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and 〈 m ([ X ]) $^{i}$〉 is the average magnitude for the i th light curve in channel X . The correlation coefficient is limited to the range − 1 ≤ r ≤ 1, where r = 1 ( − 1) means that the [3.6] and [4.5] light curves are perfectly correlated (anti-correlated). The distribution of v ([ X ]) i is dominated by contributions from noise (and any systematic errors) rather than variability, so we need to define a threshold for selecting candidate variables from these distributions. Let v$_{m}$ ([ X ]) be the median of the v ([ X ]) i as a function of magnitude and σ ([ X ]) be the standard deviation of the v ([ X ]) i values around this median. Sources are more likely to be true variables as their distance, ( v ([ X ]) i − v$_{m}$ ([ X ])) /σ ([ X ]) > 0, from the median increases, and if the [3.6] and [4.5] variabilities are strongly correlated. A reasonable metric for the joint significance of the variances in the light curves is 

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which quantifies the degree to which the object deviates from the median variances in both bands. Objects with positive excess variances (large σ$_{12}$ and v > v$_{m}$ ) and strong correlations r ≃ 1 between the two bands are the best candidates for true variables. 

Figure 3 shows the distribution of correlation strengths r for sources with σ$_{12}$ exceeding a range of thresholds. We see that there is a peak for highly correlated sources r ≃ 1 whose strength increases as the variability significance σ$_{12}$ increases. This demonstrates that the variability in the two bands is tightly correlated for real variables and that we are likely to produce a relatively clean sample of variables (the false positive rate is 6%–7% for AGNs in the Photometry Group 3 defined below) if we restrict ourselves to r > 0 . 8 and σ$_{12}$ > 2. Variability selection is always a tradeoff between completeness and contamination, so we will consider sources in three different variability levels, all restricted to have correlations r > 0 . 8 between the bands. To avoid confusion, the Variability Levels correspond directly to the values of σ$_{12}$ . Variability Levels 2, 3 and 4 correspond to sources with excess variances and significances exceeding σ$_{12}$ > 2, 3 and 4, respectively, with Variability Level 2 being the most complete and the most contaminated, and Variability Level 4 being the least complete and least contaminated. 

We estimate the fraction of false positives by assuming that the density of uncorrelated sources ( r < 0 . 5) in Figure 3 represents the density of false positives amidst the highly correlated ( r > 0 . 8) sources. If the number of identified variables is N$_{r>}$$_{0}$$_{.}$$_{8}$ , that is sources with σ$_{12}$ > 2 (or 3, 4) and r > 0 . 8, and the number of sources with σ$_{12}$ > 2 (or 3, 4) and r < 0 . 5 is N$_{r<}$$_{0}$$_{.}$$_{5}$ , then we estimate that the number of false positives is F = (0 . 2 / 1 . 5) N$_{r<}$$_{0}$$_{.}$$_{5}$ , 

