Members of a wrestling team were weighed twice. Was the standard devia

In this question, a very subtle trap is laid out in the data given in the first statement. The first statement says that half of the team lost 1 pound in the second weighing and the other half gained 1 pound. All well till now.
But, it does not tell us which half lost weight and which one gained. It does not tell us if weights of some persons are equal or not. This essentially means, each one of us can assume our own values and end up getting different answers.

For example, if we assume that there are 4 students in the team with weights of 10, 20, 30 and 40. Mean = 25. Deviations of the individual numbers from the mean are -15, -5, 5 and 15.
Square of these deviations = 225, 25, 25 and 225.

Now, in the second weighing, if the first two are reduced by 1 each and the next two are increased by 1 each, then,
Mean = 25
Deviations = -16, -6, 6 and 16. These deviations will correspond to an increase in the SD.

On the other hand, if the first two are increased by 1 each and the next two are reduced by 1 each, then,
Mean = 25
Deviations = -14, 4, 4, 14. These deviations will correspond to a decrease in the SD.

Clearly, we have a YES and NO situation for the question asked. We don’t even have to take any other case.
Statement I alone is insufficient. Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II, we can infer that the weights of the team members are all same (or ZERO, which is illogical considering that we are talking about the weights of members of a Wrestling team ). Only when all the values are equal will the SD be Zero.

But, statement II doesn’t give us any information about the second weighing. Hence, statement II alone is insufficient. Answer option B can be eliminated.

When we combine both the statements, we know that all the values in the data set are equal. Also, the SD in the first weighing is ZERO. If one half of the values is now increased by 1 and the other half reduced by 1, the deviations wrt mean will increase. This will cause the SD to increase in the second weighing.
Values in which half are increased and in which half are decreased does not matter since all the values are equal.
The combination of statements is sufficient.

The correct answer option is C.

Hope this helps!