Are there any examples of a variable being normally distributed that is *not* due to the Central Limit...
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The normal distribution seems unintuitive until you learn the CLT, which explains why it is so prevalent in real life. But does it ever arise as the "natural" distribution for some quantity?
normal-distribution central-limit-theorem
asked Apr 3 at 21:18
gardenheadgardenhead
1764
$endgroup$
add a comment |
$begingroup$
The normal distribution seems unintuitive until you learn the CLT, which explains why it is so prevalent in real life. But does it ever arise as the "natural" distribution for some quantity?
normal-distribution central-limit-theorem
asked Apr 3 at 21:18
gardenheadgardenhead
1764
$endgroup$
4
$begingroup$
The physical theory of diffusion, to the extent it is applicable to any system, predicts Normal distributions of quantities (like temperature or concentration) that originate at a point. Indeed, a great many systems are diffusive (options prices, particle transport in homogeneous media, etc.), suggesting that examples are abundant assuming one is not so naive as to suppose that a Normal distribution must hold exactly out to unrealistically large or small values--that would be a misunderstanding of all physical theory.
$endgroup$
– whuber♦
Apr 3 at 22:27
2
$begingroup$
The normal distribution seems unintuitive until you learn that it maximizes the entropy under the constraint of fixed variance.
$endgroup$
– leonbloy
Apr 4 at 3:18
add a comment |
$begingroup$
The normal distribution seems unintuitive until you learn the CLT, which explains why it is so prevalent in real life. But does it ever arise as the "natural" distribution for some quantity?
normal-distribution central-limit-theorem
asked Apr 3 at 21:18
gardenheadgardenhead
1764
$endgroup$
The normal distribution seems unintuitive until you learn the CLT, which explains why it is so prevalent in real life. But does it ever arise as the "natural" distribution for some quantity?
normal-distribution central-limit-theorem
normal-distribution central-limit-theorem
asked Apr 3 at 21:18
gardenheadgardenhead
1764
asked Apr 3 at 21:18
gardenheadgardenhead
1764
asked Apr 3 at 21:18
gardenheadgardenhead
1764
asked Apr 3 at 21:18
gardenheadgardenhead
1764
asked Apr 3 at 21:18
gardenheadgardenhead
1764
1764
4
$begingroup$
The physical theory of diffusion, to the extent it is applicable to any system, predicts Normal distributions of quantities (like temperature or concentration) that originate at a point. Indeed, a great many systems are diffusive (options prices, particle transport in homogeneous media, etc.), suggesting that examples are abundant assuming one is not so naive as to suppose that a Normal distribution must hold exactly out to unrealistically large or small values--that would be a misunderstanding of all physical theory.
$endgroup$
– whuber♦
Apr 3 at 22:27
2
$begingroup$
The normal distribution seems unintuitive until you learn that it maximizes the entropy under the constraint of fixed variance.
$endgroup$
– leonbloy
Apr 4 at 3:18
add a comment |
4
$begingroup$
The physical theory of diffusion, to the extent it is applicable to any system, predicts Normal distributions of quantities (like temperature or concentration) that originate at a point. Indeed, a great many systems are diffusive (options prices, particle transport in homogeneous media, etc.), suggesting that examples are abundant assuming one is not so naive as to suppose that a Normal distribution must hold exactly out to unrealistically large or small values--that would be a misunderstanding of all physical theory.
$endgroup$
– whuber♦
Apr 3 at 22:27
2
$begingroup$
The normal distribution seems unintuitive until you learn that it maximizes the entropy under the constraint of fixed variance.
$endgroup$
– leonbloy
Apr 4 at 3:18
4
4
$begingroup$
The physical theory of diffusion, to the extent it is applicable to any system, predicts Normal distributions of quantities (like temperature or concentration) that originate at a point. Indeed, a great many systems are diffusive (options prices, particle transport in homogeneous media, etc.), suggesting that examples are abundant assuming one is not so naive as to suppose that a Normal distribution must hold exactly out to unrealistically large or small values--that would be a misunderstanding of all physical theory.
