What is inside of the 200 star chest?

A word that means "blending into a community too much"

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

Generate basis elements of the Steenrod algebra

Let M and N be single-digit integers. If the product 2M5 x 13N is divisible by 36, how many ordered pairs (M,N) are possible?

Why can I traceroute to this IP address, but not ping?

Moving points closer to polyline using ModelBuilder?

Active low-pass filters --- good to what frequencies?

Russian word for a male zebra

Teaching a class likely meant to inflate the GPA of student athletes

A map of non-pathological topology?

New pedal fell off maybe 50 miles after installation. Should I replace the entire crank, just the arm, or repair the thread?

Why can my keyboard only digest 6 keypresses at a time?

Cascading Switches. Will it affect performance?

LuaLaTex - how to use number, computed later in the document

How did old MS-DOS games utilize various graphic cards?

Overlapping String-Blocks

Wooden cooking layout

Traversing Oceania: A Cryptic Journey

Can I utilise a baking stone to make crepes?

Longest bridge/tunnel that can be cycled over/through?

US doctor working in Tripoli wants me to open online account

60s or 70s novel about Empire of Man making 1st contact with 1st discovered alien race

Align equations within one column

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

1

$begingroup$

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 10 hours ago

Asaf Karagila♦

312k33445778

asked 10 hours ago

Money Oriented ProgrammerMoney Oriented Programmer

423210

$endgroup$

add a comment | 

2

1

$begingroup$

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 10 hours ago

Asaf Karagila♦

312k33445778

asked 10 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 10 hours ago

Asaf Karagila♦

312k33445778

asked 10 hours ago

Money Oriented ProgrammerMoney Oriented Programmer

423210

$endgroup$

share|cite|improve this question

edited 10 hours ago

Asaf Karagila♦

312k33445778

asked 10 hours ago

Money Oriented ProgrammerMoney Oriented Programmer

423210

share|cite|improve this question

edited 10 hours ago

Asaf Karagila♦

312k33445778

asked 10 hours ago

Money Oriented ProgrammerMoney Oriented Programmer

423210

edited 10 hours ago

Asaf Karagila♦

312k33445778

edited 10 hours ago

Asaf Karagila♦

312k33445778

asked 10 hours ago

Money Oriented ProgrammerMoney Oriented Programmer

423210

asked 10 hours ago

Money Oriented ProgrammerMoney Oriented Programmer

423210

4 Answers
4

active

oldest

votes

8

$begingroup$

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

share|cite|improve this answer

edited 10 hours ago

answered 10 hours ago

auscryptauscrypt

5,951212

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MoneyOrientedProgrammer No, just add a constant to all the numbers.
  $endgroup$
  – Robert Israel
  10 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
  10 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  

add a comment | 

8

$begingroup$

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 10 hours ago

David G. Stork

13.2k41937

answered 10 hours ago

Robert IsraelRobert Israel

338k23231489

$endgroup$

add a comment | 

1

$begingroup$

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 10 hours ago

answered 10 hours ago

ajotatxeajotatxe

55.6k24291

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Nitpick: the plane $x + y + z = 6$ does not pass through the center of the sphere $x^2 + y^2 + z^2 = 14$.
  $endgroup$
  – Michael Seifert
  2 hours ago
  

  
  
  
  

add a comment | 

0

$begingroup$

Let $A = x_1, x_2, dots, x_n$ add up to $nmu$.

Then $B = 2mu-x_1, 2mu-x_2, dots, 2mu-x_n$ will also add up to $nmu$.

We will also have
beginalign
sum_i=1^n (2mu - x_i)^2
&= 4nmu^2 -4musum_i=1^n x_i + sum_i=1^n x_i^2\
&= 4nmu^2 - 4nmu^2 + sum_i=1^n x_i^2\
&= sum_i=1^n x_i^2
endalign

Hence the sets $A$ and $B$ have the same mean and standard deviation.

As a side note, the $i^th$ and $(n-i)^th$ rows of the many $ntimes n$ magic squares, when $n$ is odd, have this property.

share|cite|improve this answer

edited 1 hour ago

answered 1 hour ago

steven gregorysteven gregory

18.7k32659

$endgroup$

add a comment | 

Your Answer

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3253262%2fis-it-possible-to-have-2-different-but-equal-size-real-number-sets-that-have-the%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

8

$begingroup$

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

share|cite|improve this answer

edited 10 hours ago

answered 10 hours ago

auscryptauscrypt

5,951212

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MoneyOrientedProgrammer No, just add a constant to all the numbers.
  $endgroup$
  – Robert Israel
  10 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
  10 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  

add a comment | 

8

$begingroup$

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

share|cite|improve this answer

edited 10 hours ago

answered 10 hours ago

auscryptauscrypt

5,951212

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MoneyOrientedProgrammer No, just add a constant to all the numbers.
  $endgroup$
  – Robert Israel
  10 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
  10 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  There are infinitely many sets of 4 values which have any given mean and standard deviation >0
  $endgroup$
  – auscrypt
  9 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 10 hours ago

