5 a bit more explanation

In addition to what Gerry had already mentioned: of course, if the subsamples B and C were true random samples from the full sample, call it "population", meaning they are "representative" then the correlation-coefficients of the smaller samples are always estimators for that of the "population", and if you use two or more random subsamples the estimated population-coefficient is somehow an average.

But well, as you state your problem, it looks very likely to me that B and C are not such random-samples but are taken using some criterion. If such a criterion is existent then one should determine whether it distorts the randomness of the subsamples: if you take,for instance, B from the left edge of the whole data-cloud in a scatterplot and C from the right edge then the best-fit-lines in that subsamples may have completely different slopes and variances around them.

[update2] If such an averaging of correlations is actually meaningful in your problem (your subsamples are random and not too small) then I'd recommend to average the z-transforms of the correlation-coefficients. That means $$r_{estr_{\text{est}} = \tanh(\frac{\sum_{k=1}^{s}\atanh(r_k)}{s}) tanh(\frac{\sum_{k=1}^{s}\tanh^{-1}(r_k)}{s})$$ where $s$ is the number of samples, because that fisher-transformation approximates by conversion a correlation-coefficient into a z-variable (normal distributed, mean=0, infinite range) where the averaging over the arithmetical mean is more meaningful.

[update]
Here I show examples where the subsamples were taken randomly. I generated correlated data of a population with n= 2000, normal distributed with mean=0, stddev=1, correlation r~ 0.35 . I show the variation of the occuring correlations if random samples of n=20, n=50, n=100 are drawn. For each sample-size I did 500 experiments and documented the frequencies of occuring correlations r in steps of about 0.1.

sample-n:   20          avg r:      0.37760   experiments: 500
pop-n   :   2000        pop r:      0.35247

low r       high r    freq
--------------------------------
-0.2023     -0.2023      1
-0.1807     -0.0948      8
-0.0878      0.0101      15
0.0205      0.1068      25
0.1112      0.2101      60
0.2123      0.3098      100
0.3113      0.4073      81
0.4109      0.5102      83
0.5109      0.6100      73
0.6107      0.7078      44
0.7122      0.7891      10
===================================

sample-n:   50          avg r:      0.36040
pop-n   :   2000        pop r:      0.35247

low r       high r    freq
--------------------------------
-0.1011     -0.1011      1
0.0175      0.1027      9
0.1098      0.2022      55
0.2056      0.3027      108
0.3043      0.4027      150
0.4047      0.5030      124
0.5045      0.6024      45
0.6099      0.6982      8
===================================

sample-n:   100         avg r:      0.35657
pop-n   :   2000        pop r:      0.35247

low r       high r    freq
----------------------------------
0.0504      0.0703      3
0.1139      0.2032      20
0.2054      0.3034      115
0.3055      0.4038      217
0.4047      0.4956      133
0.5046      0.5471      12
===================================


One can determine confidence-intervals for the correlations; that intervals narrow with increasing size of the samples.
But this all is only useful if the different samples are really random and not taken by some systematic criterion.

4 a bit more explanation

In addition to what Gerry had already mentioned: of course, if the subsamples B and C were true random samples from the full sample, call it "population", meaning they are "representative" then the correlation-coefficients of the smaller samples are always estimators for that of the "population", and if you use two or more random subsamples the estimated population-coefficient is somehow an average.

But well, as you state your problem, it looks very likely to me that B and C are not such random-samples but are taken using some criterion. If such a criterion is existent then one should determine whether it distorts the randomness of the subsamples: if you take,for instance, B from the left edge of the whole data-cloud in a scatterplot and C from the right edge then the best-fit-lines in that subsamples may have completely different slopes and variances around them.

[update2] If such an averaging of correlations is actually meaningful in your problem (your subsamples are random and not too small) then I'd recommend to average the z-transforms of the correlation-coefficients. That means $$r_{est} = \tanh(\frac{\sum_{k=1}^{s}\atanh(r_k)}{s})$$ where $s$ is the number of samples

[update]
Here I show examples where the subsamples were taken randomly. I generated correlated data of a population with n= 2000, normal distributed with mean=0, stddev=1, correlation r~ 0.35 . I show the variation of the occuring correlations if random samples of n=20, n=50, n=100 are drawn. For each sample-size I did 500 experiments and documented the frequencies of occuring correlations r in steps of about 0.1.

sample-n:   20          avg r:      0.37760   experiments: 500
pop-n   :   2000        pop r:      0.35247

low r       high r    freq
--------------------------------
-0.2023     -0.2023      1
-0.1807     -0.0948      8
-0.0878      0.0101      15
0.0205      0.1068      25
0.1112      0.2101      60
0.2123      0.3098      100
0.3113      0.4073      81
0.4109      0.5102      83
0.5109      0.6100      73
0.6107      0.7078      44
0.7122      0.7891      10
===================================

sample-n:   50          avg r:      0.36040
pop-n   :   2000        pop r:      0.35247

low r       high r    freq
--------------------------------
-0.1011     -0.1011      1
0.0175      0.1027      9
0.1098      0.2022      55
0.2056      0.3027      108
0.3043      0.4027      150
0.4047      0.5030      124
0.5045      0.6024      45
0.6099      0.6982      8
===================================

sample-n:   100         avg r:      0.35657
pop-n   :   2000        pop r:      0.35247

low r       high r    freq
----------------------------------
0.0504      0.0703      3
0.1139      0.2032      20
0.2054      0.3034      115
0.3055      0.4038      217
0.4047      0.4956      133
0.5046      0.5471      12
===================================


One can determine confidence-intervals for the correlations; that intervals narrow with increasing size of the samples.
But this all is only useful if the different samples are really random and not taken by some systematic criterion.

3 improving explanation, examples

[update]
Here I show examples where the subsamples were taken randomly. I generated correlated data of a population with n= 2000, normal distributed with mean=0, stddev=1, correlation r~ 0.35 . I show the variation of the occuring correlations if random samples of n=20, n=50, n=100 are drawn. For each sample-size I did 500 experiments and documented the frequencies of occuring correlations r in steps of about 0.1.

sample-n:   20          avg r:      0.37760   experiments: 500 pop-n   :   2000        pop r:      0.35247  low r       high r    freq    --------------------------------    -0.2023     -0.2023      1-0.1807     -0.0948      8-0.0878      0.0101      15 0.0205      0.1068      25 0.1112      0.2101      60 0.2123      0.3098      100 0.3113      0.4073      81 0.4109      0.5102      83 0.5109      0.6100      73 0.6107      0.7078      44 0.7122      0.7891      10sample-n:   50          avg r:      0.36040pop-n   :   2000        pop r:      0.35247  low r       high r    freq    -0.1011     -0.1011      1 0.0175      0.1027      9 0.1098      0.2022      55 0.2056      0.3027      108 0.3043      0.4027      150 0.4047      0.5030      124 0.5045      0.6024      45 0.6099      0.6982      8sample-n:   100         avg r:      0.35657pop-n   :   2000        pop r:      0.35247  low r       high r    freq     0.0504      0.0703      3 0.1139      0.2032      20 0.2054      0.3034      115 0.3055      0.4038      217 0.4047      0.4956      133 0.5046      0.5471      12One can determine confidence-intervals for the correlations; that intervals narrow with increasing size of the samples.But this all is only useful if the different samples are really random and not taken by some systematic criterion.

 
 
 
2 typo
1