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Why Standard Deviation is more popular than Mean Absolute Deviation?
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$$
Mean;Absolute;Deviation = frac1nsum_i=1^n |x_i-mean(X)|
$$
Almost all textbooks and papers are using Standard Deviation as a measurement of dispersion. And of course, almost all the models are built based on Standard Deviation.
But I don't understand how Standard Deviation has gained such popularity. I mean, Mean Absolute Deviation is a very intuitive measurement of dispersion. It tells you exact average distant that each value deviates from their mean. Standard Deviation, on the other hand, makes the result more sensitive to outliers. Why we need this sensitivity. Is there any historical reason leads us to use SD way more often than MAD?
mathematical-statistics standard-deviation
asked 4 hours ago
27182818
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up vote
1
down vote
favorite
$$
Mean;Absolute;Deviation = frac1nsum_i=1^n |x_i-mean(X)|
$$
Almost all textbooks and papers are using Standard Deviation as a measurement of dispersion. And of course, almost all the models are built based on Standard Deviation.
But I don't understand how Standard Deviation has gained such popularity. I mean, Mean Absolute Deviation is a very intuitive measurement of dispersion. It tells you exact average distant that each value deviates from their mean. Standard Deviation, on the other hand, makes the result more sensitive to outliers. Why we need this sensitivity. Is there any historical reason leads us to use SD way more often than MAD?
mathematical-statistics standard-deviation
asked 4 hours ago
27182818
61
New contributor
27182818 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
Something something absolute function not differentiable.
– user2974951
4 hours ago
Both are used in empirical research. Standard deviation is more convenient mathematically, probably among other reasons.
– Xiaomi
3 hours ago
A few potential reasons: it's differentiable, the Normal Distribution naturally parammetrises in terms of its standard deviation, the true mean minimises the expected variance (whereas the true median minimises the expected MAD)
– gazza89
3 hours ago
Possible duplicate of Is minimizing squared error equivalent to minimizing absolute error? Why squared error is more popular than the latter?
– Martijn Weterings
1 hour ago
I am voting to close this question because it has been asked several times before. However, it is useful and it could serve some purpose as helping in searches. For instance I seem not to find the post that I had in mind when marking this as duplicate (although several other posts came up that are obvious duplicates as well).
– Martijn Weterings
1 hour ago
 |Â
show 1 more comment
up vote
1
down vote
favorite
up vote
1
down vote
favorite
$$
Mean;Absolute;Deviation = frac1nsum_i=1^n |x_i-mean(X)|
$$
Almost all textbooks and papers are using Standard Deviation as a measurement of dispersion. And of course, almost all the models are built based on Standard Deviation.
But I don't understand how Standard Deviation has gained such popularity. I mean, Mean Absolute Deviation is a very intuitive measurement of dispersion. It tells you exact average distant that each value deviates from their mean. Standard Deviation, on the other hand, makes the result more sensitive to outliers. Why we need this sensitivity. Is there any historical reason leads us to use SD way more often than MAD?
mathematical-statistics standard-deviation
asked 4 hours ago
27182818
61
New contributor
27182818 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$$
Mean;Absolute;Deviation = frac1nsum_i=1^n |x_i-mean(X)|
$$
Almost all textbooks and papers are using Standard Deviation as a measurement of dispersion. And of course, almost all the models are built based on Standard Deviation.
But I don't understand how Standard Deviation has gained such popularity. I mean, Mean Absolute Deviation is a very intuitive measurement of dispersion. It tells you exact average distant that each value deviates from their mean. Standard Deviation, on the other hand, makes the result more sensitive to outliers. Why we need this sensitivity. Is there any historical reason leads us to use SD way more often than MAD?
mathematical-statistics standard-deviation
mathematical-statistics standard-deviation
asked 4 hours ago
27182818
61
New contributor
27182818 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
27182818
61
New contributor
27182818 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
27182818
61
New contributor
27182818 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
27182818
61
asked 4 hours ago
27182818
61
61
New contributor
27182818 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
27182818 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
27182818 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
1
Something something absolute function not differentiable.
– user2974951
4 hours ago
Both are used in empirical research. Standard deviation is more convenient mathematically, probably among other reasons.
– Xiaomi
3 hours ago
A few potential reasons: it's differentiable, the Normal Distribution naturally parammetrises in terms of its standard deviation, the true mean minimises the expected variance (whereas the true median minimises the expected MAD)
– gazza89
3 hours ago
Possible duplicate of Is minimizing squared error equivalent to minimizing absolute error? Why squared error is more popular than the latter?
