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Is it possible to have 2 different but equal size real number sets that have the same mean and standard deviation?

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Is it possible to have 2 different but equal size real number sets that have the same mean and standard deviation?

Is it possible to calculate the standard deviation without the samples?Calculating mean and standard deviation of very large sample sizesIf three groups of unequal sizes have the same standard deviation but different means, will the SD of the pooled data be the same?Does the value of a standard deviation have any meaningStandard deviation of the mean of sample datawhy the standard deviation is not as the same as online calculatorHow to create n datasets of size N with same mean but different standard deviation?Sample size given mean, standard deviationCan two distinct datasets of the same size have the same median and the same deviation from a real number?Mean of means and standard deviation

2

$begingroup$

By inspection I notice that

• Shifting does not change the standard deviation but change mean.
  1,3,4 has the same standard deviation as 11,13,14 for example.




• Sets with the same (or reversed) sequence of adjacent difference have the same standard deviation. For example, 1,3,4, 0,2,3, 0,1,3 have the same standard deviation. But the means are different.

My conjecture: There are no two distinct sets with the same length, mean and standard deviation.

Question

Is it possible to have 2 different but equal size real number sets that have the same mean and standard deviation?

standard-deviation means

share|cite|improve this question

edited 8 hours ago

Asaf Karagila♦

312k33445778

asked 8 hours ago

Money Oriented ProgrammerMoney Oriented Programmer

423210

$endgroup$

add a comment | 

2

$begingroup$

By inspection I notice that

• Shifting does not change the standard deviation but change mean.
  1,3,4 has the same standard deviation as 11,13,14 for example.




• Sets with the same (or reversed) sequence of adjacent difference have the same standard deviation. For example, 1,3,4, 0,2,3, 0,1,3 have the same standard deviation. But the means are different.

My conjecture: There are no two distinct sets with the same length, mean and standard deviation.

Question

Is it possible to have 2 different but equal size real number sets that have the same mean and standard deviation?

standard-deviation means

share|cite|improve this question

edited 8 hours ago

Asaf Karagila♦

312k33445778

asked 8 hours ago

Money Oriented ProgrammerMoney Oriented Programmer

423210

$endgroup$

add a comment | 

$begingroup$

By inspection I notice that

• Shifting does not change the standard deviation but change mean.
  1,3,4 has the same standard deviation as 11,13,14 for example.




• Sets with the same (or reversed) sequence of adjacent difference have the same standard deviation. For example, 1,3,4, 0,2,3, 0,1,3 have the same standard deviation. But the means are different.

My conjecture: There are no two distinct sets with the same length, mean and standard deviation.

Question

Is it possible to have 2 different but equal size real number sets that have the same mean and standard deviation?

standard-deviation means

share|cite|improve this question

edited 8 hours ago

Asaf Karagila♦

312k33445778

asked 8 hours ago

Money Oriented ProgrammerMoney Oriented Programmer

423210

$endgroup$

share|cite|improve this question

edited 8 hours ago

Asaf Karagila♦

312k33445778

asked 8 hours ago

Money Oriented ProgrammerMoney Oriented Programmer

423210

share|cite|improve this question

edited 8 hours ago

Asaf Karagila♦

312k33445778

asked 8 hours ago

Money Oriented ProgrammerMoney Oriented Programmer

423210

edited 8 hours ago

Asaf Karagila♦

312k33445778

edited 8 hours ago

Asaf Karagila♦

312k33445778

asked 8 hours ago

Money Oriented ProgrammerMoney Oriented Programmer

423210

asked 8 hours ago

Money Oriented ProgrammerMoney Oriented Programmer

423210

3 Answers
3

active

oldest

votes

6

$begingroup$

$-2,-1,3$ and $-3,1,2$ both have a mean of $0$ and a standard deviation of $sqrtfrac143$.

