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][1] of the correlation-coefficients. That means $$ r_{\text{est}} = \tanh(\frac{\sum_{k=1}^{s}\tanh^{-1}(r_k)}{s}) $$ 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.


  [1]: http://en.wikipedia.org/wiki/Fisher_transformation