The study of two characteristics X and Y on a given population provided the following results:
L’étude de deux caractères X et Y sur une population donnée a fourni les résultats suivants :

X 1 2 3 ... n
Y y1 y2 y3 ... yn

where (y1,..., yn) denotes any permutation of the set (1,2,...,n).
où (y1,..., yn) / désigne une permutation quelconque de l’ensemble (1,2,...,n). CALCULATE MEAN S

Given the problem, you are dealing with two characteristics \( X \) and \( Y \) on a population where \( Y = (y_1, y_2, ..., y_n) \) is a permutation of the set \( (1, 2, ..., n) \). To calculate the mean \( S \), let's break it down.

### Step-by-Step Calculation:

1. **X-values:** \( X = \{1, 2, 3, ..., n\} \)
   
2. **Y-values:** \( Y = (y_1, y_2, ..., y_n) \), which is some permutation of the set \( (1, 2, ..., n) \).

3. **Mean \( S \):** The mean \( S \) of the values is given by the formula:

   \[
   S = \frac{1}{n} \sum_{i=1}^{n} y_i
   \]

   Since \( Y \) is a permutation of the set \( \{1, 2, ..., n\} \), the sum of \( y_i \)'s is simply the sum of the first \( n \) natural numbers:

   \[
   \sum_{i=1}^{n} y_i = 1 + 2 + 3 + ... + n = \frac{n(n+1)}{2}
   \]

4. **Substitute into the formula for \( S \):**

   \[
   S = \frac{1}{n} \times \frac{n(n+1)}{2}
   \]

5. **Simplify the expression:**

   \[
   S = \frac{n+1}{2}
   \]

### Final Answer:
The mean \( S \) is:

\[
S = \frac{n+1}{2}
\]

This is the mean of the values in \( Y \), which corresponds to any permutation of the set \( \{1, 2, ..., n\} \).

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Do you have any questions or would you like more details?

Here are 5 related questions:
1. How do you find the median of a permutation of the set \( \{1, 2, ..., n\} \)?
2. What is the variance of a permutation of \( \{1, 2, ..., n\} \)?
3. How does the sum of squares change for different permutations of \( Y \)?
4. How can you generalize this result for other permutations or sets?
5. What is the significance of using permutations in population studies?

**Tip:** The sum of the first \( n \) natural numbers is a common result in mathematics and can often simplify calculations involving sequential data.

The study of two characteristics X and Y on a given population provided the following results: L’étude de deux caractères X et Y sur une population donnée a fourni les résultats suivants : X 1 2 3 ... n Y y1 y2 y3 ... yn where (y1,..., yn) denotes any permutation of the set (1,2,...,n). où (y1,..., yn) / désigne une permutation quelconque de l’ensemble (1,2,...,n). CALCULATE MEAN S

Learn how to calculate the mean of a permutation of the set {1, 2, ..., n} with a step-by-step breakdown. The solution involves summing the first n natural numbers and calculating the mean using statistical formulas. Suitable for students in Grades 10-12.

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