Adding averages - I'm confused
Walt987
1) Does it make sense and if so could someone explain in words what it means and how it is used?
2) Is there a term for this value?
3) I found a procedure on a web page for calculating the standard deviation (not additive) of this value but this raised a new question - how does it make sense to have a standard deviation for a total?
4) If it makes sense so far, is there a way to calculate standard error for this value?
5) If this value is not a mean, can it be used in statistical tests?
Thanks for any help.
math1234
false.
Adding or 'averaging' averages is not valid.
If I have two averages say 1 for one group and 100 for another group, then averaging these averages gives 101/2 = 50.5
But now here were the two groups 1, 2, 3 was the first group.
The Second group was 90, 95, 100, 105, 110.
If I check the average of all of them ((1+2+3)+(90+95+100+105+110))/8 = 63.5
This is no where near the 50.5 obtained by averaging their averages.
I once saw a replay of a notable astrophysicist speaking before a group of math teachers. He said (paraphasing) 'the paper said that 9 out of 10 schools are now below average. How can that be?', he exclaimed excitedly. No one said a word. I was dumbfounded. Even if you are only referring to the current year's test results consider this; on a scale of 1 to 10, nine schools score a '5' and one school scores a '6'. The average is 5.1 and nine schools are below average.
Be careful with averages!
Bruce Weaver
>> I've been told it is reasonable to add the averages
>> of several groups to get the "total mean" (for lack
>> of a better word) of the groups. For example you
>> could add the average abundance of several species of
>> fish to get a total abundance value for all species.
>> I have three questions about this procedure:
>>
>> 1) Does it make sense and if so could someone explain
>> in words what it means and how it is used?
>> 2) Is there a term for this value?
>> 3) I found a procedure on a web page for calculating
>> the standard deviation (not additive) of this value
>> but this raised a new question - how does it make
>> sense to have a standard deviation for a total?
>> 4) If it makes sense so far, is there a way to
>> calculate standard error for this value?
>> 5) If this value is not a mean, can it be used in
>> statistical tests?
>>
>> Thanks for any help.
>
>
> false.
>
> Adding or 'averaging' averages is not valid.
It can be--e.g., unweighted means analysis for factorial ANOVA.
> If I have two averages say 1 for one group and 100 for another group, then averaging these averages gives 101/2 = 50.5
>
> But now here were the two groups 1, 2, 3 was the first group.
> The Second group was 90, 95, 100, 105, 110.
>
> If I check the average of all of them ((1+2+3)+(90+95+100+105+110))/8 = 63.5
>
> This is no where near the 50.5 obtained by averaging their averages.
>
> I once saw a replay of a notable astrophysicist speaking before a group of math teachers. He said (paraphasing) 'the paper said that 9 out of 10 schools are now below average. How can that be?', he exclaimed excitedly. No one said a word. I was dumbfounded. Even if you are only referring to the current year's test results consider this; on a scale of 1 to 10, nine schools score a '5' and one school scores a '6'. The average is 5.1 and nine schools are below average.
>
> Be careful with averages!
--
Bruce Weaver
bwe...@lakeheadu.ca
http://sites.google.com/a/lakeheadu.ca/bweaver/
"When all else fails, RTFM."
Graham Ashe
aruzinsky
1) You are supposed to take a weighted average of averages, weighted
by their relative numbers, e.g., percent. You need the relative
numbers in each group.
2) Average = Sample mean.
3) Yes
4) You need the sample Std.Dev. of the individual averages and the
relative numbers per 1).
6) Yes
Rich Ulrich
<aruz...@general-cathexis.com> wrote:
>On Jul 28, 6:08�pm, Walt987 <grog...@gmail.com> wrote:
>> I've been told it is reasonable to add the averages of several groups to get the "total mean" (for lack of a better word) of the groups. For example you could add the average abundance of several species of fish to get a total abundance value for all species. �I have three questions about this procedure:
>>
>> 1) Does it make sense and if so could someone explain in words what it means and how it is used?
>> 2) Is there a term for this value?
>> 3) I found a procedure on a web page for calculating the standard deviation (not additive) of this value but this raised a new question - how does it make sense to have a standard deviation for a total?
>> 4) If it makes sense so far, is there a way to calculate �standard error for this value?
>> 5) If this value is not a mean, can it be used in statistical tests?
>>
>> Thanks for any help.
