#1




Need quick statistics help on averages
Let's say we're talking about comparing average test scores between boys (group A) and girls (group B), and the overall group (Group C, where Group C = Group A + Group B), and we're looking over the period of two tests.
Without knowing how many people are in each group, is it possible for the average scores of both Group A and Group B to go down since the previous test, while the overall average of Group C goes up at the same time? 
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#2




Also, I should mention: the number of people from Test 1 to Test 2 may change as well.

#3




Yes, see Simpson's paradox.

#4




No, there is no way for the average of the entire group both to rise when the averages of both sub groups declines. One or both of the sub group averages must increase in order for the main group average to increase.

#5




Ask yourself whether the totals could go down for Group A and for Group B (if the numbers in A and B don't change, the totals must go up and down when the averages do) and yet the total for Group C goes up (but... C = A + B), and you see that this is impossible.

#6




I don't think Simpson's paradox covers the question in the OP, where there are 2 distinct groups making up one whole. Wierd things could happen within the data sets due to the paradox, but I can't think of any way that both can decline while the main set average increases.
I am, however, prepared to be proven worng! 
#7




Ah, well then it can be thrown off: if the two groups are of widely different average scores, and the group which does worse has much larger representation the second time around.

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For test 2, there are 20 girls, and 2 boys. The both girls get a score of 85 (so the girls' average score is 85  less than the previous test result of 90), and the boys each get a 65  so the average boys' score goes down as well (from 70 to 65). The group average though goes up from 77.5 to 83.2 
#10




Yes is can happen  AndyL gave a very nice example (has a mistake in the first boy's group average, but that's obviously a typo  the result is correct). I believe this would fall under Simpson's paradox.
An average, by itself, tells you nothing. You need to know how variable the data are around the average, and the sample size. As a scientist, I deal with this sort of thing all the time during data analysis. 
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Last edited by Snarky_Kong; 02052013 at 03:55 PM.. 
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#16




Sure it is. What specifically do you take exception to?
It helps to consider a "test" that girls consistently, deterministically, do better at, and you are certain that girls will get 70%, but boys will get 50% every single time. Then the overall average can be as high as 70% (for an allgirl class) or as low as 50% (for an allboy class). Increase the proportion of girls, and even if you don't do anything else, the class average goes up solely due to the change in class composition. Simpson's paradox simply happens when a change due to the composition of the class overwhelms a change due to an actual change in the results of the test. 
#17




Here's a simple example. Population size of 3 stays the same, but between test 1 and test 2, boy 2 drops out and girl 2 is added:
Test 1: Boy 1 = 50; Boy 2 = 50; Girl 1 = 20 Average for class is 40 Test 2: Boy 1 = 55; Girl 1 = 25; Girl 2 = 25 Average for class is 35 
#18




Thanks, guys. This is really helpful. I suspected that this could work, but couldn't quite think it through.

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#21




It seems that, in order for the OP's proposed scenario to happen, group membership must change so significantly from Time 1 to Time 2 that the concepts of "Group A" and "Group B" (as entities that persist across both time points) become rather tenuous. (Or else the differences across groups and across time have to be really tiny.)

#22




That is why the groups are more nebulous things "boys" and "girls" where the groupings are still welldefined even if all the members change. The effect can not manifest if the groups are fixed in advance for all tests.

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