– Martijn Weterings
1 hour ago
I am voting to close this question because it has been asked several times before. However, it is useful and it could serve some purpose as helping in searches. For instance I seem not to find the post that I had in mind when marking this as duplicate (although several other posts came up that are obvious duplicates as well).
– Martijn Weterings
1 hour ago
 |Â
show 1 more comment
1
Something something absolute function not differentiable.
– user2974951
4 hours ago
Both are used in empirical research. Standard deviation is more convenient mathematically, probably among other reasons.
– Xiaomi
3 hours ago
A few potential reasons: it's differentiable, the Normal Distribution naturally parammetrises in terms of its standard deviation, the true mean minimises the expected variance (whereas the true median minimises the expected MAD)
– gazza89
3 hours ago
Possible duplicate of Is minimizing squared error equivalent to minimizing absolute error? Why squared error is more popular than the latter?
– Martijn Weterings
1 hour ago
I am voting to close this question because it has been asked several times before. However, it is useful and it could serve some purpose as helping in searches. For instance I seem not to find the post that I had in mind when marking this as duplicate (although several other posts came up that are obvious duplicates as well).
– Martijn Weterings
1 hour ago
1
1
Something something absolute function not differentiable.
– user2974951
4 hours ago
Something something absolute function not differentiable.
– user2974951
4 hours ago
Both are used in empirical research. Standard deviation is more convenient mathematically, probably among other reasons.
– Xiaomi
3 hours ago
Both are used in empirical research. Standard deviation is more convenient mathematically, probably among other reasons.
– Xiaomi
3 hours ago
A few potential reasons: it's differentiable, the Normal Distribution naturally parammetrises in terms of its standard deviation, the true mean minimises the expected variance (whereas the true median minimises the expected MAD)
– gazza89
3 hours ago
A few potential reasons: it's differentiable, the Normal Distribution naturally parammetrises in terms of its standard deviation, the true mean minimises the expected variance (whereas the true median minimises the expected MAD)
– gazza89
3 hours ago
Possible duplicate of Is minimizing squared error equivalent to minimizing absolute error? Why squared error is more popular than the latter?
– Martijn Weterings
1 hour ago
Possible duplicate of Is minimizing squared error equivalent to minimizing absolute error? Why squared error is more popular than the latter?
– Martijn Weterings
1 hour ago
I am voting to close this question because it has been asked several times before. However, it is useful and it could serve some purpose as helping in searches. For instance I seem not to find the post that I had in mind when marking this as duplicate (although several other posts came up that are obvious duplicates as well).
– Martijn Weterings
1 hour ago
I am voting to close this question because it has been asked several times before. However, it is useful and it could serve some purpose as helping in searches. For instance I seem not to find the post that I had in mind when marking this as duplicate (although several other posts came up that are obvious duplicates as well).
– Martijn Weterings
1 hour ago
 |Â
show 1 more comment
2 Answers
2
active
oldest
votes
up vote
2
down vote
Historically, Laplace started with the expected absolute deviation from the expectation and got mired into computational issues, beyond the Laplace (or double exponential) distribution, while Legendre and Gauss advocated the expected square difference from the expectation, which is more naturally connected with the Normal or Gaussian distribution. Portnoy and Koenker wrote a nice paper called the Gaussian Hare and the Laplacian Tortoise (!) on that issue, including a parody of the Hare and the Tortoise with Laplace's and Gauss' heads:
The issue is covered in depth in this earlier (2015) X Validated question. (Which makes the current one a potential duplicate.)
answered 2 hours ago
Xi'an
50.5k686335
add a comment |Â
up vote
1
down vote
One possible answer, among many others, is that the mean is precisely defined from the standard deviation as
$meanleft( X right) = mathop arg min limits_x frac1nsumlimits_i = 1^n left( x_i - x right)^2 $
Hence, mean and standard deviation come together. This is not the case for the MAD.
answered 2 hours ago
Fabrice Pautot
664
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
Historically, Laplace started with the expected absolute deviation from the expectation and got mired into computational issues, beyond the Laplace (or double exponential) distribution, while Legendre and Gauss advocated the expected square difference from the expectation, which is more naturally connected with the Normal or Gaussian distribution. Portnoy and Koenker wrote a nice paper called the Gaussian Hare and the Laplacian Tortoise (!) on that issue, including a parody of the Hare and the Tortoise with Laplace's and Gauss' heads:
The issue is covered in depth in this earlier (2015) X Validated question. (Which makes the current one a potential duplicate.)
