therefore, “that” cannot be justification for not calling the median “average” unless it is also justification for not calling the mean “average”. But your whole argument is that the mean should be called “average” and the median shouldn’t.
No (and I don’t really understand how you could get that idea from anything I wrote). I am saying:
The word “average” is used in a variety of ways. I’d much prefer to see it used much less, and more specific terms used instead.
When it is used with a specific meaning, it is (as you say) usually used to mean the arithmetic mean but (I say) it can also legitimately be used to mean other measures of central tendency.
One should say which one.
(I didn’t say this, but:) I don’t claim to have a precise definition of the characteristics a thing should have for it to be reasonable to call it an average. In practice, since children are taught in school—at least where I come from—that “the” “three” averages are the mean, the median and the mode, it is probably best to avoid the term for other kinds of average unless there’s an especially strong reason.
As I already said, in these exact words: I think it would be better if GWWC had said “median” rather than “average”. But the question isn’t whether “median” is a good word to use—we are agreed that it is—but whether “average” is a legitimate word to use. I say it is; you have offered no actual grounds for disagreeing with that. Do you disagree, or are you just arguing for the sake of arguing?
Yes, I think so. We seem to have persistent difficulties in being clear to each other.
Your line of reasoning looks to me like this: We can call small felines (mean) pussycats (average). The small felines are mammals (location parameters). Ninjas (medians) are also mammals. Therefore we can can call ninjas pussycats.
Yes, and I thought I was pretty explicit about that:
I wasn’t taught in school that there are three averages. To me “average” is a colloquial term for the mean with the implied handwaviness of “something something middle, we don’t care to specify precisely”. I do not think that that the word “average” should be used in the meaning of “median” (or “mode”).
No. I am not saying “We can call ninjas pussycats because they are mammals”. I am saying “The fact that ninjas are mammals is not a reason not to call them pussycats”. There are other reasons for not calling ninjas pussycats, and those (not the fact that ninjas are mammals) are why we shouldn’t call ninjas pussycats. Which is why I was puzzled that you wrote “In normal, even academic, speech, ‘pussycat’ means Felis catus. A ninja is a mammal (as is a dog or cat).” And you didn’t state any actual reasons for not calling ninjas pussycats.
Fair enough; I was and my 10-year-old daughter was. (For the avoidance of doubt, I am not saying ”… and therefore that is the correct usage” but ”… which tells us something about how the term is likely to be understood by generally informed readers without specialist knowledge of statistics”. Of course schools in different places may do different things.)
Here’s the Shorter Oxford[1]. Its meaning I is an older but obscure one to do with shipping.
(I promise the bits I have omitted don’t change the meaning or implications of what I quoted.) That word “medial”, as defined in the same dictionary, has the same double use: it can mean specifically “equal to the arithmetic mean” but can also mean “typical”, “central”, “kinda in the middle”, etc.
As a further indication of how the word is used casually by a mathematically literate writer, here’s an extract from Darrell Huff’s famous “How to lie with statistics”:
This sort of usage really isn’t uncommon, and it’s why I think saying “average” when you mean the median (or even, for nice unimodal distributions, the mode) is reasonable—at least if, as GWWC did, you say somewhere what sort of average you are using.
[1] For the avoidance of doubt: Not because I think dictionaries determine meanings, but because good dictionaries record actual usages; the SOED is a very good dictionary.
The problem is that if you write for a general audience many people might not know what median means. I think in the average person the words that are chosen make the person think of the right concept. I think that the “average person” think’s that the “average citizen of the world” has a “average income”.
People who don’t know what median is will certainly understand “average” as “sum up all the incomes and divide by the number of people”.
Under the assumption that your audience doesn’t know what a median is, using the word “average” to refer to median would be deliberately misleading.
If I speak of what the “average person on the street” thinks then the word average doesn’t equal “sum/amount”. The word is understand to point to to a quality that’s not defined by a fixed mathematical formula. It takes math training for people to associate the word with the fixed.
In our statistics for bioinformatics class the general idea that they taught us was that it’s usually a bad idea to use the straight arithmetic mean as is, as using it means one measurement errors can throw of your whole data set. Data-cleaning is usually needed to get useful statistics.
When I see the word average I don’t associate it with a specific formula but with ‘we want to know a statistics that represents “the middle” of a data set’. A middle that’s appropriately calculated for the context in question. In think that’s the sense that most people who are not well educated in math use. They don’t focus on a specific formula.
The notion that the GWWC calculator is firmly aimed at mathematical illiterates has the slight problem that they put a note right there which says “We use equivalised income” with a Wikipedia link.
So you are saying they bothered to explain “equivalised” but didn’t bother to explain “average”?
I think it’s quite obvious which what’s meant with “average”, if you talk about the fact that many people are poor. On the other hand it’s less obvious what’s meant with income.
Not to me it isn’t. I would normally take it to mean the total divided by the number of people, not the 50th percentile.
In my experience, most articles comparing income use median, because that’s the value that makes more sense in this context.
This is true. For skewed distributions (and the income distribution is quite skewed) mean and median are different and if you have to choose you pick the one that serves your purposes better. For the comparisons of population well-being the median is, indeed, the preferred metric.
If you demand a definition from the median user, the definition would be the mean. But the actual usage is closer to the median or mode.
In mathematical speech it doesn’t.
As far as normal speech I’m not sure. If you ask a group of people how much the average wealth of the people in a bar changes when Bill Gates walks into a city of 10,000 citzens, I’m not sure that a majority will tell you that the average wealth shoot up a great deal.
Would you like to go ask Wolfram Alpha about it?
My math books and stat classes used to define average as the hypernym of mean and mode. Wikipedia has the same terminology.
Wolfram Alpha says that it assumes you mean the “arithmetic mean”. It’s likely useful to make that assumption but that doesn’t mean it’s the only way.
Today I learned the words “hypernym” and “hyponym”!
(Wikipedia: “Hyponymy and hypernymy”; oxforddictionaries dot com: “hypernym”, “hypernymy”, “hyponym”, “hyponymy”.)
Related useful words are meronym and holonym.
But saying “the global average” suggests a meaning of “average” such that there is only one of. So it cannot generically mean “measure of central tendency” there.
English Wikipedia says:
Basically it says that “average” is not a mathematical or statistical term but means “arithmetic mean” which is the proper expression to use in math/stat context.