Middle digit mean

If the three digit number is "$abc$", then we need $b = \frac{a+c}{2}$.

As $a$ and $c$ are chosen to be between $0$ and $9$, $b$ will be certainly, as it is the mean. This means we only need to check which cases make $b$ an integer.

$b$ is an integer exactly when $a+c$ is even.

If $a$ is even, $c$ needs to be even also. If $a$ is odd, $c$ needs to be odd also.

If $a$ and $c$ are both even, $a$ can be any of $2$, $4$, $6$ and $8$, so four options. $c$ can be any of $0$, $2$, $4$, $6$ and $8$, so five options. This means there are $4 \times 5 = 20$ combinations.

If $a$ and $c$ are both odd, they can be any of $1$, $3$, $5$, $7$ and $9$. This gives $5$ options for each, so there are $5 \times 5 = 25$ combinations.

Thus there are $20 + 25 = 45$ three digit numbers with this property.