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This is how I understood the concept of weight.

In deriving statistics, we talk about "weight" to assign to each observation.  When I learned it the first time, it took me a while to get it, but I felt I understood it completely when I realized that even without weight, we are assigning a weight of 1 to observations, i.e., we are saying to our statistical software or Excel, "please treat each individual case as one case."  For example, let's say we want to get an average score of three persons' test score:
  Test score
John 50
Mary 60
David 40
Average 50
What I did was 50 + 60 + 40 and divide the result by 3 (count of observation).  In this calculation, you don't see any weights used.  But in fact they are there:
(50*1) + (60*1) + (40*1) and divide the result by (1+1+1, the number of weights).
Notice I now call count of observation "the number of weights."
So if for some reason if you want to treat each observation (each person) with different weights, that is possible.  For example, for a strange reason you want to count John's response twice.  You assign a weight of 2 to Mr. John (treating him as if he is worth two persons), but treat everyone else as 1 (one person).
(50*2) + (60*1) + (40*1) and divide the result by (2+1+1, the number of weights).
What about the case when you don't want to include John into calculation.  John is a new student to the class, so, say, a teacher didn't want to include his score to the average test score that he needs to report to the school district.  Then you assign 0 to John:
(50*0) + (60*1) + (40*1) and divide the result by (0+1+1, the number of weights).
From this you realize you are already doing a lot of weighting.  You only considered John, Mary, and David, but you are ignoring the rest of classmates.  You are assigning 0 to all these people:
(50*1) + (60*1) + (40*1) + (35*0) + (26*0) + 25(*0) ...
and divide the result by (1+1+1+0+0+0...., the number of weights).
You are already using weights; you may not be aware of it.

What kind weights are available?
When I started learning statistics and actually using them, I didn't know there were different types of weight.  I thought there was only sample weight that comes with a data set like NELS and TIMSS (these are nationally representative data sets).  I didn't realize for a long time that there are lots of weights.  Also for a longer time, I didn't realize that you can create any type of weights on your own. 
     For example, above I said for some strange reason you want to give a weight of 2 to John.  This is perfectly okay if you just feel like it.  For no reason you want to treat this guy specially.  Maybe you are trying to contaminate the data.  Of course you don't use this for research purpose.  I use this example, so you know you can do anything you want and still you get results.  Such results are simply wrong.
    John's example was an extreme example.  My point is that weight isn't something that is fixed in stone.  Of course, there are some established ways to construct weight, but often you can create your own weight to satisfy your own reasons.  In my case, just by recognizing this, I feel like I understand what weight is. 
But anyways of course you will need good reasons for constructing specific types of weight.  Before I review some famous types of weight, let me mention two things.  I said weight isn't something that is fixed in stone.  But there are two things you always have to do.  One is to use the sum of weights as a diviser.  The other is after you create weights, you may have to adjust the weights, so the sum of weights will equal the number of observations.  This latter point isn't always true, but in practice this makes subsequent analysis easier.  I will explain all these later.
Sample weight
A natinal representative data set (e.g., NELS, TIMSS) comes with sample weights.  Such weights are created to reflect the inverse of probability that a subject gets selected into the sample.  If everyone has equal chance of ending up being included in the sample is the same, we don' t need weight.  But sometimes we "oversample" so we make sure

proc sql;

create table newdata as

select *,

f1pnlwt * (count(f1pnlwt)/Sum(f1pnlwt)) as F1weight

from here.all; /*proc SQL does not require run statement*/

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