answered 10 hours ago

auscryptauscrypt

5,951212

$endgroup$

share|cite|improve this answer

edited 10 hours ago

answered 10 hours ago

auscryptauscrypt

5,951212

answered 10 hours ago

auscryptauscrypt

5,951212

answered 10 hours ago

auscryptauscrypt

5,951212

8

$begingroup$

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 10 hours ago

David G. Stork

13.2k41937

answered 10 hours ago

Robert IsraelRobert Israel

338k23231489

$endgroup$

add a comment | 

8

$begingroup$

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 10 hours ago

David G. Stork

13.2k41937

answered 10 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 10 hours ago

David G. Stork

13.2k41937

answered 10 hours ago

Robert IsraelRobert Israel

338k23231489

$endgroup$

share|cite|improve this answer

edited 10 hours ago

David G. Stork

13.2k41937

answered 10 hours ago

Robert IsraelRobert Israel

338k23231489

edited 10 hours ago

David G. Stork

13.2k41937

edited 10 hours ago

David G. Stork

13.2k41937

answered 10 hours ago

Robert IsraelRobert Israel

338k23231489

answered 10 hours ago

Robert IsraelRobert Israel

338k23231489

1

$begingroup$

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 10 hours ago

answered 10 hours ago

ajotatxeajotatxe

55.6k24291

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Nitpick: the plane $x + y + z = 6$ does not pass through the center of the sphere $x^2 + y^2 + z^2 = 14$.
  $endgroup$
  – Michael Seifert
  2 hours ago
  

  
  
  
  

add a comment | 

1

$begingroup$

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 10 hours ago

answered 10 hours ago

ajotatxeajotatxe

55.6k24291

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Nitpick: the plane $x + y + z = 6$ does not pass through the center of the sphere $x^2 + y^2 + z^2 = 14$.
  $endgroup$
  – Michael Seifert
  2 hours ago
  

  
  
  
  

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 10 hours ago

answered 10 hours ago

ajotatxeajotatxe

55.6k24291

$endgroup$

share|cite|improve this answer

edited 10 hours ago

answered 10 hours ago

ajotatxeajotatxe

55.6k24291

answered 10 hours ago

ajotatxeajotatxe

55.6k24291

answered 10 hours ago

ajotatxeajotatxe

55.6k24291

0

$begingroup$

Let $A = x_1, x_2, dots, x_n$ add up to $nmu$.

Then $B = 2mu-x_1, 2mu-x_2, dots, 2mu-x_n$ will also add up to $nmu$.

We will also have
beginalign
sum_i=1^n (2mu - x_i)^2
&= 4nmu^2 -4musum_i=1^n x_i + sum_i=1^n x_i^2\
&= 4nmu^2 - 4nmu^2 + sum_i=1^n x_i^2\
&= sum_i=1^n x_i^2
endalign

Hence the sets $A$ and $B$ have the same mean and standard deviation.

As a side note, the $i^th$ and $(n-i)^th$ rows of the many $ntimes n$ magic squares, when $n$ is odd, have this property.

share|cite|improve this answer

edited 1 hour ago

answered 1 hour ago

steven gregorysteven gregory

18.7k32659

$endgroup$

add a comment | 

0

$begingroup$

Let $A = x_1, x_2, dots, x_n$ add up to $nmu$.

Then $B = 2mu-x_1, 2mu-x_2, dots, 2mu-x_n$ will also add up to $nmu$.

We will also have
beginalign
sum_i=1^n (2mu - x_i)^2
&= 4nmu^2 -4musum_i=1^n x_i + sum_i=1^n x_i^2\
&= 4nmu^2 - 4nmu^2 + sum_i=1^n x_i^2\
&= sum_i=1^n x_i^2
endalign

Hence the sets $A$ and $B$ have the same mean and standard deviation.

As a side note, the $i^th$ and $(n-i)^th$ rows of the many $ntimes n$ magic squares, when $n$ is odd, have this property.

share|cite|improve this answer

edited 1 hour ago

answered 1 hour ago

steven gregorysteven gregory

18.7k32659

$endgroup$

add a comment | 

$begingroup$

Let $A = x_1, x_2, dots, x_n$ add up to $nmu$.

Then $B = 2mu-x_1, 2mu-x_2, dots, 2mu-x_n$ will also add up to $nmu$.

We will also have
beginalign
sum_i=1^n (2mu - x_i)^2
&= 4nmu^2 -4musum_i=1^n x_i + sum_i=1^n x_i^2\
&= 4nmu^2 - 4nmu^2 + sum_i=1^n x_i^2\
&= sum_i=1^n x_i^2
endalign

Hence the sets $A$ and $B$ have the same mean and standard deviation.

As a side note, the $i^th$ and $(n-i)^th$ rows of the many $ntimes n$ magic squares, when $n$ is odd, have this property.

share|cite|improve this answer

edited 1 hour ago

answered 1 hour ago

steven gregorysteven gregory

18.7k32659

$endgroup$

share|cite|improve this answer

edited 1 hour ago

answered 1 hour ago

steven gregorysteven gregory

18.7k32659

answered 1 hour ago

steven gregorysteven gregory

18.7k32659

answered 1 hour ago

steven gregorysteven gregory

18.7k32659