Five street vendors sell oranges. Four of them ask \$1 each, and one asks \$3.
The first approach to a price average would be ( 1 + 1 + 1 + 1 + 3 ) / 5 = 1.4.
The second approach, only considering unique prices, would be ( 1 + 3 ) / 2 = 2.
What are the correct terms for these two different averages, and in what situations is one a better approach than the other?
On
I do not think there is a one unique 'correct way'. Different ways to calculate the mean tell you different things. The first one tells you about the mean price over the sample of shops/vendors. The second one tells you about the mean price level.
There are other possibilities. Say each of the cheap vendors sells $\frac{1}{4}$ of all sold oranges. Then $(1\frac{1}{4}+1\frac{1}{4}+1\frac{1}{4}+1\frac{1}{4})$ is the mean price at which oranges are transacted.
The first one is the only one that makes sense. It's just the average. The second one doesn't have a name I know, and doesn't deserve one.
You could rewrite the first one as $$ \frac{4 \times 1 + 1 \times 3}{4 + 1} $$ and call it a weighted average.
This is the average vendor price. If the vendors had different numbers of oranges you could compute the weighted average price per orange.