Sampling Theory and Distributions

Sampling Theory

Data scientists are required to draw conclusions about a group, a.k.a population from a few samples of it because getting the entire population is intractable. This process of drawing samples is called sampling. There are different kinds of sampling , few of which are:

• Random sampling
• Clustered sampling
• Stratified sampling
• Systematic sampling You can read about them over here The drawing of conclusions or inference about the population from the samples is called statistical inference.

In this section we will consider two different types of samples:

• Sampling with Replacement
• Sampling without Replacement

Random Sampling with Replacement

As the name suggests, this is a type of sampling where each member of the population may be included more than once. It’s like picking a ball from an urn and then putting it back into the urn.

Random Sampling without Replacement

In this type of sampling, each member of the population can be included atmost once. A similar example for this type of sampling would be picking a ball from the urn and not putting it back inside the urn.

Sampling statistics

A quantity obtained from the sample for the purpose estimating a population parameter is called a sample statistic or briefly statistic. Mathematically, a sample statistic for a sample of size $n$ can be defined as a function of the random variables $X_1, X_2,…,X_n$ i.e., $g(X_1, X_2,…,X_n)$. The function $g(X_1, X_2,…,X_n)$ is another random variable whose values can be represented by $g(x_1, x_2,…,x_n)$.

Sampling Distributions

Note: This chapter has sections taken from the book and Statistical Methods course offered at IIT Kharagpur