$endgroup$
– whuber♦
Apr 3 at 22:27
$begingroup$
The physical theory of diffusion, to the extent it is applicable to any system, predicts Normal distributions of quantities (like temperature or concentration) that originate at a point. Indeed, a great many systems are diffusive (options prices, particle transport in homogeneous media, etc.), suggesting that examples are abundant assuming one is not so naive as to suppose that a Normal distribution must hold exactly out to unrealistically large or small values--that would be a misunderstanding of all physical theory.
$endgroup$
– whuber♦
Apr 3 at 22:27
2
2
$begingroup$
The normal distribution seems unintuitive until you learn that it maximizes the entropy under the constraint of fixed variance.
$endgroup$
– leonbloy
Apr 4 at 3:18
$begingroup$
The normal distribution seems unintuitive until you learn that it maximizes the entropy under the constraint of fixed variance.
$endgroup$
– leonbloy
Apr 4 at 3:18
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
To an extent I think this this may be a philosophical issue as much as a statistical one.
A lot of naturally occurring phenomena are approximately normally distributed. One can argue
whether the underlying cause of that may be something like the CLT:
Heights of people may be considered as the the sum of many smaller causes (perhaps independent, unlikely identically distributed): lengths of various bones, or results of various gene expressions, or results of many dietary
influences, or some combination of all of the above.Test scores may be considered as the sums of scores on many individual test questions (possibly identically distributed, unlikely entirely independent).
Distance a particle travels in one dimension as a result of Brownian motion in a fluid: Motion may be considered abstractly as a random walk resulting from IID random hits by molecules.
One example where the CLT is not necessarily involved is the dispersion of shots around a bull's eye: The distance from the bull's eye can be modeled as a Rayleigh
distribution (proportional to square root of chi-sq with 2 DF) and the counterclockwise angle from the the positive horizontal axis can be modeled as uniform on $(0, 2pi).$ Then after changing from polar to rectangular coordinates, distances in horizontal (x) and
vertical (y) directions turn out to be uncorrelated bivariate normal. [This is the essence of the Box-Muller transformation, which you can google.] However, the normal x and y coordinates might be considered as the sum of many small inaccuracies in targeting, which might justify a CLT-related mechanism in the background.
In a historical sense, the widespread use of normal (Gaussian) distributions instead of double-exponential (Laplace) distributions to model astronomical observations may be partly due to the CLT. In the early days of modeling errors of such observations, there was a debate between Gauss and Laplace, each arguing for his own favorite distribution. For various reasons, the normal model has won out. One can argue that one reason for the eventual success of the normal distribution was mathematical convenience based on normal limits of the CLT. This seems to be true even when it is unclear which family of distributions provides the better fit. (Even now, there are still astronomers who feel that the "one best observation" made by
a meticulous, respected astronomer is bound to be a better value than than the average of many observations made by presumably less-gifted observers. In effect, they would prefer no intervention at all by statisticians.)
answered Apr 3 at 21:53
BruceETBruceET
6,4481721
$endgroup$
$begingroup$
Yep. Still fixing typos. Thanks for noticing this one. Same error in 'test scores' also fixed.
$endgroup$
– BruceET
Apr 3 at 22:20
add a comment |
$begingroup$
Lots of naturally occurring variables are normally distributed. Heights of humans? Size of animal colonies?
answered Apr 3 at 21:20
HappyHappy
112
$endgroup$
5
$begingroup$
But are those examples really normally distributed, or is that just a useful approximation? True normally distributed random variables take negative values with positive probability.
$endgroup$
– Artem Mavrin
Apr 3 at 21:27
1
$begingroup$
@Happy Actually neither example given here is normally distributed because the support of the normal distribution is -infinity to +infinity and the examples given can never be zero or less. In each case the normal distribution might be a useful approximation, but not if you were interested in the tails of the distribution.
$endgroup$
– JeremyC
Apr 3 at 21:28
2
$begingroup$
Human height is the result of the sum of (approximately) independent genes, so they actually are due to the CLT.
$endgroup$
– gardenhead
Apr 3 at 22:09
2
$begingroup$
@ArtemMavrin: getting a negative height would be something like 8+ standard deviations. If one objects to a normal approximation not being valid because it places zero probability mass beyond 8 sd's, you might as well also complain that a truly Normally distributed value is irrational with probability 1, yet all our measurements are rational numbers.
