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Single-pass standard deviation in PHP

I recently had a need for a standard deviation function for a project. It wasn’t until after I’d already implemented it that I discovered the existence of a virtually-undocumented stats_standard_deviation() in PHP’s Statistics package.

Anyway, I did some research and found Don Knuth’s on-line, single-pass standard deviation function and implemented it in PHP.

function stddev($array){
  //Don Knuth is the $deity of algorithms
  $n = 0;
  $mean = 0;
  $M2 = 0;
  foreach($array as $x){
      $delta = $x - $mean;
      $mean = $mean + $delta/$n;
      $M2 = $M2 + $delta*($x - $mean);
  $variance = $M2/($n - 1);
  return sqrt($variance);

Please note that there are more efficient ways of doing this if you have have written a mean() or variance() function, as you should. There’s also a stats_variance() function in PHP.

I think I might do some serious benchmarking with PHP’s implementation of stddev and mine and see which is faster. I’ll be that the PHP one is faster because it’s written in C and mine’s being interpreted at runtime.

Update 19:40 stats is undocumented because it’s a PECL extension which isn’t included by default, and it’s not terribly clear to install. Install php-dev then use pecl install stats to get it. Don’t forget to sudo if you need to.

Update 2009/11/02 14:52 I had a missing -1 on the variance calculation for nearly a year and finally someone caught it. Thanks, Shane.


  1. Gene McKenna:

    On line 14 you have
    $variance = $MS/$n;
    shouldn’t $MS be $M2?

  2. Colin Dean:

    You are correct, Gene. That’s what I get for retyping it instead of copy and pasting. Thanks. I fixed it.

  3. Shane:

    I believe (and Excel seems to confirm) that

    $variance = $M2/$n

    Should be:

    $variance = $M2/($n-1);

    Std Deviation is the square root of the sum divided by ONE LESS than the count.

  4. Colin Dean:

    Blah. You’re right. I’ve modified it.

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