Mixing
Exclude worst value from weighted arithmetic mean
$begingroup$
I'm working on a tournament rating, that is calculated as a weighted arithmetic mean, i.e. the formula is:
$$
R = frac{a_1 x_1+a_2 x_2+cdots+a_n x_n}{a_1+a_2+cdots+a_n}
$$
$a_i$ is a positive tournament weight.
$x_i$ is a non-negative tournament score.
The formula has one particularity, though: one or a few worst results (let's say 10% of them) are excluded from it. In other words, I need to find which values to exclude (i.e. set $a_i=0$) to maximize the $R$ value.
How can I find which values to exclude without brute-force search?
statistics average means
edited Jun 16 '18 at 15:10
Michael Hardy
1
asked Jun 16 '18 at 12:37
Pavel BogachevPavel Bogachev
112
$endgroup$
add a comment |
$begingroup$
I'm working on a tournament rating, that is calculated as a weighted arithmetic mean, i.e. the formula is:
$$
R = frac{a_1 x_1+a_2 x_2+cdots+a_n x_n}{a_1+a_2+cdots+a_n}
$$
$a_i$ is a positive tournament weight.
$x_i$ is a non-negative tournament score.
The formula has one particularity, though: one or a few worst results (let's say 10% of them) are excluded from it. In other words, I need to find which values to exclude (i.e. set $a_i=0$) to maximize the $R$ value.
How can I find which values to exclude without brute-force search?
statistics average means
edited Jun 16 '18 at 15:10
Michael Hardy
1
asked Jun 16 '18 at 12:37
Pavel BogachevPavel Bogachev
112
$endgroup$
$begingroup$
Unless $n$ is very large it can't to be too 'brutal' a search to find the scores in the lowest 10% and exclude them. Some scoring methods (e.g., Olympic) just eliminate the highest and lowest (without weights). Unweighted averages that disregard the lowest and highest 5% of the data are called 5% trimmed means. Here are 20 observations, sorted in order:20 20 22 23 25 25 27 28 28 28 29 29 31 32 32 33 34 38 38 46From R statistical software, the mean, 5% trimmed mean, and median are 29.4, 29.0, 28.5, respectively.
$endgroup$
– BruceET
Jun 16 '18 at 16:24
add a comment |
$begingroup$
I'm working on a tournament rating, that is calculated as a weighted arithmetic mean, i.e. the formula is:
$$
R = frac{a_1 x_1+a_2 x_2+cdots+a_n x_n}{a_1+a_2+cdots+a_n}
$$
$a_i$ is a positive tournament weight.
$x_i$ is a non-negative tournament score.
The formula has one particularity, though: one or a few worst results (let's say 10% of them) are excluded from it. In other words, I need to find which values to exclude (i.e. set $a_i=0$) to maximize the $R$ value.
How can I find which values to exclude without brute-force search?
statistics average means
edited Jun 16 '18 at 15:10
Michael Hardy
1
asked Jun 16 '18 at 12:37
Pavel BogachevPavel Bogachev
112
$endgroup$
I'm working on a tournament rating, that is calculated as a weighted arithmetic mean, i.e. the formula is:
$$
R = frac{a_1 x_1+a_2 x_2+cdots+a_n x_n}{a_1+a_2+cdots+a_n}
$$
$a_i$ is a positive tournament weight.
$x_i$ is a non-negative tournament score.
The formula has one particularity, though: one or a few worst results (let's say 10% of them) are excluded from it. In other words, I need to find which values to exclude (i.e. set $a_i=0$) to maximize the $R$ value.
How can I find which values to exclude without brute-force search?
statistics average means
statistics average means
edited Jun 16 '18 at 15:10
Michael Hardy
1
asked Jun 16 '18 at 12:37
Pavel BogachevPavel Bogachev
112
edited Jun 16 '18 at 15:10
Michael Hardy
1
asked Jun 16 '18 at 12:37
Pavel BogachevPavel Bogachev
112
edited Jun 16 '18 at 15:10
Michael Hardy
1
edited Jun 16 '18 at 15:10
Michael Hardy
1
edited Jun 16 '18 at 15:10
Michael Hardy
1
1
asked Jun 16 '18 at 12:37
Pavel BogachevPavel Bogachev
112
asked Jun 16 '18 at 12:37
Pavel BogachevPavel Bogachev
112
asked Jun 16 '18 at 12:37
Pavel BogachevPavel Bogachev
112
112
$begingroup$
Unless $n$ is very large it can't to be too 'brutal' a search to find the scores in the lowest 10% and exclude them. Some scoring methods (e.g., Olympic) just eliminate the highest and lowest (without weights). Unweighted averages that disregard the lowest and highest 5% of the data are called 5% trimmed means. Here are 20 observations, sorted in order:20 20 22 23 25 25 27 28 28 28 29 29 31 32 32 33 34 38 38 46From R statistical software, the mean, 5% trimmed mean, and median are 29.4, 29.0, 28.5, respectively.
$endgroup$
– BruceET
Jun 16 '18 at 16:24
add a comment |
$begingroup$
Unless $n$ is very large it can't to be too 'brutal' a search to find the scores in the lowest 10% and exclude them. Some scoring methods (e.g., Olympic) just eliminate the highest and lowest (without weights). Unweighted averages that disregard the lowest and highest 5% of the data are called 5% trimmed means. Here are 20 observations, sorted in order:20 20 22 23 25 25 27 28 28 28 29 29 31 32 32 33 34 38 38 46From R statistical software, the mean, 5% trimmed mean, and median are 29.4, 29.0, 28.5, respectively.
$endgroup$
– BruceET
Jun 16 '18 at 16:24
$begingroup$
Unless $n$ is very large it can't to be too 'brutal' a search to find the scores in the lowest 10% and exclude them. Some scoring methods (e.g., Olympic) just eliminate the highest and lowest (without weights). Unweighted averages that disregard the lowest and highest 5% of the data are called 5% trimmed means. Here are 20 observations, sorted in order:
20 20 22 23 25 25 27 28 28 28 29 29 31 32 32 33 34 38 38 46 From R statistical software, the mean, 5% trimmed mean, and median are 29.4, 29.0, 28.5, respectively.$endgroup$
– BruceET
Jun 16 '18 at 16:24
$begingroup$
Unless $n$ is very large it can't to be too 'brutal' a search to find the scores in the lowest 10% and exclude them. Some scoring methods (e.g., Olympic) just eliminate the highest and lowest (without weights). Unweighted averages that disregard the lowest and highest 5% of the data are called 5% trimmed means. Here are 20 observations, sorted in order:
20 20 22 23 25 25 27 28 28 28 29 29 31 32 32 33 34 38 38 46 From R statistical software, the mean, 5% trimmed mean, and median are 29.4, 29.0, 28.5, respectively.$endgroup$
– BruceET
Jun 16 '18 at 16:24
add a comment |
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$begingroup$
Unless $n$ is very large it can't to be too 'brutal' a search to find the scores in the lowest 10% and exclude them. Some scoring methods (e.g., Olympic) just eliminate the highest and lowest (without weights). Unweighted averages that disregard the lowest and highest 5% of the data are called 5% trimmed means. Here are 20 observations, sorted in order:
20 20 22 23 25 25 27 28 28 28 29 29 31 32 32 33 34 38 38 46From R statistical software, the mean, 5% trimmed mean, and median are 29.4, 29.0, 28.5, respectively.$endgroup$
– BruceET
Jun 16 '18 at 16:24