$endgroup$
– Cliff AB
Apr 3 at 22:15
2
$begingroup$
@gardenhead have a read of the statement of the central limit theorem and note where it doesn't match what you said (number of genes is finite while CLT is about the limiting distribution of a standardized sum as $ntoinfty$. (By the way, there's another theorem that says that a finite sum of iid non-normal variates cannot be normal; this theorem doesn't contradict the actual CLT!)
$endgroup$
– Glen_b♦
Apr 4 at 0:36
|
show 7 more comments
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2 Answers
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2 Answers
2
active
oldest
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active
oldest
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active
oldest
votes
$begingroup$
To an extent I think this this may be a philosophical issue as much as a statistical one.
A lot of naturally occurring phenomena are approximately normally distributed. One can argue
whether the underlying cause of that may be something like the CLT:
Heights of people may be considered as the the sum of many smaller causes (perhaps independent, unlikely identically distributed): lengths of various bones, or results of various gene expressions, or results of many dietary
influences, or some combination of all of the above.Test scores may be considered as the sums of scores on many individual test questions (possibly identically distributed, unlikely entirely independent).
Distance a particle travels in one dimension as a result of Brownian motion in a fluid: Motion may be considered abstractly as a random walk resulting from IID random hits by molecules.
One example where the CLT is not necessarily involved is the dispersion of shots around a bull's eye: The distance from the bull's eye can be modeled as a Rayleigh
distribution (proportional to square root of chi-sq with 2 DF) and the counterclockwise angle from the the positive horizontal axis can be modeled as uniform on $(0, 2pi).$ Then after changing from polar to rectangular coordinates, distances in horizontal (x) and
vertical (y) directions turn out to be uncorrelated bivariate normal. [This is the essence of the Box-Muller transformation, which you can google.] However, the normal x and y coordinates might be considered as the sum of many small inaccuracies in targeting, which might justify a CLT-related mechanism in the background.
In a historical sense, the widespread use of normal (Gaussian) distributions instead of double-exponential (Laplace) distributions to model astronomical observations may be partly due to the CLT. In the early days of modeling errors of such observations, there was a debate between Gauss and Laplace, each arguing for his own favorite distribution. For various reasons, the normal model has won out. One can argue that one reason for the eventual success of the normal distribution was mathematical convenience based on normal limits of the CLT. This seems to be true even when it is unclear which family of distributions provides the better fit. (Even now, there are still astronomers who feel that the "one best observation" made by
a meticulous, respected astronomer is bound to be a better value than than the average of many observations made by presumably less-gifted observers. In effect, they would prefer no intervention at all by statisticians.)
answered Apr 3 at 21:53
BruceETBruceET
6,4481721
$endgroup$
$begingroup$
Yep. Still fixing typos. Thanks for noticing this one. Same error in 'test scores' also fixed.
$endgroup$
– BruceET
Apr 3 at 22:20
add a comment |
$begingroup$
To an extent I think this this may be a philosophical issue as much as a statistical one.
A lot of naturally occurring phenomena are approximately normally distributed. One can argue
whether the underlying cause of that may be something like the CLT:
Heights of people may be considered as the the sum of many smaller causes (perhaps independent, unlikely identically distributed): lengths of various bones, or results of various gene expressions, or results of many dietary
influences, or some combination of all of the above.Test scores may be considered as the sums of scores on many individual test questions (possibly identically distributed, unlikely entirely independent).
Distance a particle travels in one dimension as a result of Brownian motion in a fluid: Motion may be considered abstractly as a random walk resulting from IID random hits by molecules.
One example where the CLT is not necessarily involved is the dispersion of shots around a bull's eye: The distance from the bull's eye can be modeled as a Rayleigh
distribution (proportional to square root of chi-sq with 2 DF) and the counterclockwise angle from the the positive horizontal axis can be modeled as uniform on $(0, 2pi).$ Then after changing from polar to rectangular coordinates, distances in horizontal (x) and
vertical (y) directions turn out to be uncorrelated bivariate normal. [This is the essence of the Box-Muller transformation, which you can google.] However, the normal x and y coordinates might be considered as the sum of many small inaccuracies in targeting, which might justify a CLT-related mechanism in the background.
