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Locaha comments on Open Thread for January 17 - 23 2014 - Less Wrong Discussion
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I want to study probability and statistics in a deeper way than the Probability and Statistics course I had to take in the university. The problem is, my mathematical education isn't very good (on the level of Calculus 101). I'm not afraid of math, but so far all the books I could find are either about pure application, with barely any explanations, or they start with a lot of assumptions about my knowledge and introduce reams of unfamiliar notation.
I want a deeper understanding of the basic concepts. Like, mean is an indicator of the central tendency of a sample. Intuitively, it makes sense. But why this particular formula of sum/n? You can apply all kinds of mathematical stuff to the sample. And it's even worse with variance...
Any ideas how to proceed?
What do you want to learn?
Do you want to learn to do statistical analysis with a tool like R and interpret data? Do you want to learn mathematical axioms and theorems about probability and statistics and how to prove them.
Depending on who you ask it's not. sum/n is the arithmetic mean. There are also other mean's like the geometric mean and the harmonic mean. Depending on the context different mean's can be used. It's just a convention that one usually means the arithmetic mean if one says mean. It has the advantage that's relatively easy to calculate by hand and therefore people liked it. It's not complicated to do mathematical proofs with it.
One of the key reasons why few people use robust statistics is that the math is more complicated.
This is even a political issue. In soviet Russia the usage of the arithmetic mean was very looked down upon. People were supposed to use the median. Sometimes that communist thought got in the way for cases where the arithmetic mean was really appropriate.
When it comes to learning notation. Anki is quite good.
That, um, sounds like an urban legend to me. Links?
Unfortunately I don't have a link to a source available. It might be an urban legend from an untrustworthy source. A bunch of my knowledge of statistics comes from formal education without online sources but that doesn't even mean that it's necessarily trustworthy as I wouldn't be surprised if my statistics professor would have a retold a urban myth about a case like this.
Given how Russians handled the issue of Mendelian genetics I wouldn't be too surprised if it's true ( http://en.wikipedia.org/wiki/Trofim_Lysenko ).
If you want to convince people that the communist economic model is better than the Western one, thinking in terms on median income instead of mean income as important helps. I think I picked up that meme somewhere in the context of possibilities of shaping data to your liking.
Means, motive and capabilities are all there.
Income (and wealth) in societies tends to be distributed according to a power law. That makes the mean a bad estimator regardless of your ideology. The Western economic literature almost universally uses the median when discussing income and wealth comparisons.
Given that the utility of money is (assumed to be) logarithmic, what I'd be curious to know is the geometric mean income of countries.
I am sure Google can point you in the right direction.
I see a lot of data expressed in GDP per capita to compare the wealth of different nations. http://www.gapminder.org/ for example uses it. The CIA Worldbook does so as well.
If the economics literature really does things different this seems to be a neat way that the CIA is actually using to push it's policy agenda.
If you have nearly data over the median income of countries all over the world over the last 100 years I would be interested in the data set.
GDP per capita is the aggregated Gross Domestic Product of the entire country divided by its population. It says nothing about income or wealth distribution within this country. It is also NOT the mean of personal income in that country.
GDP per capita basically tells you how much does a country produce, normalized for its population.
<rolls eyes>
I don't and I doubt it exists. You can find estimates of median income for developed countries during the last couple of decades easily enough in the usual places, but beyond that the data is likely to be sparse to absent.
If you don't have the data how would you go about comparing the wealth of different countries based on it? I don't see how those claims fit together.
I wouldn't. At least until the concept of "wealth of a country" gets defined.
But to answer your question, reread the last paragraph, particularly the part which starts "you can find...".
While the basic fact is that Christian is wrong, I think your response is an over-reaction. For example, the main point of PPP is to use GDP per capita as a typical income.
Why do you think so? As far as I am concerned, the main point of PPP is to adjust the FX rates to make comparisons (of incomes, costs, living standards, etc.) between countries more meaningful.
PPP has nothing to do with the relationship between GDP per capita and personal (or household) income.
Yes, PPP is logically independent of GDP, but PPP GDP per capita is quite popular, probably the most common use of PPP.
What is an example where the mean is better than the median?
I care more about my daily mean income in the last year than my median income over the last year.
Ideally, I'd like to learn both.
I know about different means. And I know that sometimes mean isn't appropriate at all (bimodal...) The context is one of the things I'd like to understand.
Just used it to learn the Greek alphabet. :-)