There are $17$ students in Marek's class, and all of them took a test. Marek's score was $17$ points higher than the arithmetic mean of the scores of the other students. By how many points is Marek's score higher than the arithmetic mean of the scores of the entire class? Justify your answer.
1. Let the scores of the 16 other students be \( s_1, s_2, \ldots, s_{16} \).
2. Let \( S \) be the sum of the scores of these 16 students. Thus, \( S = s_1 + s_2 + \cdots + s_{16} \).
3. The arithmetic mean of the scores of the other students is given by:
   \[
   \text{Mean}_{\text{others}} = \frac{S}{16}
   \]
4. Let Marek's score be \( M \). According to the problem, Marek's score is 17 points higher than the arithmetic mean of the scores of the other students:
   \[
   M = \frac{S}{16} + 17
   \]
5. The total score of the entire class, including Marek, is:
   \[
   \text{Total score} = S + M
   \]
6. Substituting \( M \) from step 4 into the total score:
   \[
   \text{Total score} = S + \left( \frac{S}{16} + 17 \right)
   \]
7. Simplify the expression:
   \[
   \text{Total score} = S + \frac{S}{16} + 17 = \frac{16S + S}{16} + 17 = \frac{17S}{16} + 17
   \]
8. The arithmetic mean of the scores of the entire class is:
   \[
   \text{Mean}_{\text{class}} = \frac{\text{Total score}}{17} = \frac{\frac{17S}{16} + 17}{17} = \frac{17S}{16 \cdot 17} + \frac{17}{17} = \frac{S}{16} + 1
   \]
9. Marek's score \( M \) is:
   \[
   M = \frac{S}{16} + 17
   \]
10. The difference between Marek's score and the arithmetic mean of the scores of the entire class is:
    \[
    M - \text{Mean}_{\text{class}} = \left( \frac{S}{16} + 17 \right) - \left( \frac{S}{16} + 1 \right) = 17 - 1 = 16
    \]

The final answer is \(\boxed{16}\).