share|cite|improve this answer

edited 8 hours ago

answered 8 hours ago

auscryptauscrypt

5,836212

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MoneyOrientedProgrammer No, just add a constant to all the numbers.
  $endgroup$
  – Robert Israel
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MoneyOrientedProgrammer No, simply add $4$ to all numbers in the example given. Both means are now $4$ and the standard deviation is left unchanged.
  $endgroup$
  – auscrypt
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  There are infinitely many sets of 4 values which have any given mean and standard deviation >0
  $endgroup$
  – auscrypt
  7 hours ago
  

  
  
  
  

add a comment | 

4

$begingroup$

The example by auscrypt settles the question, but maybe it's worth mentioning why this should be obvious by considering degrees of freedom.

Mean and standard deviation are two quantities. A collection of $m$ real numbers has $m$ degrees of freedom. Specifying the mean and standard deviation removes two degrees of freedom, leaving $m-2$. So as long as $m > 2$, there should still be lots of room to have
different sets with the same mean and standard deviation.

share|cite|improve this answer

edited 8 hours ago

David G. Stork

13.2k41937

answered 8 hours ago

Robert IsraelRobert Israel

338k23231489

$endgroup$

add a comment | 

0

$begingroup$

Yes. Two sets of numbers has the same mean and the same SD iff their sum and the sum of their squares match.

The set $1,2,3$ has sum $6$ and squares' sum $14$. The set $x,y,z$ the same mean and SD iff
$$begincasesx+y+z=6\x^2+y^2+z^2=14endcases$$
This is the intersection of a spherical surface and a plane passing through the center, that has certainly infinitely many points.

share|cite|improve this answer

edited 8 hours ago

answered 8 hours ago

ajotatxeajotatxe

55.6k24291

$endgroup$

add a comment | 

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6

$begingroup$

$-2,-1,3$ and $-3,1,2$ both have a mean of $0$ and a standard deviation of $sqrtfrac143$.

share|cite|improve this answer

edited 8 hours ago

answered 8 hours ago

auscryptauscrypt

5,836212

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MoneyOrientedProgrammer No, just add a constant to all the numbers.
  $endgroup$
  – Robert Israel
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MoneyOrientedProgrammer No, simply add $4$ to all numbers in the example given. Both means are now $4$ and the standard deviation is left unchanged.
  $endgroup$
  – auscrypt
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  There are infinitely many sets of 4 values which have any given mean and standard deviation >0
  $endgroup$
  – auscrypt
  7 hours ago
  

  
  
  
  

add a comment | 

6

$begingroup$

$-2,-1,3$ and $-3,1,2$ both have a mean of $0$ and a standard deviation of $sqrtfrac143$.

share|cite|improve this answer

edited 8 hours ago

answered 8 hours ago

auscryptauscrypt

5,836212

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MoneyOrientedProgrammer No, just add a constant to all the numbers.
  $endgroup$
  – Robert Israel
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MoneyOrientedProgrammer No, simply add $4$ to all numbers in the example given. Both means are now $4$ and the standard deviation is left unchanged.
  $endgroup$
  – auscrypt
  8 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  There are infinitely many sets of 4 values which have any given mean and standard deviation >0
  $endgroup$
  – auscrypt
  7 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

$-2,-1,3$ and $-3,1,2$ both have a mean of $0$ and a standard deviation of $sqrtfrac143$.

share|cite|improve this answer

edited 8 hours ago

answered 8 hours ago

auscryptauscrypt

5,836212

$endgroup$

share|cite|improve this answer

edited 8 hours ago

answered 8 hours ago

auscryptauscrypt

5,836212

answered 8 hours ago

auscryptauscrypt

5,836212

answered 8 hours ago

auscryptauscrypt

5,836212

4

$begingroup$

The example by auscrypt settles the question, but maybe it's worth mentioning why this should be obvious by considering degrees of freedom.

