## Simple Random Sampling

### Probability of selection

#### Simple random sampling without replacement (SRSWOR)

If the sample is to be drawn without replacement, after each subsequent draw, the units likely to be selected will reduce by 1; for example, after the first draw, we will have N-1 units to be a candidate for the second draw. Hence the probability of selection will be now 1/(N-1). And if we repeat this process till nth selection, the probabilities of selection at each subsequent draw will be 1/(N-2), 1/(N-3)….1/(N-n) respectively. And in such a case, each unit selected is dependent on the previous draw. It can easily be shown that the two draws are not independent.

Consider A_1,A_2 be the two draws to select 2 units out of N units.

Then

Hence these two events are not independent.

Simple random sampling with replacement (SRSWR)

If the sample is to be drawn with replacement, then at each draw, we will have all N units as an equal candidates to be selected. Hence the probability of selection is the same at all draws regardless of the units to be selected. And in such a case, each unit selected is independent of the previous draw.

Consider A_1,A_2 be the two draws to select 2 units out of N units with replacement.

Then

Hence these two events are independent.

Unbiasedness

One advantage of SRS is that the sample statistics used to estimate a parameter is unbiased; for example,the sample means unbiased regardless of SRSWOR or SRSWR.

This means if we take all possible samples of size n and calculate the mean of all the means of the samples, it will be equal to Y ̅. It can be proved that both SRSWOR and SRSWR give us unbiased estimates.

### Methodology

There are a number of ways to generate a random sample from any population. One way is to use random numbers to generate a sample. Suppose there are N numbers of units in the population. And we want to draw a sample of n units with the SRSWOR scheme.

Algorithm:

Number the population units as 1 to N and generate a random number between k_1 such that 1≤k_1≤N.

Select the k_1^(th )unit in the sample.

Select a second random number k_2 such that 1≤k_2≤N and k_1≠k_2 and select the k_2^(th )in the sample.

Repeat the above steps n times.

Similarly, for the SRSWR scheme to select n units out N, the algorithm below can be used:

Algorithm:

Number the population units as 1 to N and generate a random number between k_1 such that 1≤k_1≤N.

Select the k_1^(th )unit in the sample.

Select a second random number k_2 such that 1≤k_2≤N and select the k_2^(th )in the sample.

Repeat the above steps n times.

Computer-based random numbers

Random numbers can be generated through computers using different programming languages. This is an easy way to generate. SAS homework help can be taken to generate random numbers. Nowadays, there are plenty of online websites that generate random numbers if N, the size of the population, is provided.

#### Random number tables

These tables are easily available in books and on the internet. The random numbers are given column-wise in the table, with each number having 4 digits. They can be helpful in SAS assignment help. The number of digits to be taken for each random number will depend upon the number of units in the population. There is a possibility of more rejections while selecting numbers from a random number table. For example, if we have N= 373, we will have the probability of rejection as 1- 373/999 which is very high. To overcome it, we will take every number between 1 to 999 and divide it by 3 and take the remainder as a random number. Now we do not need to reject any number selected from the table. Hence, it reduces operational time.

### Simple random sampling and other sampling schemes

Simple random sampling is the easiest way to select a sample out of the population. In fact, it is used in other sampling schemes as well. Stratified sampling is, in a way, simple random sampling in different stratum,i.e., we select a sample with simple random sampling from each stratum. In a similar way, systematic sampling is SRS at the first draw, i.e., the first unit is selected with simple random sampling, and other units are drawn in a systematic way. It can be said that even some of the sampling schemes, such as stratified sampling, increase the precision of an estimator, but they are relatively cumbersome to perform than simple random sampling. Other estimation procedures where simple random sampling is used are the ratio method of estimation and the regression method of estimation. In the Mizuno-Sen sampling scheme, the first unit is selected proportional to the size rest (n-1) units are sampled as simple random sampling out of the remaining (N-1) units.

### Advantages of simple random sampling

As we discussed earlier, it is relatively easy to implement than other sampling schemes. Various news channels and governmental organizations used it repeatedly for various studies for policymaking. We just need to list out every unit in the population and keep on generating random numbers. No other information is required. This sampling scheme is widely used and mostly applicable to practical scenarios. In terms of efficiency, stratified sampling overtakes simple random sampling when the variation is more among stratum and less between different stratum. Similarly, we have such cases when systematic sampling is better in terms of efficiency than simple random sampling.

#### Disadvantages of simple random sampling

One of the serious concerns with simple random sampling is that it does not give any weight to the size of the unit in population. Consider a population of villages in a district that is required to be sampled for wheat crop output estimation. It will treat all villages equally regardless of the area under the wheat crop in a village. It is sometimes desirable that a village with more area under wheat cultivation have more chances of selection than other villages. In those cases, we may use probability proportional to size (PPS) to select the sample of villages. It also cannot be useful in cases where stratified sampling is used.