Calculate standard deviation given frequency counts rather than sample

Given a sample x

x = [1 1 1 1 2 2 2 3 3 3 3 4 5 6 6];

it is trivial to calculate the standard deviation:

s = std(x);

Suppose instead of x, I have the an array with the frequency counts of x:

 A = [4 1;
      3 2;
      4 3;
      1 4;
      1 5;
      2 6];

What's an elegant way to calculate the standard deviation, without needing to reconstruct x along the way?

Answer 1

[dpb]

One way,

>> m=dot(a(:,1)',a(:,2))/sum(a(:,1))
m =
  2.8667
>> s=sqrt(dot([[a(:,2)-m]'].^2,a(:,1))/(sum(a(:,1))-1))
s =
  1.7265
>> [mean(x) std(x)]
ans =
  2.8667    1.7265
>>

[the cyclist]

Thanks!

[dpb]

No accept??? :)

[the cyclist]

Patience! Not every pounces on the answers within minutes. :-)

[dpb]

Chuckles...

This implementation is, of course, straightforward and for small sample sizes and well-behaved inputs should be fine. You're at the mercy of data order for computation of course, so isn't as robust as might be (and as I presume the builtin mean/std functions are)

Answer 2

[Iain]

 mu = (A(:,1).*A(:,2)) ./ sum(A(:,1));

 std =sqrt(sum(A(:,1).*(A(:,2)-mu).^2)) ./(sum(A(:,1))-1));

[the cyclist]

This gives vector results for mu and std, so guessing you are missing an operation.

[Iain]

Yup, I'm missing a sum here or there:

mu = sum(A(:,1).*A(:,2)) ./ sum(A(:,1));