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==Covariance==
Very similar algorithms can be used to compute the [[covariance]].  

===Na?ve algorithm===
The na?ve algorithm is
:<math>\operatorname{Cov}(X,Y) = \frac {\sum_{i=1}^n x_i y_i - (\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)/n}{n}. </math>

For the algorithm above, one could use the following Python code:
<syntaxhighlight lang="python">
def naive_covariance(data1, data2):
    n = len(data1)
    sum1 = sum(data1)
    sum2 = sum(data2)
    sum12 = sum([i1 * i2 for i1, i2 in zip(data1, data2)])

    covariance = (sum12 - sum1 * sum2 / n) / n
    return covariance
</syntaxhighlight>

===With estimate of the mean===
As for the variance, the covariance of two random variables is also shift-invariant, so given any two constant values <math>k_x</math> and <math>k_y,</math> it can be written:

:<math>\operatorname{Cov}(X,Y) = \operatorname{Cov}(X-k_x,Y-k_y) = \dfrac {\sum_{i=1}^n (x_i-k_x) (y_i-k_y) - (\sum_{i=1}^n (x_i-k_x))(\sum_{i=1}^n (y_i-k_y))/n}{n}. </math>

and again choosing a value inside the range of values will stabilize the formula against catastrophic cancellation as well as make it more robust against big sums. Taking the first value of each data set, the algorithm can be written as:

<syntaxhighlight lang="python">
def shifted_data_covariance(data_x, data_y):
    n = len(data_x)
    if n < 2:
        return 0
    kx = data_x[0]
    ky = data_y[0]
    Ex = Ey = Exy = 0
    for ix, iy in zip(data_x, data_y):
        Ex += ix - kx
        Ey += iy - ky
        Exy += (ix - kx) * (iy - ky)
    return (Exy - Ex * Ey / n) / n
</syntaxhighlight>

===Two-pass===
The two-pass algorithm first computes the sample means, and then the covariance:
:<math>\bar x = \sum_{i=1}^n x_i/n</math>
:<math>\bar y = \sum_{i=1}^n y_i/n</math>
:<math>\operatorname{Cov}(X,Y) = \frac {\sum_{i=1}^n (x_i - \bar x)(y_i - \bar y)}{n}. </math>

The two-pass algorithm may be written as:
<syntaxhighlight lang="python">
def two_pass_covariance(data1, data2):
    n = len(data1)
    mean1 = sum(data1) / n
    mean2 = sum(data2) / n

    covariance = 0
    for i1, i2 in zip(data1, data2):
        a = i1 - mean1
        b = i2 - mean2
        covariance += a * b / n
    return covariance
</syntaxhighlight>

A slightly more accurate compensated version performs the full naive algorithm on the residuals.  The final sums <math display="inline">\sum_i x_i</math> and <math display="inline">\sum_i y_i</math> ''should'' be zero, but the second pass compensates for any small error.

===Online===

A stable one-pass algorithm exists, similar to the online algorithm for computing the variance, that computes co-moment <math display="inline"> C_n = \sum_{i=1}^n (x_i - \bar x_n)(y_i - \bar y_n)</math>:
:<math>\begin{alignat}{2}
\bar x_n &= \bar x_{n-1} &\,+\,& \frac{x_n - \bar x_{n-1}}{n} \\[5pt]
\bar y_n &= \bar y_{n-1} &\,+\,& \frac{y_n - \bar y_{n-1}}{n} \\[5pt]
C_n      &= C_{n-1}      &\,+\,& (x_n - \bar x_n)(y_n - \bar y_{n-1}) \\[5pt]
         &= C_{n-1}      &\,+\,& (x_n - \bar x_{n-1})(y_n - \bar y_n)
\end{alignat}</math>
The apparent asymmetry in that last equation is due to the fact that <math display="inline"> (x_n - \bar x_n) = \frac{n-1}{n}(x_n - \bar x_{n-1})</math>, so both update terms are equal to <math display="inline"> \frac{n-1}{n}(x_n - \bar x_{n-1})(y_n - \bar y_{n-1})</math>.  Even greater accuracy can be achieved by first computing the means, then using the stable one-pass algorithm on the residuals.

Thus the covariance can be computed as
:<math>\begin{align}
\operatorname{Cov}_N(X,Y) = \frac{C_N}{N} &= \frac{\operatorname{Cov}_{N-1}(X,Y)\cdot(N-1) + (x_n - \bar x_n)(y_n - \bar y_{n-1})}{N}\\
   &= \frac{\operatorname{Cov}_{N-1}(X,Y)\cdot(N-1) + (x_n - \bar x_{n-1})(y_n - \bar y_n)}{N}\\
   &= \frac{\operatorname{Cov}_{N-1}(X,Y)\cdot(N-1) + \frac{N-1}{N}(x_n - \bar x_{n-1})(y_n - \bar y_{n-1})}{N}\\
   &= \frac{\operatorname{Cov}_{N-1}(X,Y)\cdot(N-1) + \frac{N}{N-1}(x_n - \bar x_{n})(y_n - \bar y_{n})}{N}.
\end{align}</math>

<syntaxhighlight lang="python">
def online_covariance(data1, data2):
    meanx = meany = C = n = 0
    for x, y in zip(data1, data2):
        n += 1
        dx = x - meanx
        meanx += dx / n
        meany += (y - meany) / n
        C += dx * (y - meany)

    population_covar = C / n
    # Bessel's correction for sample variance
    sample_covar = C / (n - 1)
</syntaxhighlight>

A small modification can also be made to compute the weighted covariance:

<syntaxhighlight lang="python">
def online_weighted_covariance(data1, data2, data3):
    meanx = meany = 0
    wsum = wsum2 = 0
    C = 0
    for x, y, w in zip(data1, data2, data3):
        wsum += w
        wsum2 += w * w
        dx = x - meanx
        meanx += (w / wsum) * dx
        meany += (w / wsum) * (y - meany)
        C += w * dx * (y - meany)

    population_covar = C / wsum
    # Bessel's correction for sample variance
    # Frequency weights
    sample_frequency_covar = C / (wsum - 1)
    # Reliability weights
    sample_reliability_covar = C / (wsum - wsum2 / wsum)
</syntaxhighlight>

Likewise, there is a formula for combining the covariances of two sets that can be used to parallelize the computation:<ref name=":1" />

:<math>C_X = C_A + C_B + (\bar x_A - \bar x_B)(\bar y_A - \bar y_B)\cdot\frac{n_A n_B}{n_X}. </math>

===Weighted batched version===

A version of the weighted online algorithm that does batched updated also exists: let <math>w_1, \dots w_N</math> denote the weights, and write

:<math>\begin{alignat}{2}
\bar x_{n+k} &= \bar x_n &\,+\,& \frac{\sum_{i=n+1}^{n+k} w_i (x_i - \bar x_n)}{\sum_{i=1}^{n+k} w_i} \\
\bar y_{n+k} &= \bar y_n &\,+\,& \frac{\sum_{i=n+1}^{n+k} w_i (y_i - \bar y_n)}{\sum_{i=1}^{n+k} w_i} \\
C_{n+k}      &= C_n      &\,+\,& \sum_{i=n+1}^{n+k} w_i (x_i - \bar x_{n+k})(y_i - \bar y_n) \\
             &= C_n      &\,+\,& \sum_{i=n+1}^{n+k} w_i (x_i - \bar x_n)(y_i - \bar y_{n+k}) \\
\end{alignat}</math>

The covariance can then be computed as

:<math>\operatorname{Cov}_N(X,Y) = \frac{C_N}{\sum_{i=1}^{N} w_i}</math>