Why does the function VarianceMLE give a different result from Variance?












3












$begingroup$


Why does the function VarianceMLE give a different result from Variance?



And what is it in Mathematica 11.3?



enter image description here



Please see the picture above t665he MLE is 621 and the other is 665.






probability-or-statistics







share|improve this question









New contributor




Facet is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$endgroup$












  • $begingroup$
    In Mathematica 11.3, << Statistics` produces an error message and Variance@data // N gives 665.524
    $endgroup$
    – m_goldberg
    21 hours ago








  • 2




    $begingroup$
    I'm going to guess that VarianceMLE is the maximum likelihood variance estimator rather than the unbiased one (assuming normally distributed data). The difference between the two is that Variance divides by N-1 (N == Length[data]) while the MLE estimator divides by N.
    $endgroup$
    – Sjoerd Smit
    19 hours ago






  • 1




    $begingroup$
    And for future reference: please post copyable code in your question rather than a screenshot. This makes it much easier for someone else to copy your code and try things out.
    $endgroup$
    – Sjoerd Smit
    19 hours ago












  • $begingroup$
    Which book did you see this in? It looks like a scan.
    $endgroup$
    – Szabolcs
    17 hours ago
















3












$begingroup$


Why does the function VarianceMLE give a different result from Variance?



And what is it in Mathematica 11.3?



enter image description here



Please see the picture above t665he MLE is 621 and the other is 665.






probability-or-statistics







share|improve this question









New contributor




Facet is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$












  • $begingroup$
    In Mathematica 11.3, << Statistics` produces an error message and Variance@data // N gives 665.524
    $endgroup$
    – m_goldberg
    21 hours ago








  • 2




    $begingroup$
    I'm going to guess that VarianceMLE is the maximum likelihood variance estimator rather than the unbiased one (assuming normally distributed data). The difference between the two is that Variance divides by N-1 (N == Length[data]) while the MLE estimator divides by N.
    $endgroup$
    – Sjoerd Smit
    19 hours ago






  • 1




    $begingroup$
    And for future reference: please post copyable code in your question rather than a screenshot. This makes it much easier for someone else to copy your code and try things out.
    $endgroup$
    – Sjoerd Smit
    19 hours ago












  • $begingroup$
    Which book did you see this in? It looks like a scan.
    $endgroup$
    – Szabolcs
    17 hours ago














3












3








3





$begingroup$


Why does the function VarianceMLE give a different result from Variance?



And what is it in Mathematica 11.3?



enter image description here



Please see the picture above t665he MLE is 621 and the other is 665.






probability-or-statistics







share|improve this question









New contributor




Facet is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




Why does the function VarianceMLE give a different result from Variance?



And what is it in Mathematica 11.3?



enter image description here



Please see the picture above t665he MLE is 621 and the other is 665.






probability-or-statistics




probability-or-statistics






share|improve this question









New contributor




Facet is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|improve this question









New contributor




Facet is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|improve this question




share|improve this question








edited 20 hours ago









m_goldberg

84.7k872196




84.7k872196






New contributor




Facet is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked 21 hours ago









FacetFacet

254




254




New contributor




Facet is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





Facet is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Facet is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












  • $begingroup$
    In Mathematica 11.3, << Statistics` produces an error message and Variance@data // N gives 665.524
    $endgroup$
    – m_goldberg
    21 hours ago








  • 2




    $begingroup$
    I'm going to guess that VarianceMLE is the maximum likelihood variance estimator rather than the unbiased one (assuming normally distributed data). The difference between the two is that Variance divides by N-1 (N == Length[data]) while the MLE estimator divides by N.
    $endgroup$
    – Sjoerd Smit
    19 hours ago






  • 1




    $begingroup$
    And for future reference: please post copyable code in your question rather than a screenshot. This makes it much easier for someone else to copy your code and try things out.
    $endgroup$
    – Sjoerd Smit
    19 hours ago












  • $begingroup$
    Which book did you see this in? It looks like a scan.
    $endgroup$
    – Szabolcs
    17 hours ago


















