A group of 210 students appeared in some test. The mean of \(\frac{1}{3}\)rd of students is found to be 60. The mean of the remaining students is found to be 78. The mean of the whole group will be:
72
A combined mean is found by adding the totals of the two groups and dividing by the total number of members, not by averaging the two means.
First split the group. One-third of 210 is 70 students, so the remaining students number 210 - 70 = 140.
Next find each group's total. The first group has mean 60, so its total is 70 x 60 = 4200. The second group has mean 78, so its total is 140 x 78 = 10920. The grand total is 4200 + 10920 = 15120.
The combined mean is therefore 15120 divided by 210, which equals 72.
The common error is to take the simple average of 60 and 78, giving 69, or to lean towards the midpoint. The correct value must be closer to 78 than to 60, because the group with mean 78 is twice as large and therefore carries twice the weight. Indeed 72 sits two-thirds of the way from 60 to 78, exactly as the 70:140 split requires. A quick check: 60 + (2/3) x 18 = 60 + 12 = 72.
Hence the mean of the whole group is 72.
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