In a historical sense, the widespread use of normal (Gaussian) distributions instead of double-exponential (Laplace) distributions to model astronomical observations may be partly due to the CLT. In the early days of modeling errors of such observations, there was a debate between Gauss and Laplace, each arguing for his own favorite distribution. For various reasons, the normal model has won out. One can argue that one reason for the eventual success of the normal distribution was mathematical convenience based on normal limits of the CLT. This seems to be true even when it is unclear which family of distributions provides the better fit. (Even now, there are still astronomers who feel that the "one best observation" made by
a meticulous, respected astronomer is bound to be a better value than than the average of many observations made by presumably less-gifted observers. In effect, they would prefer no intervention at all by statisticians.)
answered Apr 3 at 21:53
BruceETBruceET
6,4481721
$endgroup$
$begingroup$
Yep. Still fixing typos. Thanks for noticing this one. Same error in 'test scores' also fixed.
$endgroup$
– BruceET
Apr 3 at 22:20
add a comment |
$begingroup$
To an extent I think this this may be a philosophical issue as much as a statistical one.
A lot of naturally occurring phenomena are approximately normally distributed. One can argue
whether the underlying cause of that may be something like the CLT:
Heights of people may be considered as the the sum of many smaller causes (perhaps independent, unlikely identically distributed): lengths of various bones, or results of various gene expressions, or results of many dietary
influences, or some combination of all of the above.Test scores may be considered as the sums of scores on many individual test questions (possibly identically distributed, unlikely entirely independent).
Distance a particle travels in one dimension as a result of Brownian motion in a fluid: Motion may be considered abstractly as a random walk resulting from IID random hits by molecules.
One example where the CLT is not necessarily involved is the dispersion of shots around a bull's eye: The distance from the bull's eye can be modeled as a Rayleigh
distribution (proportional to square root of chi-sq with 2 DF) and the counterclockwise angle from the the positive horizontal axis can be modeled as uniform on $(0, 2pi).$ Then after changing from polar to rectangular coordinates, distances in horizontal (x) and
vertical (y) directions turn out to be uncorrelated bivariate normal. [This is the essence of the Box-Muller transformation, which you can google.] However, the normal x and y coordinates might be considered as the sum of many small inaccuracies in targeting, which might justify a CLT-related mechanism in the background.
In a historical sense, the widespread use of normal (Gaussian) distributions instead of double-exponential (Laplace) distributions to model astronomical observations may be partly due to the CLT. In the early days of modeling errors of such observations, there was a debate between Gauss and Laplace, each arguing for his own favorite distribution. For various reasons, the normal model has won out. One can argue that one reason for the eventual success of the normal distribution was mathematical convenience based on normal limits of the CLT. This seems to be true even when it is unclear which family of distributions provides the better fit. (Even now, there are still astronomers who feel that the "one best observation" made by
a meticulous, respected astronomer is bound to be a better value than than the average of many observations made by presumably less-gifted observers. In effect, they would prefer no intervention at all by statisticians.)
answered Apr 3 at 21:53
BruceETBruceET
6,4481721
$endgroup$
To an extent I think this this may be a philosophical issue as much as a statistical one.
A lot of naturally occurring phenomena are approximately normally distributed. One can argue
whether the underlying cause of that may be something like the CLT:
Heights of people may be considered as the the sum of many smaller causes (perhaps independent, unlikely identically distributed): lengths of various bones, or results of various gene expressions, or results of many dietary
influences, or some combination of all of the above.Test scores may be considered as the sums of scores on many individual test questions (possibly identically distributed, unlikely entirely independent).
Distance a particle travels in one dimension as a result of Brownian motion in a fluid: Motion may be considered abstractly as a random walk resulting from IID random hits by molecules.