>
>1) You are supposed to take a weighted average of averages, weighted
>by their relative numbers, e.g., percent. You need the relative
>numbers in each group.
Also -- Averages mainly make sense in situations where
they estimate a central and modal value (though, perhaps, in a
biased way - like using a mean of a log-normal situation); or where
they estimate a parameter for a process.
>2) Average = Sample mean.
>3) Yes
>4) You need the sample Std.Dev. of the individual averages and the
>relative numbers per 1).
>6) Yes
["6" should be "5"]
Amplifying on (5) --
What an ANOVA for groups says, as a result of a test, is whether
you learn something by using separate averages to represent
separate groups. As the examples of several posters have shown,
it can be silly or totally misleading to use an overall average to
describe mixed groups when the separate groups are thoroughly
distinct from each other.
The original mention of "average abundance" raises, for me,
the problem of a terminology that is potentially ambiguous and
confusing. Is an "abundance of fish" satisfied by having a huge
school of one sort of fish? Isn't it also satisfied (perhaps) by
having a huge diversity of fishes with a few of each of many sorts?
But I'm not an ecologist saying that. For all I know, "abundance"
could be part of the proper, technically precise description.
--
Rich Ulrich
Walt987
For example, four samples each in which 3 species showed up - see below. He was taking the average count for each species then adding them, and calling this the total number of fish (or the average number of fish?). To me it doesn't make sense.
Species A
5
7
9
3
Species B
23
11
21
16
Species C
14
11
12
19
I'm not a statistician (ecologist!) so I'm not sure I'm following the forum responses. Perhaps I started a debate? I would appreciate a simpler answer about whether it's valid to add averages, and if so, what it means.
Walt987
My method makes standard deviation and SE straightforward to calculate . . . so I'm thinking his question was really about how to calculate a standard error (as I said we found a method for calculating the std dev on the web) when he uses this way to calculate the sample mean . . .
Graham Ashe
> I'm following the forum responses. Perhaps I started
> a debate? I would appreciate a simpler answer about
> whether it's valid to add averages, and if so, what
> it means.
I hear you. It's one thing to understand statistics (even have a Ph.D. in it), but to be able to explain it in a way that makes sense to the layman in statistics (like you and I), is quite another.
I wouldn't presume to be able to advise you correctly on this problem of yours, but I'm glad you asked it because it lead me to the issue of "average of averages", which I had taken for granted in some of my own work where going back to every single datum of the samples did not seem suitable or pragmatic. Based on what I've read so far and the circumstances surrounding my data, I think my results are still valid.
Walt987
This would happen for example when you're measuring some property of a species (like how many individuals or their size). You need each sample unit to provide a value that applies to a larger area than a single point, so you define an area for the sample unit and "subsample" it because there are too many individuals to count and measure them all. You might use animal traps or square frames for subsampling to limit how many individuals you need to count or measure. You then average these subsample values to give an estimate for that sample unit.
Your overall goal is usuall an estimate for a larger area (say a lake or a forest), so you subsample at numerous locations within the larger area. To get the estimate for the larger area, you average the sample unit means for an overall mean.
The unsatisfying part of this is that while you have variability estimates for the overall mean (for the mean of the sample unit means), and for the sample unit means, you have no overall picture of all this in your final numers - you're left to your judgement regarding the quality of the individual sample unit means. You have to assume that the investigator looked them over critically (e.g., for wildly high SD) before averaging them into the larger mean.
Note that averaging all of the measurements in one calculation is regarded as "pseudoreplication" because it artificially inflates sample size and therefore depresses variability of the overall mean, compromising tests.
I haven't heard yet of a way to capture all this in one clean operation without psedoreplicating, but I'd love to hear any thoughts or examples of how this problem (essentially a sampling question) is handled is other fields.
aruzinsky
I am going to guess these numbers are per something you left out.
Let's pretend the data are the numbers of each species caught in a net
and you want the average number of fish of those combine species
caught in a net. This is more algebra than statistics and something
you should have learned in high school:
(5+7+9+3)/4 + (23+11+21+16)/4 + (14+11+12+19)/4 =
(5+7+9+3+23+11+21+16+14+11+12+19)/4
Graham Ashe
> of averages for years. Typically in the context of
> subsamples. That is, when a value for a sample unit
> has to be a mean because the sample unit has to be
> sampled itself. You then have the matter of how to
> use these means from multiple sample units to give an
> estiimate for an overall mean.