  • $begingroup$
    In Mathematica 11.3, << Statistics` produces an error message and Variance@data // N gives 665.524
    $endgroup$
    – m_goldberg
    21 hours ago








  • 2




    $begingroup$
    I'm going to guess that VarianceMLE is the maximum likelihood variance estimator rather than the unbiased one (assuming normally distributed data). The difference between the two is that Variance divides by N-1 (N == Length[data]) while the MLE estimator divides by N.
    $endgroup$
    – Sjoerd Smit
    19 hours ago






  • 1




    $begingroup$
    And for future reference: please post copyable code in your question rather than a screenshot. This makes it much easier for someone else to copy your code and try things out.
    $endgroup$
    – Sjoerd Smit
    19 hours ago












  • $begingroup$
    Which book did you see this in? It looks like a scan.
    $endgroup$
    – Szabolcs
    17 hours ago
















$begingroup$
In Mathematica 11.3, << Statistics` produces an error message and Variance@data // N gives 665.524
$endgroup$
– m_goldberg
21 hours ago






$begingroup$
In Mathematica 11.3, << Statistics` produces an error message and Variance@data // N gives 665.524
$endgroup$
– m_goldberg
21 hours ago






2




2




$begingroup$
I'm going to guess that VarianceMLE is the maximum likelihood variance estimator rather than the unbiased one (assuming normally distributed data). The difference between the two is that Variance divides by N-1 (N == Length[data]) while the MLE estimator divides by N.
$endgroup$
– Sjoerd Smit
19 hours ago




$begingroup$
I'm going to guess that VarianceMLE is the maximum likelihood variance estimator rather than the unbiased one (assuming normally distributed data). The difference between the two is that Variance divides by N-1 (N == Length[data]) while the MLE estimator divides by N.
$endgroup$
– Sjoerd Smit
19 hours ago




1




1




$begingroup$
And for future reference: please post copyable code in your question rather than a screenshot. This makes it much easier for someone else to copy your code and try things out.
$endgroup$
– Sjoerd Smit
19 hours ago






$begingroup$
And for future reference: please post copyable code in your question rather than a screenshot. This makes it much easier for someone else to copy your code and try things out.
$endgroup$
– Sjoerd Smit
19 hours ago














$begingroup$
Which book did you see this in? It looks like a scan.
$endgroup$
– Szabolcs
17 hours ago




$begingroup$
Which book did you see this in? It looks like a scan.
$endgroup$
– Szabolcs
17 hours ago










2 Answers
2






active

oldest

votes


















8












$begingroup$

I just checked my guess in my comment and I was right. VarianceMLE is the maximum likelihood variance estimator (see, e.g. here).



data = {34, 56, 28, 62, 32, 90, 20, 10, 12, 35, 63, 78, 12, 25, 68};

Variance[data]




13976/21





myVariance[lst_List] := Total[(lst - Mean[lst])^2]/(Length[lst] - 1);

myVarianceMLE[lst_List] := Total[(lst - Mean[lst])^2]/Length[lst];



myVariance[data]

myVarianceMLE[data]




13976/21



27952/45








share|improve this answer









$endgroup$













  • $begingroup$
    Thank you for your kind help, I will remember to post code next time.
    $endgroup$
    – Facet
    14 hours ago



















3












$begingroup$

VarianceMLE computes a biased, maximum likelihood estimate of the population variance. Variance computes an unbiased estimate of the population variance. It can be shown that VarianceMLE underestimates the variance of the population.



Let ${y_i : 1 leq i leq n}$ be a sample of $n$ values from a population. The variance (central second moment) of the sample is
$$ sigma_y^2 = frac{1}{n} sum_{i=1}^n (y_i - bar{y}) text{,} $$
where $bar{y} = frac{1}{n} sum_{i=1}^n y_i$ is the sample mean. This $sigma_y^2$ is computed by VarianceMLE.