Mean and standard deviation are two quantities. A collection of $m$ real numbers has $m$ degrees of freedom. Specifying the mean and standard deviation removes two degrees of freedom, leaving $m-2$. So as long as $m > 2$, there should still be lots of room to have
different sets with the same mean and standard deviation.

share|cite|improve this answer

edited 8 hours ago

David G. Stork

13.2k41937

answered 8 hours ago

Robert IsraelRobert Israel

338k23231489

$endgroup$

add a comment | 

4

$begingroup$

The example by auscrypt settles the question, but maybe it's worth mentioning why this should be obvious by considering degrees of freedom.

Mean and standard deviation are two quantities. A collection of $m$ real numbers has $m$ degrees of freedom. Specifying the mean and standard deviation removes two degrees of freedom, leaving $m-2$. So as long as $m > 2$, there should still be lots of room to have
different sets with the same mean and standard deviation.

share|cite|improve this answer

edited 8 hours ago

David G. Stork

13.2k41937

answered 8 hours ago

Robert IsraelRobert Israel

338k23231489

$endgroup$

add a comment | 

$begingroup$

The example by auscrypt settles the question, but maybe it's worth mentioning why this should be obvious by considering degrees of freedom.

Mean and standard deviation are two quantities. A collection of $m$ real numbers has $m$ degrees of freedom. Specifying the mean and standard deviation removes two degrees of freedom, leaving $m-2$. So as long as $m > 2$, there should still be lots of room to have
different sets with the same mean and standard deviation.

share|cite|improve this answer

edited 8 hours ago

David G. Stork

13.2k41937

answered 8 hours ago

Robert IsraelRobert Israel

338k23231489

$endgroup$

share|cite|improve this answer

edited 8 hours ago

David G. Stork

13.2k41937

answered 8 hours ago

Robert IsraelRobert Israel

338k23231489

edited 8 hours ago

David G. Stork

13.2k41937

edited 8 hours ago

David G. Stork

13.2k41937

answered 8 hours ago

Robert IsraelRobert Israel

338k23231489

answered 8 hours ago

Robert IsraelRobert Israel

338k23231489

0

$begingroup$

Yes. Two sets of numbers has the same mean and the same SD iff their sum and the sum of their squares match.

The set $1,2,3$ has sum $6$ and squares' sum $14$. The set $x,y,z$ the same mean and SD iff
$$begincasesx+y+z=6\x^2+y^2+z^2=14endcases$$
This is the intersection of a spherical surface and a plane passing through the center, that has certainly infinitely many points.

share|cite|improve this answer

edited 8 hours ago

answered 8 hours ago

ajotatxeajotatxe

55.6k24291

$endgroup$

add a comment | 

0

$begingroup$

Yes. Two sets of numbers has the same mean and the same SD iff their sum and the sum of their squares match.

The set $1,2,3$ has sum $6$ and squares' sum $14$. The set $x,y,z$ the same mean and SD iff
$$begincasesx+y+z=6\x^2+y^2+z^2=14endcases$$
This is the intersection of a spherical surface and a plane passing through the center, that has certainly infinitely many points.

share|cite|improve this answer

edited 8 hours ago

answered 8 hours ago

ajotatxeajotatxe

55.6k24291

$endgroup$

add a comment | 

$begingroup$

Yes. Two sets of numbers has the same mean and the same SD iff their sum and the sum of their squares match.

The set $1,2,3$ has sum $6$ and squares' sum $14$. The set $x,y,z$ the same mean and SD iff
$$begincasesx+y+z=6\x^2+y^2+z^2=14endcases$$
This is the intersection of a spherical surface and a plane passing through the center, that has certainly infinitely many points.

share|cite|improve this answer

edited 8 hours ago

answered 8 hours ago

ajotatxeajotatxe

55.6k24291

$endgroup$

share|cite|improve this answer

edited 8 hours ago

answered 8 hours ago

ajotatxeajotatxe

55.6k24291

answered 8 hours ago

ajotatxeajotatxe

55.6k24291

answered 8 hours ago

ajotatxeajotatxe

55.6k24291