> This would happen for example when you're measuring
> some property of a species (like how many individuals
> or their size). You need each sample unit to provide
> a value that applies to a larger area than a single
> point, so you define an area for the sample unit and
> "subsample" it because there are too many individuals
> to count and measure them all. You might use animal
> traps or square frames for subsampling to limit how
> many individuals you need to count or measure. You
> then average these subsample values to give an
> estimate for that sample unit.
>
> Your overall goal is usuall an estimate for a larger
> area (say a lake or a forest), so you subsample at
> numerous locations within the larger area. To get the
> estimate for the larger area, you average the sample
> unit means for an overall mean.
This is just the kind fieldwork that some statisticians cannot grasp or excuse based on their textbook methods for reasons like, "... your sub-samples may not all have been of the same size." Unfortunately, in the real world, we have to make do. I'm not quite sure it means our results are any less valid.
> The unsatisfying part of this is that while you have
> variability estimates for the overall mean (for the
> mean of the sample unit means), and for the sample
> unit means, you have no overall picture of all this
> in your final numers - you're left to your judgement
> regarding the quality of the individual sample unit
> means. You have to assume that the investigator
> looked them over critically (e.g., for wildly high
> SD) before averaging them into the larger mean.
>
> Note that averaging all of the measurements in one
> calculation is regarded as "pseudoreplication"
> because it artificially inflates sample size and
> therefore depresses variability of the overall mean,
> compromising tests.
>
> I haven't heard yet of a way to capture all this in
> one clean operation without psedoreplicating, but I'd
> love to hear any thoughts or examples of how this
> problem (essentially a sampling question) is handled
> is other fields.
Well, I've also done some research in AI and games and I take samples and sub-samples there too. As you might have guessed, the number of possible (legal) games that could occur on say, a chessboard, is pretty large. Several orders of magnitude more than all the living (and dead) creatures that have ever existed on the planet, in fact.
It never occurred to me that my samples of games (however large and stratified) represented only an infinitesimal portion of the actual "population". Or that the sub-sample means I had been using and re-using were any less representative of the "truth". In fact, I've often considered a sub-sample mean to be the best way to approximate a statistic rather than the data themselves.
Yet, I am still usually able to make reasonable predictions based on the experimental results and statistical analysis I think is best. Due to the different domains of our work, I don't think I have the problem of pseudo-replication here (even if I couldn't ensure that all my relevant sub-samples were of the same size and that data collection circumstances were optimal). Sometimes, perhaps it is best to concede that our current understanding of statistics simply cannot provide all the answers we need. This is usually better than thinking (or being made to think) that the work we do is pointless.
Rich Ulrich
[snip]
>
>For example, four samples each in which 3 species showed up - see
>below. He was taking the average count for each species then adding
>them, and calling this the total number of fish (or the average number
>of fish?). To me it doesn't make sense.
He can say whatever he wants if it communicates well with
his audience -- You say that it didn't make sense to you, so
he failed, at least a little.
What about --
"For four samples, the typical number 6 of A, 18 of B, and
13 of C." That seems meaningful, perhaps, if everyone
understands something about the reason for the sampling.
It seems to me that it might take some more information to
*justify* making a statement about the variance or SD.
Was sampling of fixed duration for each site?
The SD of the four totals would probably be the easiest
number to justify as the SD of the overall total.
>
>Species A
>5
>7
>9
>3
>Species B
>23
>11
>21
>16
>Species C
>14
>11
>12
>19
>
>
[snip]
--
Rich Ulrich
Walt987
aruzinsky
> Yes, and ((5+23+14)+(7+11+11)+(9+21+12)+(3+16+19))/4 is also = 37.75, which is what I said in the message immediately following the message you replied to. But thanks for the algebra lesson anyway!
According to my browser's search function, that was the first time you
mentioned "37.75".
In the previous fish net example, the sum of averages might be
construed as an estimate of the total number of fish caught in a net,
albeit, a procedurally bad estimate because the average numbers of the
individual species could be correlated (fish tend to travel in
schools).
rith...@gmail.com
You cannot add averages if for example your one class got an average of 50% on their test and you have another class that got 75% on their test, you can't add their averages and say the combined average for your two classes is 50% +75%=125%. (It doesn't make sense to add averages in this case)
Let me know what you think...
Rich Ulrich
a) You have replied to a post from 2009.
b) There are only a few posts here per month, these days.
--
Rich Ulrich