# Need quick statistics help on averages
**URL:** https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417
**Category:** Factual Questions
**Created:** [February 5, 2013, 8:30pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417 "2013-02-05T20:30:21Z")
**Posts on this page:** 20
**Page:** 1
### Author:  [@Cagey\_Drifter](https://boards.straightdope.com/u/Cagey_Drifter)
#### Post date: [February 5, 2013, 8:30pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/1 "2013-02-05T20:30:21Z")
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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### Author:  [@Cagey\_Drifter](https://boards.straightdope.com/u/Cagey_Drifter)
#### Post date: [February 5, 2013, 8:50pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/2 "2013-02-05T20:50:54Z")
Also, I should mention: the number of people from Test 1 to Test 2 may change as well.
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### Author:  [@RedSwinglineOne](https://boards.straightdope.com/u/RedSwinglineOne)
#### Post date: [February 5, 2013, 9:10pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/3 "2013-02-05T21:10:51Z")
Yes, see [Simpson’s paradox](http://en.wikipedia.org/wiki/Simpson%27s_paradox).
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### Author:  [@Doctor\_Jackson](https://boards.straightdope.com/u/Doctor_Jackson)
#### Post date: [February 5, 2013, 9:17pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/4 "2013-02-05T21:17:32Z")
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.
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### Author:  [@Bob\_X](https://boards.straightdope.com/u/Bob_X)
#### Post date: [February 5, 2013, 9:22pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/5 "2013-02-05T21:22:02Z")
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.
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### Author:  [@Doctor\_Jackson](https://boards.straightdope.com/u/Doctor_Jackson)
#### Post date: [February 5, 2013, 9:24pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/6 "2013-02-05T21:24:24Z")
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!
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### Author:  [@Bob\_X](https://boards.straightdope.com/u/Bob_X)
#### Post date: [February 5, 2013, 9:25pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/7 "2013-02-05T21:25:13Z")
> [@Cagey\_Drifter](#):
>
> Also, I should mention: the number of people from Test 1 to Test 2 may change as well.
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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### Author:  [@Evil\_Economist](https://boards.straightdope.com/u/Evil_Economist)
#### Post date: [February 5, 2013, 9:35pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/8 "2013-02-05T21:35:40Z")
> [@RedSwinglineOne](#):
>
> Yes, see [Simpson’s paradox](http://en.wikipedia.org/wiki/Simpson%27s_paradox).
> [@Doctor\_Jackson](#):
>
> 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.
> [@Bob\_X](#):
>
> 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.
> [@Doctor\_Jackson](#):
>
> 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!
I do not understand how this sequence of answers could happen.
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### Author:  [@Andy\_L](https://boards.straightdope.com/u/Andy_L)
#### Post date: [February 5, 2013, 9:42pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/9 "2013-02-05T21:42:34Z")
> [@Cagey\_Drifter](#):
>
> Also, I should mention: the number of people from Test 1 to Test 2 may change as well.
Let’s start with 3 girls and 5 boys: all the girls have test scores of 90, so the average score for the girls is 90, and all the boys have test scores of 70, so the average for the boys is 80. The average score for boys and girls is 77.5.
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
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### Author:  [@mozchron](https://boards.straightdope.com/u/mozchron)
#### Post date: [February 5, 2013, 9:51pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/10 "2013-02-05T21:51:37Z")
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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### Author:  [@leahcim](https://boards.straightdope.com/u/leahcim)
#### Post date: [February 5, 2013, 9:51pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/11 "2013-02-05T21:51:50Z")
> [@Andy\_L](#):
>
> 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
I think the confusion is due to the fact that the fact that the populations are different sizes is _necessary_. When I hear “average test scores”, I implicitly think about a class where the boys and girls doing the second test are pretty much the same as the boys and girls doing the first test.
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### Author:  [@Snarky\_Kong](https://boards.straightdope.com/u/Snarky_Kong)
#### Post date: [February 5, 2013, 9:51pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/12 "2013-02-05T21:51:50Z")
> [@Andy\_L](#):
>
> Let’s start with 3 girls and 5 boys: all the girls have test scores of 90, so the average score for the girls is 90, and all the boys have test scores of 70, so the average for the boys is 80. The average score for boys and girls is 77.5.
>
> 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
Whoops.
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### Author:  [@Andy\_L](https://boards.straightdope.com/u/Andy_L)
#### Post date: [February 5, 2013, 9:52pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/13 "2013-02-05T21:52:40Z")
> [@mozchron](#):
>
> 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.
Oops. Thanks for noting the typo - I modified the example midstream, but missed a spot.
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### Author:  [@mozchron](https://boards.straightdope.com/u/mozchron)
#### Post date: [February 5, 2013, 9:55pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/14 "2013-02-05T21:55:07Z")
> [@Snarky\_Kong](#):
>
> That’s not how averages work.
???
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### Author:  [@Doctor\_Jackson](https://boards.straightdope.com/u/Doctor_Jackson)
#### Post date: [February 5, 2013, 9:58pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/15 "2013-02-05T21:58:44Z")
> [@leahcim](#):
>
> I think the confusion is due to the fact that the fact that the populations are different sizes is _necessary_. When I hear “average test scores”, I implicitly think about a class where the boys and girls doing the second test are pretty much the same as the boys and girls doing the first test.
Yep, my bad. I missed post #2.
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### Author:  [@leahcim](https://boards.straightdope.com/u/leahcim)
#### Post date: [February 5, 2013, 10:02pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/16 "2013-02-05T22:02:59Z")
> [@Snarky\_Kong](#):
>
> That’s not how averages work.
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 all-girl class) or as low as 50% (for an all-boy 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.
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### Author:  [@Giles](https://boards.straightdope.com/u/Giles)
#### Post date: [February 5, 2013, 10:13pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/17 "2013-02-05T22:13:04Z")
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
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### Author:  [@Cagey\_Drifter](https://boards.straightdope.com/u/Cagey_Drifter)
#### Post date: [February 5, 2013, 10:47pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/18 "2013-02-05T22:47:09Z")
Thanks, guys. This is really helpful. I suspected that this could work, but couldn’t quite think it through.
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### Author:  [@Andy\_L](https://boards.straightdope.com/u/Andy_L)
#### Post date: [February 5, 2013, 11:55pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/19 "2013-02-05T23:55:55Z")
> [@Giles](#):
>
> 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
Nice example - just about the minimum case for demonstrating the issue.
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### Author:  [@Snarky\_Kong](https://boards.straightdope.com/u/Snarky_Kong)
#### Post date: [February 6, 2013, 12:36pm UTC](https://boards.straightdope.com/t/need-quick-statistics-help-on-averages/649417/20 "2013-02-06T12:36:10Z")
> [@leahcim](#):
>
> 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 all-girl class) or as low as 50% (for an all-boy 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.
Nothing, which is why I edited my post. I misread it and thought he was taking an average of averages.
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