answered 2 hours ago
Xi'an
50.5k686335
add a comment |Â
up vote
2
down vote
Historically, Laplace started with the expected absolute deviation from the expectation and got mired into computational issues, beyond the Laplace (or double exponential) distribution, while Legendre and Gauss advocated the expected square difference from the expectation, which is more naturally connected with the Normal or Gaussian distribution. Portnoy and Koenker wrote a nice paper called the Gaussian Hare and the Laplacian Tortoise (!) on that issue, including a parody of the Hare and the Tortoise with Laplace's and Gauss' heads:
The issue is covered in depth in this earlier (2015) X Validated question. (Which makes the current one a potential duplicate.)
answered 2 hours ago
Xi'an
50.5k686335
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Historically, Laplace started with the expected absolute deviation from the expectation and got mired into computational issues, beyond the Laplace (or double exponential) distribution, while Legendre and Gauss advocated the expected square difference from the expectation, which is more naturally connected with the Normal or Gaussian distribution. Portnoy and Koenker wrote a nice paper called the Gaussian Hare and the Laplacian Tortoise (!) on that issue, including a parody of the Hare and the Tortoise with Laplace's and Gauss' heads:
The issue is covered in depth in this earlier (2015) X Validated question. (Which makes the current one a potential duplicate.)
answered 2 hours ago
Xi'an
50.5k686335
Historically, Laplace started with the expected absolute deviation from the expectation and got mired into computational issues, beyond the Laplace (or double exponential) distribution, while Legendre and Gauss advocated the expected square difference from the expectation, which is more naturally connected with the Normal or Gaussian distribution. Portnoy and Koenker wrote a nice paper called the Gaussian Hare and the Laplacian Tortoise (!) on that issue, including a parody of the Hare and the Tortoise with Laplace's and Gauss' heads:
The issue is covered in depth in this earlier (2015) X Validated question. (Which makes the current one a potential duplicate.)
answered 2 hours ago
Xi'an
50.5k686335
edited 1 hour ago
answered 2 hours ago
Xi'an
50.5k686335
answered 2 hours ago
Xi'an
50.5k686335
answered 2 hours ago
Xi'an
50.5k686335
50.5k686335
add a comment |Â
add a comment |Â
up vote
1
down vote
One possible answer, among many others, is that the mean is precisely defined from the standard deviation as
$meanleft( X right) = mathop arg min limits_x frac1nsumlimits_i = 1^n left( x_i - x right)^2 $
Hence, mean and standard deviation come together. This is not the case for the MAD.
answered 2 hours ago
Fabrice Pautot
664
add a comment |Â
up vote
1
down vote
One possible answer, among many others, is that the mean is precisely defined from the standard deviation as
$meanleft( X right) = mathop arg min limits_x frac1nsumlimits_i = 1^n left( x_i - x right)^2 $
Hence, mean and standard deviation come together. This is not the case for the MAD.
answered 2 hours ago
Fabrice Pautot
664
add a comment |Â
up vote
1
down vote
up vote
1
down vote
One possible answer, among many others, is that the mean is precisely defined from the standard deviation as
$meanleft( X right) = mathop arg min limits_x frac1nsumlimits_i = 1^n left( x_i - x right)^2 $
Hence, mean and standard deviation come together. This is not the case for the MAD.
answered 2 hours ago
Fabrice Pautot
664
One possible answer, among many others, is that the mean is precisely defined from the standard deviation as
$meanleft( X right) = mathop arg min limits_x frac1nsumlimits_i = 1^n left( x_i - x right)^2 $
Hence, mean and standard deviation come together. This is not the case for the MAD.
answered 2 hours ago
Fabrice Pautot
664
answered 2 hours ago
Fabrice Pautot
664
answered 2 hours ago
Fabrice Pautot
664
answered 2 hours ago
Fabrice Pautot
664
664
add a comment |Â
add a comment |Â
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1
Something something absolute function not differentiable.
– user2974951
4 hours ago
Both are used in empirical research. Standard deviation is more convenient mathematically, probably among other reasons.
– Xiaomi
3 hours ago
A few potential reasons: it's differentiable, the Normal Distribution naturally parammetrises in terms of its standard deviation, the true mean minimises the expected variance (whereas the true median minimises the expected MAD)
– gazza89
3 hours ago
Possible duplicate of Is minimizing squared error equivalent to minimizing absolute error? Why squared error is more popular than the latter?
– Martijn Weterings
1 hour ago
I am voting to close this question because it has been asked several times before. However, it is useful and it could serve some purpose as helping in searches. For instance I seem not to find the post that I had in mind when marking this as duplicate (although several other posts came up that are obvious duplicates as well).
– Martijn Weterings
1 hour ago