One example where the CLT is not necessarily involved is the dispersion of shots around a bull's eye: The distance from the bull's eye can be modeled as a Rayleigh
distribution (proportional to square root of chi-sq with 2 DF) and the counterclockwise angle from the the positive horizontal axis can be modeled as uniform on $(0, 2pi).$ Then after changing from polar to rectangular coordinates, distances in horizontal (x) and
vertical (y) directions turn out to be uncorrelated bivariate normal. [This is the essence of the Box-Muller transformation, which you can google.] However, the normal x and y coordinates might be considered as the sum of many small inaccuracies in targeting, which might justify a CLT-related mechanism in the background.
In a historical sense, the widespread use of normal (Gaussian) distributions instead of double-exponential (Laplace) distributions to model astronomical observations may be partly due to the CLT. In the early days of modeling errors of such observations, there was a debate between Gauss and Laplace, each arguing for his own favorite distribution. For various reasons, the normal model has won out. One can argue that one reason for the eventual success of the normal distribution was mathematical convenience based on normal limits of the CLT. This seems to be true even when it is unclear which family of distributions provides the better fit. (Even now, there are still astronomers who feel that the "one best observation" made by
a meticulous, respected astronomer is bound to be a better value than than the average of many observations made by presumably less-gifted observers. In effect, they would prefer no intervention at all by statisticians.)
answered Apr 3 at 21:53
BruceETBruceET
6,4481721
edited Apr 4 at 0:56
answered Apr 3 at 21:53
BruceETBruceET
6,4481721
answered Apr 3 at 21:53
BruceETBruceET
6,4481721
answered Apr 3 at 21:53
BruceETBruceET
6,4481721
6,4481721
$begingroup$
Yep. Still fixing typos. Thanks for noticing this one. Same error in 'test scores' also fixed.
$endgroup$
– BruceET
Apr 3 at 22:20
add a comment |
$begingroup$
Yep. Still fixing typos. Thanks for noticing this one. Same error in 'test scores' also fixed.
$endgroup$
– BruceET
Apr 3 at 22:20
$begingroup$
Yep. Still fixing typos. Thanks for noticing this one. Same error in 'test scores' also fixed.
$endgroup$
– BruceET
Apr 3 at 22:20
$begingroup$
Yep. Still fixing typos. Thanks for noticing this one. Same error in 'test scores' also fixed.
$endgroup$
– BruceET
Apr 3 at 22:20
add a comment |
$begingroup$
Lots of naturally occurring variables are normally distributed. Heights of humans? Size of animal colonies?
answered Apr 3 at 21:20
HappyHappy
112
$endgroup$
5
$begingroup$
But are those examples really normally distributed, or is that just a useful approximation? True normally distributed random variables take negative values with positive probability.
$endgroup$
– Artem Mavrin
Apr 3 at 21:27
1
$begingroup$
@Happy Actually neither example given here is normally distributed because the support of the normal distribution is -infinity to +infinity and the examples given can never be zero or less. In each case the normal distribution might be a useful approximation, but not if you were interested in the tails of the distribution.
$endgroup$
– JeremyC
Apr 3 at 21:28
2
$begingroup$
Human height is the result of the sum of (approximately) independent genes, so they actually are due to the CLT.
$endgroup$
– gardenhead
Apr 3 at 22:09
2
$begingroup$
@ArtemMavrin: getting a negative height would be something like 8+ standard deviations. If one objects to a normal approximation not being valid because it places zero probability mass beyond 8 sd's, you might as well also complain that a truly Normally distributed value is irrational with probability 1, yet all our measurements are rational numbers.
$endgroup$
– Cliff AB
Apr 3 at 22:15
2
$begingroup$
@gardenhead have a read of the statement of the central limit theorem and note where it doesn't match what you said (number of genes is finite while CLT is about the limiting distribution of a standardized sum as $ntoinfty$. (By the way, there's another theorem that says that a finite sum of iid non-normal variates cannot be normal; this theorem doesn't contradict the actual CLT!)
$endgroup$
– Glen_b♦
Apr 4 at 0:36
|
show 7 more comments
$begingroup$
Lots of naturally occurring variables are normally distributed. Heights of humans? Size of animal colonies?
answered Apr 3 at 21:20
HappyHappy
112
$endgroup$
5
$begingroup$
But are those examples really normally distributed, or is that just a useful approximation? True normally distributed random variables take negative values with positive probability.
