Mean of a Set of Numbers

The mean of a set of numbers is a measure of central tendency that represents the average value of the numbers in the set. It is calculated by summing up all the numbers in the set and dividing the sum by the total number of elements in the set.

Formula for Mean:
The formula for calculating the mean of a set of numbers is as follows:

x bar = (x1 + x2 + x3 + ... + xn) / n

where x bar represents the mean, x1, x2, ..., xn are the individual numbers in the set, and n is the total number of elements in the set.

Mean of the Numbers xi^2:

To find the mean of the numbers xi^2, we need to calculate the sum of the squares of each number and divide it by the total number of elements. Let's denote the mean of xi^2 as y bar.

y bar = (x1^2 + x2^2 + x3^2 + ... + xn^2) / n

Now, let's express each xi^2 term in terms of xi by expanding the equation:

y bar = [(xi)^2 + (2xi)^2 + (3xi)^2 + ... + (nxi)^2] / n

= [(xi^2) + (4x^2i) + (9x^2i) + ... + (n^2x^2i)] / n

= (xi^2 + 4x^2i + 9x^2i + ... + n^2x^2i) / n

= (x1^2 + x2^2 + x3^2 + ... + xn^2) / n + (4x^2i + 9x^2i + ... + n^2x^2i) / n

= (x1^2 + x2^2 + x3^2 + ... + xn^2) / n + x^2i(4 + 9 + ... + n^2) / n

= (x1^2 + x2^2 + x3^2 + ... + xn^2) / n + (4 + 9 + ... + n^2) * (xi^2) / n

The sum of the squares of the numbers, (x1^2 + x2^2 + x3^2 + ... + xn^2), is equal to n times the mean of the numbers, x bar, because each number appears n times in the sum. Therefore, we can simplify the equation further:

y bar = x bar + (4 + 9 + ... + n^2) * (xi^2) / n

Now, let's consider the sum of squares of consecutive numbers:

4 + 9 + ... + n^2 = (1^2 + 2^2 + 3^2 + ... + n^2) - (1^2 + 2^2 + 3^2 + ... + (n-1)^2)

The sum of squares of consecutive numbers can be expressed as the sum of the first n natural numbers and the sum of the squares of the first (n-1) natural numbers:

= [n(n + 1)(2n + 1) / 6] - [(n -