Are there any examples of a variable being normally distributed that is *not* due to the Central Limit Theorem?Central limit theorem and the law of large numbersCentral limit theorem when the mean is not constantWhy does the central limit theorem work with a single sample?The central limit theorem, What it meansUnderstanding the Central Limit Theorem (CLT)How does the Central Limit Theorem show that the Binomial Distribution is approximately Normal for a large value of n?For which parameters does the Central Limit Theorem work?What distributions don't follow the central limit theorem?How can the central limit theorem hold for distributions which have limits on the random variable?Are there any examples of where the central limit theorem does not hold?
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Are there any examples of a variable being normally distributed that is *not* due to the Central Limit Theorem?
Central limit theorem and the law of large numbersCentral limit theorem when the mean is not constantWhy does the central limit theorem work with a single sample?The central limit theorem, What it meansUnderstanding the Central Limit Theorem (CLT)How does the Central Limit Theorem show that the Binomial Distribution is approximately Normal for a large value of n?For which parameters does the Central Limit Theorem work?What distributions don't follow the central limit theorem?How can the central limit theorem hold for distributions which have limits on the random variable?Are there any examples of where the central limit theorem does not hold?
$begingroup$
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 1 hour ago
gardenheadgardenhead
1463
$endgroup$
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 1 hour ago
gardenheadgardenhead
1463
$endgroup$
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 1 hour ago
gardenheadgardenhead
1463
$endgroup$
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 1 hour ago
gardenheadgardenhead
1463
asked 1 hour ago
gardenheadgardenhead
1463
asked 1 hour ago
gardenheadgardenhead
1463
asked 1 hour ago
gardenheadgardenhead
1463
asked 1 hour ago
gardenheadgardenhead
1463
1463
add a comment |
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 is 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 the modeling errors of such observations, there was a debate between Gauss and Laplace, each in favor of 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 32 mins ago
BruceETBruceET
6,1881720
$endgroup$
$begingroup$
Yep. Still fixing typos. Thanks for noticing this one. Same error in 'test scores' also fixed.
$endgroup$
– BruceET
6 mins ago
add a comment |
$begingroup$
Lots of naturally occurring variables are normally distributed. Heights of humans? Size of animal colonies?
answered 1 hour ago
HappyHappy
112
New contributor
Happy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
2
$begingroup$
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
58 mins ago
$begingroup$
@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
58 mins ago
$begingroup$
Human height is the result of the sum of (approximately) independent genes, so they actually are due to the CLT.
$endgroup$
– gardenhead
17 mins ago
1
$begingroup$
@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
10 mins ago
$begingroup$
@CliffAB that's a fair point, but I never said anything about it not being a valid approximation. In fact, I said that it's a useful approximation, but of course heights are not truly normally distributed
$endgroup$
– Artem Mavrin
7 mins ago
|
show 2 more comments
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2 Answers
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active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
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 is 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 the modeling errors of such observations, there was a debate between Gauss and Laplace, each in favor of 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 32 mins ago
BruceETBruceET
6,1881720
$endgroup$
$begingroup$
Yep. Still fixing typos. Thanks for noticing this one. Same error in 'test scores' also fixed.
$endgroup$
– BruceET
6 mins ago
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 is 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 the modeling errors of such observations, there was a debate between Gauss and Laplace, each in favor of 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 32 mins ago
BruceETBruceET
6,1881720
$endgroup$
$begingroup$
Yep. Still fixing typos. Thanks for noticing this one. Same error in 'test scores' also fixed.
$endgroup$
– BruceET
6 mins ago
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 is 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 the modeling errors of such observations, there was a debate between Gauss and Laplace, each in favor of 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 32 mins ago
BruceETBruceET
6,1881720
$endgroup$
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 is 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 the modeling errors of such observations, there was a debate between Gauss and Laplace, each in favor of 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 32 mins ago
BruceETBruceET
6,1881720
edited 7 mins ago
answered 32 mins ago
BruceETBruceET
6,1881720
answered 32 mins ago
BruceETBruceET
6,1881720
answered 32 mins ago
BruceETBruceET
6,1881720
6,1881720
$begingroup$
Yep. Still fixing typos. Thanks for noticing this one. Same error in 'test scores' also fixed.
$endgroup$
– BruceET
6 mins ago
add a comment |
$begingroup$
Yep. Still fixing typos. Thanks for noticing this one. Same error in 'test scores' also fixed.