$endgroup$
– Artem Mavrin
Apr 3 at 21:27
1
$begingroup$
@Happy Actually neither example given here is normally distributed because the support of the normal distribution is -infinity to +infinity and the examples given can never be zero or less. In each case the normal distribution might be a useful approximation, but not if you were interested in the tails of the distribution.
$endgroup$
– JeremyC
Apr 3 at 21:28
2
$begingroup$
Human height is the result of the sum of (approximately) independent genes, so they actually are due to the CLT.
$endgroup$
– gardenhead
Apr 3 at 22:09
2
$begingroup$
@ArtemMavrin: getting a negative height would be something like 8+ standard deviations. If one objects to a normal approximation not being valid because it places zero probability mass beyond 8 sd's, you might as well also complain that a truly Normally distributed value is irrational with probability 1, yet all our measurements are rational numbers.
$endgroup$
– Cliff AB
Apr 3 at 22:15
2
$begingroup$
@gardenhead have a read of the statement of the central limit theorem and note where it doesn't match what you said (number of genes is finite while CLT is about the limiting distribution of a standardized sum as $ntoinfty$. (By the way, there's another theorem that says that a finite sum of iid non-normal variates cannot be normal; this theorem doesn't contradict the actual CLT!)
$endgroup$
– Glen_b♦
Apr 4 at 0:36
|
show 7 more comments
$begingroup$
Lots of naturally occurring variables are normally distributed. Heights of humans? Size of animal colonies?
answered Apr 3 at 21:20
HappyHappy
112
$endgroup$
Lots of naturally occurring variables are normally distributed. Heights of humans? Size of animal colonies?
answered Apr 3 at 21:20
HappyHappy
112
answered Apr 3 at 21:20
HappyHappy
112
answered Apr 3 at 21:20
HappyHappy
112
answered Apr 3 at 21:20
HappyHappy
112
112
5
$begingroup$
But are those examples really normally distributed, or is that just a useful approximation? True normally distributed random variables take negative values with positive probability.
$endgroup$
– Artem Mavrin
Apr 3 at 21:27
1
$begingroup$
@Happy Actually neither example given here is normally distributed because the support of the normal distribution is -infinity to +infinity and the examples given can never be zero or less. In each case the normal distribution might be a useful approximation, but not if you were interested in the tails of the distribution.
$endgroup$
– JeremyC
Apr 3 at 21:28
2
$begingroup$
Human height is the result of the sum of (approximately) independent genes, so they actually are due to the CLT.
$endgroup$
– gardenhead
Apr 3 at 22:09
2
$begingroup$
@ArtemMavrin: getting a negative height would be something like 8+ standard deviations. If one objects to a normal approximation not being valid because it places zero probability mass beyond 8 sd's, you might as well also complain that a truly Normally distributed value is irrational with probability 1, yet all our measurements are rational numbers.
$endgroup$
– Cliff AB
Apr 3 at 22:15
2
$begingroup$
@gardenhead have a read of the statement of the central limit theorem and note where it doesn't match what you said (number of genes is finite while CLT is about the limiting distribution of a standardized sum as $ntoinfty$. (By the way, there's another theorem that says that a finite sum of iid non-normal variates cannot be normal; this theorem doesn't contradict the actual CLT!)
$endgroup$
– Glen_b♦
Apr 4 at 0:36
|
show 7 more comments
5
$begingroup$
But are those examples really normally distributed, or is that just a useful approximation? True normally distributed random variables take negative values with positive probability.
$endgroup$
– Artem Mavrin
Apr 3 at 21:27
1
$begingroup$
@Happy Actually neither example given here is normally distributed because the support of the normal distribution is -infinity to +infinity and the examples given can never be zero or less. In each case the normal distribution might be a useful approximation, but not if you were interested in the tails of the distribution.
$endgroup$
– JeremyC
Apr 3 at 21:28
2
$begingroup$
Human height is the result of the sum of (approximately) independent genes, so they actually are due to the CLT.
