Let SD = standard deviation.

SD describes how much the values deviate from the mean.
If SD = 0, then the values do not deviate at all from the mean. In other words, all the values are equal.
A large SD indicates that the values are deviating significantly from the mean. In other words, the values are spread far from the mean.
A small SD indicates that the values are deviating only slightly from the mean. In other words, the values are clustered close to the mean.

Let's see how a change in the data can affect the SD.

Given {10, 20, 30}.

If 5 is added to each value, the new values are {15, 25, 35}. The values are not becoming more or less spread out. No change in the SD.
If 5 is subtracted from each value, the new values are {5, 15, 25}. The values are not becoming more or less spread out. No change in the SD.
If each value is increased by 50%, the new values are {15, 30, 45). The values are now more spread out. The SD increases.
If each value is decreased by 50%, the new values are {5, 10, 15}. The values are now less spread out. The SD decreases.

So here are the take-aways:

If a constant is added to or subtracted from each value, no change in the SD.
If each value is increased or decreased by the same percentage, the SD changes. If the original SD is known, the new SD can be determined. (When the SD changes, we'll never be asked to determine the actual value of the new SD, but we need to know that it can be determined.)

Now onto the problem above:

During an experiment, some water was removed from each of 6 water tanks. If the standard deviation of the volumes of water in the tanks at the beginning of the experiment was 10 gallons, what was the standard deviation of the volume of water in the tanks at the end of the experiment?

1. For each tank, 30 percent of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment.

2. The average volume of the water in the tanks at the end of the experiment was 63 gallons.

Statement 1: For each tank, 30 percent of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment.
Since each value is decreased by 30%, the SD decreases. Since the original SD is known, the new SD can be determined.
Sufficient.

Statement 2: The average volume of the water in the tanks at the end of the experiment was 63 gallons.
No way to determine how much the volumes are deviating from the mean of 63, so the SD cannot be determined.
Insufficient.

The correct answer is A.