Statistics And Probability

Step 1: Formula for the average of squares.
The average of the squares of the first natural numbers is given by the formula: The sum of the squares of the first natural numbers is given by:

Step 2: Apply the formula for .
Substitute into the formula: Now, the average is:

Step 3: Conclusion.
Thus, the average of the squares of the first 50 natural numbers is .

Final Answer: .

Step 1: Organize the data.
The given data set is: 89, 45, 56, 67, 89, 45, 78, 45, 78, 45, 67. Arrange the data in ascending order:

Step 2: Find the median.
The median is the middle number of a data set. For an odd number of values, it is the middle one. For an even number of values, it is the average of the two middle values. Here, we have 11 values, so the middle value is the 6th value. The 6th value is 67.

Step 3: Conclusion.
Therefore, the median is 67.

Final Answer: 67.