• Total number of samples: ( n = 100 )
• Mean of all samples: ( \mu = 50 )
• Number of remaining samples: ( n_1 = 80 )
• Mean of remaining samples: ( \mu_1 = 52 )

The lost samples number ( n_2 = n - n_1 = 100 - 80 = 20 ).

Step 1: Calculate the total sum of all samples

The total sum of all samples is given by: [ \text{Total sum} = n \times \mu = 100 \times 50 = 5000 ]

Step 2: Calculate the sum of the remaining samples

The sum of the remaining 80 samples is: [ \text{Sum of remaining} = n_1 \times \mu_1 = 80 \times 52 = 4160 ]

Step 3: Calculate the sum of the lost samples

The sum of the lost samples is the difference between the total sum and the sum of the remaining samples: [ \text{Sum of lost} = \text{Total sum} - \text{Sum of remaining} = 5000 - 4160 = 840 ]

Step 4: Calculate the mean of the lost samples

The mean of the lost samples is: [ \mu_2 = \frac{\text{Sum of lost}}{n_2} = \frac{840}{20} = 42 ]

Verification of Consistency

The overall standard deviation (( \sigma = 10 )) and the standard deviation of the remaining samples (( \sigma_1 = 12 )) are provided. However, these lead to an inconsistency when calculating the variance of the lost samples, as the resulting variance would be negative, which is impossible. This indicates that the given data (overall standard deviation of 10) is inconsistent with the group statistics. Despite this, the mean of the lost samples is directly computable from the means and counts, and the calculation for the mean is valid.

Final Answer

The mean of the lost 20 samples is 42.