Calculate standard deviation given frequency counts rather than sample

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the cyclist
the cyclist 2013-8-27
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?

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dpb
dpb 2013-8-27
编辑：dpb 2013-8-27
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
>>
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dpb
dpb 2013-8-27
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)

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Iain
Iain 2013-8-27
编辑：Iain 2013-8-27
mu = (A(:,1).*A(:,2)) ./ sum(A(:,1));
std =sqrt(sum(A(:,1).*(A(:,2)-mu).^2)) ./(sum(A(:,1))-1));

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