Let $P=\{p_1, p_2,\cdots, p_n\}$ be population where $p_i \in \mathbb{R}, 1 \le i \le n$.
Let $S_k \subset P$ is a sample from population $P$ of size $1 \le k < n$.
My professor told that
we use measures calculated from samples to guess the measures of population
Then I assumed that we take a subset of population and perform some function on the values of that subset to guess measures about $P$.
Then later he proved and showed us that
Expectation for probability distribution over means of all samples of fixed length is equal to the expectation over population
Any way we need to consider all elements from the population. In-fact sampling increases our time for computation also.
Then what is the need for sampling here? or where I am going wrong?
Example he explained is
$P= \{1,2,3,4\}$
Expectation is 2.5.
By sampling of size 3 we get $\{1,2,3\},\{1,2,4\}\{1,3,4\},\{2,3,4\}$ Probabilities of corresponding samples are $1/4$ with means $6/3, 7/3, 8,3, 9/3$ and hence expectation is 2.5.