$endgroup$
– gardenhead
Apr 3 at 22:09
2
$begingroup$
@ArtemMavrin: getting a negative height would be something like 8+ standard deviations. If one objects to a normal approximation not being valid because it places zero probability mass beyond 8 sd's, you might as well also complain that a truly Normally distributed value is irrational with probability 1, yet all our measurements are rational numbers.
$endgroup$
– Cliff AB
Apr 3 at 22:15
2
$begingroup$
@gardenhead have a read of the statement of the central limit theorem and note where it doesn't match what you said (number of genes is finite while CLT is about the limiting distribution of a standardized sum as $ntoinfty$. (By the way, there's another theorem that says that a finite sum of iid non-normal variates cannot be normal; this theorem doesn't contradict the actual CLT!)
$endgroup$
– Glen_b♦
Apr 4 at 0:36
5
5
$begingroup$
But are those examples really normally distributed, or is that just a useful approximation? True normally distributed random variables take negative values with positive probability.
$endgroup$
– Artem Mavrin
Apr 3 at 21:27
$begingroup$
But are those examples really normally distributed, or is that just a useful approximation? True normally distributed random variables take negative values with positive probability.
$endgroup$
– Artem Mavrin
Apr 3 at 21:27
1
1
$begingroup$
@Happy Actually neither example given here is normally distributed because the support of the normal distribution is -infinity to +infinity and the examples given can never be zero or less. In each case the normal distribution might be a useful approximation, but not if you were interested in the tails of the distribution.
$endgroup$
– JeremyC
Apr 3 at 21:28
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@Happy Actually neither example given here is normally distributed because the support of the normal distribution is -infinity to +infinity and the examples given can never be zero or less. In each case the normal distribution might be a useful approximation, but not if you were interested in the tails of the distribution.
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– JeremyC
Apr 3 at 21:28
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Human height is the result of the sum of (approximately) independent genes, so they actually are due to the CLT.
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– gardenhead
Apr 3 at 22:09
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Human height is the result of the sum of (approximately) independent genes, so they actually are due to the CLT.
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– gardenhead
Apr 3 at 22:09
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@ArtemMavrin: getting a negative height would be something like 8+ standard deviations. If one objects to a normal approximation not being valid because it places zero probability mass beyond 8 sd's, you might as well also complain that a truly Normally distributed value is irrational with probability 1, yet all our measurements are rational numbers.
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– Cliff AB
Apr 3 at 22:15
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@ArtemMavrin: getting a negative height would be something like 8+ standard deviations. If one objects to a normal approximation not being valid because it places zero probability mass beyond 8 sd's, you might as well also complain that a truly Normally distributed value is irrational with probability 1, yet all our measurements are rational numbers.
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– Cliff AB
Apr 3 at 22:15
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@gardenhead have a read of the statement of the central limit theorem and note where it doesn't match what you said (number of genes is finite while CLT is about the limiting distribution of a standardized sum as $ntoinfty$. (By the way, there's another theorem that says that a finite sum of iid non-normal variates cannot be normal; this theorem doesn't contradict the actual CLT!)
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– Glen_b♦
Apr 4 at 0:36
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@gardenhead have a read of the statement of the central limit theorem and note where it doesn't match what you said (number of genes is finite while CLT is about the limiting distribution of a standardized sum as $ntoinfty$. (By the way, there's another theorem that says that a finite sum of iid non-normal variates cannot be normal; this theorem doesn't contradict the actual CLT!)
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– Glen_b♦
Apr 4 at 0:36
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The physical theory of diffusion, to the extent it is applicable to any system, predicts Normal distributions of quantities (like temperature or concentration) that originate at a point. Indeed, a great many systems are diffusive (options prices, particle transport in homogeneous media, etc.), suggesting that examples are abundant assuming one is not so naive as to suppose that a Normal distribution must hold exactly out to unrealistically large or small values--that would be a misunderstanding of all physical theory.
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– whuber♦
Apr 3 at 22:27
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The normal distribution seems unintuitive until you learn that it maximizes the entropy under the constraint of fixed variance.
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– leonbloy
Apr 4 at 3:18