I just enrolled in AP Statistics course this week. And, one thing that popped up a lot in descriptive statistics is the idea of the mean. I understand mean as the closest data point to all of the data points. But, I simply don't understand the rationale behind it. How the one who made this formula come to think about it? What's intuition? Can anyone resolve that for me?
Why mean = Sum of observations / Number of observations?
Imagine that the $n$ data points are on a see-saw. It is possible to place a pivot to balance the see-saw, call this position $y$.
There are $k$ data points to the left, these contribute
$$\sum_{i=1}^k (y-x_i) = ky - \sum_{i=1}^k x_i$$
On the right, the $n-k$ data points contribute
$$\sum_{i=k+1}^n (x_i - y) = \sum_{i=k+1}^n x_i - (n-k)y$$
As the smallest the difference can be is zero, this is achieved when
$$ky - \sum_{i=1}^k x_i = \sum_{i=k+1}^n x_i - (n-k)y$$$
$$ky + (n-k)y = \sum_{i=1}^k x_i + \sum_{i=k+1}^n x_i$$
$$ny = \sum_{i=1}^n x_i$$
$$y = \frac1n\sum_{i=1}^n x_i$$