If $sigma^2$ is the population variance, with some work, one can show that the expected value of $sigma_y^2$ is $frac{n-1}{n} sigma^2$, so the sample variance is a biased estimator of the population variance. We can make this an unbiased estimator via
$$ s^2 = frac{n}{n-1} sigma_y^2 = frac{1}{n-1}sum_{i=1}^n (y_i - bar{y}) text{.} $$
This $s^2$ is computed by Variance. From the documentation (in the Details):



"Variance[list] is equivalent to Total[(list-Mean[list])^2]/(Length[list]-1) for real-valued data."



The very sparse documentation for VarianceMLE indicates that it is implemented in terms of Variance:



"VarianceMLE[data_] := Variance[data] (Length[data] - 1)/Length[data]"






share|improve this answer









$endgroup$













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    2 Answers
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    2 Answers
    2






    active

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    active

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    active

    oldest

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    8












    $begingroup$

    I just checked my guess in my comment and I was right. VarianceMLE is the maximum likelihood variance estimator (see, e.g. here).



    data = {34, 56, 28, 62, 32, 90, 20, 10, 12, 35, 63, 78, 12, 25, 68};
    
Variance[data]




    13976/21





    myVariance[lst_List] := Total[(lst - Mean[lst])^2]/(Length[lst] - 1);
    
myVarianceMLE[lst_List] := Total[(lst - Mean[lst])^2]/Length[lst];
    

    
myVariance[data]
    
myVarianceMLE[data]




    13976/21



    27952/45








    share|improve this answer









    $endgroup$













    • $begingroup$
      Thank you for your kind help, I will remember to post code next time.
      $endgroup$
      – Facet
      14 hours ago
















    8












    $begingroup$

    I just checked my guess in my comment and I was right. VarianceMLE is the maximum likelihood variance estimator (see, e.g. here).



    data = {34, 56, 28, 62, 32, 90, 20, 10, 12, 35, 63, 78, 12, 25, 68};
    
Variance[data]




    13976/21





    myVariance[lst_List] := Total[(lst - Mean[lst])^2]/(Length[lst] - 1);
    
myVarianceMLE[lst_List] := Total[(lst - Mean[lst])^2]/Length[lst];
    

    
myVariance[data]
    
myVarianceMLE[data]




    13976/21



    27952/45








    share|improve this answer









    $endgroup$













    • $begingroup$
      Thank you for your kind help, I will remember to post code next time.
      $endgroup$
      – Facet
      14 hours ago














    8












    8








    8





    $begingroup$

    I just checked my guess in my comment and I was right. VarianceMLE is the maximum likelihood variance estimator (see, e.g. here).



    data = {34, 56, 28, 62, 32, 90, 20, 10, 12, 35, 63, 78, 12, 25, 68};
    
Variance[data]




    13976/21





    myVariance[lst_List] := Total[(lst - Mean[lst])^2]/(Length[lst] - 1);
    
myVarianceMLE[lst_List] := Total[(lst - Mean[lst])^2]/Length[lst];
    

    
myVariance[data]
    
myVarianceMLE[data]




    13976/21



    27952/45








    share|improve this answer









    $endgroup$



    I just checked my guess in my comment and I was right. VarianceMLE is the maximum likelihood variance estimator (see, e.g. here).



    data = {34, 56, 28, 62, 32, 90, 20, 10, 12, 35, 63, 78, 12, 25, 68};
    
Variance[data]




    13976/21





    myVariance[lst_List] := Total[(lst - Mean[lst])^2]/(Length[lst] - 1);
    
myVarianceMLE[lst_List] := Total[(lst - Mean[lst])^2]/Length[lst];
    

    
myVariance[data]
    
myVarianceMLE[data]




    13976/21



    27952/45









    share|improve this answer












    share|improve this answer



    share|improve this answer










    answered 19 hours ago









    Sjoerd SmitSjoerd Smit

    3,410715




    3,410715












    • $begingroup$
      Thank you for your kind help, I will remember to post code next time.
      $endgroup$
      – Facet
      14 hours ago


















