# Sampling

### Statistical Inference

Statistical inference studies extract conclusions for a whole population from only a sample of that population.

## Probability Sampling

Probability sampling consists of choosing a sample from the population at random. Several types can be distinguished:

### Simple Random Sampling

To obtain a sample, the elements of the population are numbered and then the elements that make up the sample are selected at random.

### Systematic Sampling

To obtain a sample, the elements of the population are numbered and then a single element is chosen at random. Then, the other elements that make up the sample are selected at constant intervals.

For example, if there is a population formed by 100 elements and a sample of 25 elements is needed, the interval of selection must be established, which equals 100/25 = 4. Then, the starting element is chosen which is a number randomly selected between 1 and 4. From this element the remaining elements of the sample are obtained in equal intervals.

2, 6, 10, 14,..., 98

### Stratified Sampling

To obtain a sample, the population is divided into classes or strata. Generally, these groups can be clearly identified (i.e. gender, age, profession etc.). Then, the number of elements that are randomly selected from each stratum is proportional to the size of each stratum with respect to the population.

In a factory of 600 workers, a sample of 20 employees is needed. It is known that there are 200 employees in Section A, 150 in B, 150 in C and 100 in D. Therefore:

A sampling can be done with or without replacement, and the starting population can be infinite or finite.

If all possible samples of size n in a population are considered, a statistic can be calculated for each sample (mean, standard deviation, proportion, ...) and each will vary from one to the other.

So, the distribution of the statistic is obtained which is called the sampling distribution.