$endgroup$
– BruceET
6 mins ago
$begingroup$
Yep. Still fixing typos. Thanks for noticing this one. Same error in 'test scores' also fixed.
$endgroup$
– BruceET
6 mins ago
$begingroup$
Yep. Still fixing typos. Thanks for noticing this one. Same error in 'test scores' also fixed.
$endgroup$
– BruceET
6 mins ago
add a comment |
$begingroup$
Lots of naturally occurring variables are normally distributed. Heights of humans? Size of animal colonies?
answered 1 hour ago
HappyHappy
112
New contributor
Happy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
2
$begingroup$
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
58 mins ago
$begingroup$
@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
58 mins ago
$begingroup$
Human height is the result of the sum of (approximately) independent genes, so they actually are due to the CLT.
$endgroup$
– gardenhead
17 mins ago
1
$begingroup$
@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
10 mins ago
$begingroup$
@CliffAB that's a fair point, but I never said anything about it not being a valid approximation. In fact, I said that it's a useful approximation, but of course heights are not truly normally distributed
$endgroup$
– Artem Mavrin
7 mins ago
|
show 2 more comments
$begingroup$
Lots of naturally occurring variables are normally distributed. Heights of humans? Size of animal colonies?
answered 1 hour ago
HappyHappy
112
New contributor
Happy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
2
$begingroup$
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
58 mins ago
$begingroup$
@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
58 mins ago
$begingroup$
Human height is the result of the sum of (approximately) independent genes, so they actually are due to the CLT.
$endgroup$
– gardenhead
17 mins ago
1
$begingroup$
@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
10 mins ago
$begingroup$
@CliffAB that's a fair point, but I never said anything about it not being a valid approximation. In fact, I said that it's a useful approximation, but of course heights are not truly normally distributed
$endgroup$
– Artem Mavrin
7 mins ago
|
show 2 more comments
$begingroup$
Lots of naturally occurring variables are normally distributed. Heights of humans? Size of animal colonies?
answered 1 hour ago
HappyHappy
112
New contributor
Happy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Lots of naturally occurring variables are normally distributed. Heights of humans? Size of animal colonies?
answered 1 hour ago
HappyHappy
112
New contributor
Happy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 1 hour ago
HappyHappy
112
New contributor
Happy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 1 hour ago
HappyHappy
112
answered 1 hour ago
HappyHappy
112
112
New contributor
Happy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Happy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Happy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
2
$begingroup$
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
58 mins ago
$begingroup$
@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
58 mins ago
$begingroup$
Human height is the result of the sum of (approximately) independent genes, so they actually are due to the CLT.
$endgroup$
– gardenhead
17 mins ago
1
$begingroup$
@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
10 mins ago
$begingroup$
@CliffAB that's a fair point, but I never said anything about it not being a valid approximation. In fact, I said that it's a useful approximation, but of course heights are not truly normally distributed
$endgroup$
– Artem Mavrin
7 mins ago
|
show 2 more comments
2
$begingroup$
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
58 mins ago
$begingroup$
@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
58 mins ago
$begingroup$
Human height is the result of the sum of (approximately) independent genes, so they actually are due to the CLT.
$endgroup$
– gardenhead
17 mins ago
1
$begingroup$
@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
10 mins ago
$begingroup$
@CliffAB that's a fair point, but I never said anything about it not being a valid approximation. In fact, I said that it's a useful approximation, but of course heights are not truly normally distributed
$endgroup$
– Artem Mavrin
7 mins ago
2
2
$begingroup$
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
58 mins ago
$begingroup$
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
58 mins ago
$begingroup$
@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
58 mins ago
$begingroup$
@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
58 mins ago
$begingroup$
Human height is the result of the sum of (approximately) independent genes, so they actually are due to the CLT.
$endgroup$
– gardenhead
17 mins ago
$begingroup$
Human height is the result of the sum of (approximately) independent genes, so they actually are due to the CLT.
$endgroup$
– gardenhead
17 mins ago
1
1
$begingroup$
@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
10 mins ago
$begingroup$
@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.
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– Cliff AB
10 mins ago
$begingroup$
@CliffAB that's a fair point, but I never said anything about it not being a valid approximation. In fact, I said that it's a useful approximation, but of course heights are not truly normally distributed
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– Artem Mavrin
7 mins ago
$begingroup$
@CliffAB that's a fair point, but I never said anything about it not being a valid approximation. In fact, I said that it's a useful approximation, but of course heights are not truly normally distributed
$endgroup$
– Artem Mavrin
7 mins ago
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