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











$endgroup$












  • $begingroup$
    In Mathematica 11.3, << Statistics` produces an error message and Variance@data // N gives 665.524
    $endgroup$
    – m_goldberg
    Jan 14 at 7:47








  • 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
    Jan 14 at 9:22






  • 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
    Jan 14 at 9:23












  • $begingroup$
    Which book did you see this in? It looks like a scan.
    $endgroup$
    – Szabolcs
    Jan 14 at 11:45
















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











$endgroup$












  • $begingroup$
    In Mathematica 11.3, << Statistics` produces an error message and Variance@data // N gives 665.524
    $endgroup$
    – m_goldberg
    Jan 14 at 7:47








  • 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
    Jan 14 at 9:22






  • 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
    Jan 14 at 9:23












  • $begingroup$
    Which book did you see this in? It looks like a scan.
    $endgroup$
    – Szabolcs
    Jan 14 at 11:45














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











$endgroup$




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















share|improve this question













share|improve this question




share|improve this question








edited Jan 14 at 8:01









m_goldberg

85k872196




85k872196










asked Jan 14 at 7:24









FacetFacet

254




254












  • $begingroup$
    In Mathematica 11.3, << Statistics` produces an error message and Variance@data // N gives 665.524
    $endgroup$
    – m_goldberg
    Jan 14 at 7:47








  • 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
    Jan 14 at 9:22






  • 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
    Jan 14 at 9:23












  • $begingroup$
    Which book did you see this in? It looks like a scan.
    $endgroup$
    – Szabolcs
    Jan 14 at 11:45


















  • $begingroup$
    In Mathematica 11.3, << Statistics` produces an error message and Variance@data // N gives 665.524
    $endgroup$
    – m_goldberg
    Jan 14 at 7:47








  • 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
    Jan 14 at 9:22






  • 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
    Jan 14 at 9:23












  • $begingroup$
    Which book did you see this in? It looks like a scan.
    $endgroup$
    – Szabolcs
    Jan 14 at 11:45
















$begingroup$
In Mathematica 11.3, << Statistics` produces an error message and Variance@data // N gives 665.524
$endgroup$
– m_goldberg
Jan 14 at 7:47






$begingroup$
In Mathematica 11.3, << Statistics` produces an error message and Variance@data // N gives 665.524
$endgroup$
– m_goldberg
Jan 14 at 7:47






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
Jan 14 at 9:22




$begingroup$
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
Jan 14 at 9:22




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
Jan 14 at 9:23






$begingroup$
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
Jan 14 at 9:23














$begingroup$
Which book did you see this in? It looks like a scan.
$endgroup$
– Szabolcs
Jan 14 at 11:45




$begingroup$
Which book did you see this in? It looks like a scan.
$endgroup$
– Szabolcs
Jan 14 at 11:45










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
    Jan 14 at 14:47










  • $begingroup$
    Wolfram Research article also mentions a replacement for VarianceMLE: reference.wolfram.com/language/Compatibility/tutorial/… .
    $endgroup$
    – kirma
    Jan 18 at 17:31



















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
    2






    active

    oldest

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    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    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
      Jan 14 at 14:47










    • $begingroup$
      Wolfram Research article also mentions a replacement for VarianceMLE: reference.wolfram.com/language/Compatibility/tutorial/… .
      $endgroup$
      – kirma
      Jan 18 at 17:31
















    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
      Jan 14 at 14:47










    • $begingroup$
      Wolfram Research article also mentions a replacement for VarianceMLE: reference.wolfram.com/language/Compatibility/tutorial/… .
      $endgroup$
      – kirma
      Jan 18 at 17:31














    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 Jan 14 at 9:28









    Sjoerd SmitSjoerd Smit

    3,410715




    3,410715












    • $begingroup$
      Thank you for your kind help, I will remember to post code next time.
      $endgroup$
      – Facet
      Jan 14 at 14:47










    • $begingroup$
      Wolfram Research article also mentions a replacement for VarianceMLE: reference.wolfram.com/language/Compatibility/tutorial/… .
      $endgroup$
      – kirma
      Jan 18 at 17:31


















    • $begingroup$
      Thank you for your kind help, I will remember to post code next time.
      $endgroup$
      – Facet
      Jan 14 at 14:47










    • $begingroup$
      Wolfram Research article also mentions a replacement for VarianceMLE: reference.wolfram.com/language/Compatibility/tutorial/… .
      $endgroup$
      – kirma
      Jan 18 at 17:31
















    $begingroup$
    Thank you for your kind help, I will remember to post code next time.
    $endgroup$
    – Facet
    Jan 14 at 14:47




    $begingroup$
    Thank you for your kind help, I will remember to post code next time.
    $endgroup$
    – Facet
    Jan 14 at 14:47












    $begingroup$
    Wolfram Research article also mentions a replacement for VarianceMLE: reference.wolfram.com/language/Compatibility/tutorial/… .
    $endgroup$
    – kirma
    Jan 18 at 17:31




    $begingroup$
    Wolfram Research article also mentions a replacement for VarianceMLE: reference.wolfram.com/language/Compatibility/tutorial/… .
    $endgroup$
    – kirma
    Jan 18 at 17:31











    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 Jan 14 at 15:15









        Eric TowersEric Towers

        2,306613




        2,306613






























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            0 I want to add Tabulator js to my java project. I added tabulator.min.js, tabulator.min.css to resources folder. After that added to my html page these dependencies as recommending in readme.md. <link rel="stylesheet" th:href="@{/css/tabulator.min.css}"> <script type="text/javascript" th:src="@{/js/tabulator.min.js}"></script> Now my tables view all elements, that I have in the html page, but not correct. In the documentation of Tabulator is example how to create header and table, var table = new Tabulator("#example-table", { columns:[ {title:"Name", field:"name", sorter:"string", width:200, editor:true}, {title:"Age", field:"age", sorter:"
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