Why both averages rise when one number moves
An average is the total of a group’s values divided by how many values it contains. Moving a value changes both parts of that calculation: the group it leaves loses the value, while the group it enters gains it.
Suppose one group has values 2 and 8, for an average of 5. A second group has values 10 and 12, for an average of 11. Move the 8 from the first group to the second. The first group is left with 2, so its average falls—not rises. That example shows the crucial condition: for both averages to rise, the moved value must be below the first group’s average but above the second group’s average. That can happen only if the supposed “lower-average” group is not actually lower than the other one, so the claim as stated cannot hold for ordinary averages.
The direction follows from the group averages
In fact, when a value moves from a group with a lower average to one with a higher average, the source group’s average rises because removing a below-average value leaves a relatively larger share of its total behind. But the receiving group’s average falls: adding a value below its existing average pulls that average down.
For instance, move 2 from a group averaging 5 into a group averaging 11. The remaining source values average more than 5, while the destination’s average drops below 11. Both averages cannot rise under these conditions. The movement can make the two averages closer together, but it cannot push them both upward.
Why the idea can sound plausible
Averages are often described as if each group has a fixed “level,” but they are recalculated after every change. Removing a low value helps the group it leaves; adding that same value has the opposite effect on the group it joins. If both averages appear to rise, some other change—such as values being added or removed elsewhere—must also be involved.