    • $begingroup$
      Thank you for your kind help, I will remember to post code next time.
      $endgroup$
      – Facet
      14 hours ago
















    $begingroup$
    Thank you for your kind help, I will remember to post code next time.
    $endgroup$
    – Facet
    14 hours ago




    $begingroup$
    Thank you for your kind help, I will remember to post code next time.
    $endgroup$
    – Facet
    14 hours ago











    3












    $begingroup$

    VarianceMLE computes a biased, maximum likelihood estimate of the population variance. Variance computes an unbiased estimate of the population variance. It can be shown that VarianceMLE underestimates the variance of the population.



    Let ${y_i : 1 leq i leq n}$ be a sample of $n$ values from a population. The variance (central second moment) of the sample is
    $$ sigma_y^2 = frac{1}{n} sum_{i=1}^n (y_i - bar{y}) text{,} $$
    where $bar{y} = frac{1}{n} sum_{i=1}^n y_i$ is the sample mean. This $sigma_y^2$ is computed by VarianceMLE.



    If $sigma^2$ is the population variance, with some work, one can show that the expected value of $sigma_y^2$ is $frac{n-1}{n} sigma^2$, so the sample variance is a biased estimator of the population variance. We can make this an unbiased estimator via
    $$ s^2 = frac{n}{n-1} sigma_y^2 = frac{1}{n-1}sum_{i=1}^n (y_i - bar{y}) text{.} $$
    This $s^2$ is computed by Variance. From the documentation (in the Details):



    "Variance[list] is equivalent to Total[(list-Mean[list])^2]/(Length[list]-1) for real-valued data."



    The very sparse documentation for VarianceMLE indicates that it is implemented in terms of Variance:



    "VarianceMLE[data_] := Variance[data] (Length[data] - 1)/Length[data]"






    share|improve this answer









    $endgroup$


















      3












      $begingroup$

      VarianceMLE computes a biased, maximum likelihood estimate of the population variance. Variance computes an unbiased estimate of the population variance. It can be shown that VarianceMLE underestimates the variance of the population.



      Let ${y_i : 1 leq i leq n}$ be a sample of $n$ values from a population. The variance (central second moment) of the sample is
      $$ sigma_y^2 = frac{1}{n} sum_{i=1}^n (y_i - bar{y}) text{,} $$
      where $bar{y} = frac{1}{n} sum_{i=1}^n y_i$ is the sample mean. This $sigma_y^2$ is computed by VarianceMLE.



      If $sigma^2$ is the population variance, with some work, one can show that the expected value of $sigma_y^2$ is $frac{n-1}{n} sigma^2$, so the sample variance is a biased estimator of the population variance. We can make this an unbiased estimator via
      $$ s^2 = frac{n}{n-1} sigma_y^2 = frac{1}{n-1}sum_{i=1}^n (y_i - bar{y}) text{.} $$
      This $s^2$ is computed by Variance. From the documentation (in the Details):



      "Variance[list] is equivalent to Total[(list-Mean[list])^2]/(Length[list]-1) for real-valued data."



      The very sparse documentation for VarianceMLE indicates that it is implemented in terms of Variance:



      "VarianceMLE[data_] := Variance[data] (Length[data] - 1)/Length[data]"






      share|improve this answer









      $endgroup$
















        3












        3








        3





        $begingroup$

        VarianceMLE computes a biased, maximum likelihood estimate of the population variance. Variance computes an unbiased estimate of the population variance. It can be shown that VarianceMLE underestimates the variance of the population.



        Let ${y_i : 1 leq i leq n}$ be a sample of $n$ values from a population. The variance (central second moment) of the sample is
        $$ sigma_y^2 = frac{1}{n} sum_{i=1}^n (y_i - bar{y}) text{,} $$
        where $bar{y} = frac{1}{n} sum_{i=1}^n y_i$ is the sample mean. This $sigma_y^2$ is computed by VarianceMLE.



        If $sigma^2$ is the population variance, with some work, one can show that the expected value of $sigma_y^2$ is $frac{n-1}{n} sigma^2$, so the sample variance is a biased estimator of the population variance. We can make this an unbiased estimator via
        $$ s^2 = frac{n}{n-1} sigma_y^2 = frac{1}{n-1}sum_{i=1}^n (y_i - bar{y}) text{.} $$
        This $s^2$ is computed by Variance. From the documentation (in the Details):



        "Variance[list] is equivalent to Total[(list-Mean[list])^2]/(Length[list]-1) for real-valued data."



        The very sparse documentation for VarianceMLE indicates that it is implemented in terms of Variance:



        "VarianceMLE[data_] := Variance[data] (Length[data] - 1)/Length[data]"






        share|improve this answer









        $endgroup$



        VarianceMLE computes a biased, maximum likelihood estimate of the population variance. Variance computes an unbiased estimate of the population variance. It can be shown that VarianceMLE underestimates the variance of the population.



        Let ${y_i : 1 leq i leq n}$ be a sample of $n$ values from a population. The variance (central second moment) of the sample is
        $$ sigma_y^2 = frac{1}{n} sum_{i=1}^n (y_i - bar{y}) text{,} $$
        where $bar{y} = frac{1}{n} sum_{i=1}^n y_i$ is the sample mean. This $sigma_y^2$ is computed by VarianceMLE.



        If $sigma^2$ is the population variance, with some work, one can show that the expected value of $sigma_y^2$ is $frac{n-1}{n} sigma^2$, so the sample variance is a biased estimator of the population variance. We can make this an unbiased estimator via
        $$ s^2 = frac{n}{n-1} sigma_y^2 = frac{1}{n-1}sum_{i=1}^n (y_i - bar{y}) text{.} $$
        This $s^2$ is computed by Variance. From the documentation (in the Details):



        "Variance[list] is equivalent to Total[(list-Mean[list])^2]/(Length[list]-1) for real-valued data."



        The very sparse documentation for VarianceMLE indicates that it is implemented in terms of Variance:



        "VarianceMLE[data_] := Variance[data] (Length[data] - 1)/Length[data]"







        share|improve this answer












        share|improve this answer



        share|improve this answer










        answered 13 hours ago









        Eric TowersEric Towers

        2,286613




        2,286613






















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            .mw-parser-output .nota-disambigua{clear:both;margin-bottom:.5em;border:1px solid #CCC;padding-left:4px}.mw-parser-output .nota-disambigua i{vertical-align:middle} Disambiguazione – Se stai cercando il teorema, vedi Teorema di deduzione . Disambiguazione – Se stai cercando il concetto di deduzione in economia, vedi deduzione fiscale . Il ragionamento deduttivo umano assimilato a un meccanismo Il metodo deduttivo o deduzione è il procedimento razionale che fa derivare una certa conclusione da premesse più generiche, dentro cui quella conclusione è implicita. Il termine significa letteralmente «condurre da», perché proviene dal latino "de" (traducibile con da , preposizione indicante provenienza, o moto di discesa dall'alto verso il basso) e "ducere" ( condurre ). Questo metodo parte da postulati e princìpi primi e, attraverso una serie di rigorose concatenazioni logiche, procede verso determinazioni più particolari attinenti alla realtà tangi
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            Mysql.sock missing - “Can't connect to local MySQL server through socket”

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            .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; } 0 Operating System - OSX 10.11.2 MAMP version - 3.5 I'm following along with this guide, setting up wordpress with Xammp on PC and MAMP on Mac, where I'm in step "Mac - 4" is asked to input: "In the Terminal run: /Applications/MAMP/Library/bin/mysqladmin -u root -p password {Your Windows MySQL password here}" However, this (with out bracket text) results in error message: error: 'Can't connect to local MySQL server through socket '/Applications/MAMP/tmp/mysql/mysql.sock' (2)' Check that mysqld is running and that the socket: '/Applications/MAMP/tmp/mysql/mysql.sock'
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