Simple Random Sampling

Simple random sample (SRS) is a special case of a random sampling. A sample is called simple random sample if each unit of the population has an equal chance of being selected for the sample. Whenever a unit is selected for the sample, the units of the population are equally likely to be selected. It must be noted that the probability of selecting the first element is not to be compared with the probability of selecting the second unit. When the first unit is selected, all the units of the population have an equal chance of selection, which is $1/N$. When the second unit is selected, all the remaining $\left( {N – 1} \right)$ units of the population have $1/\left( {N – 1} \right)$ chance of selection.

Another way of defining a simple random sample is that if we consider all possible samples of size $n$, and then each possible sample has an equal probability of being selected.

If sampling is done with replacement, there are ${N^n}$ possible samples and each sample has a probability of selection equal to $\frac{1}{{{N^n}}}$. If sampling is done without replacement with the help of combinations, then there are $^N{C_n}$ possible samples and each sample has a probability of selection equal to $\frac{1}{{^N{C_n}}}$. If samples are made with permutations, each sample has a probability of selection equal to $\frac{1}{{^N{P_n}}}$ Strictly speaking, the sample selected without replacement is called a simple random sample.

The Difference Between Random Sample and Sample Random Sample

If each unit of the population has a known (equal or un-equal) probability of selection in the sample, the sample is called a random sample. If each unit of the population has an equal probability of being selected for the sample, the sample obtained is called a simple random sample.

Selection of a Sample Random Sample

A simple random sample is usually selected without replacement. The following methods are used for the selection of a simple random sample:

• Lottery Method. This is a classic method but it is a powerful technique, and modern methods of selection are very close to this method. All the units of the population are numbered from $1$ to $N$. This is called a sampling frame. These numbers are written on small slips of paper or small metallic balls. The paper slips or the metallic balls should be of the same size, otherwise the selected sample will not be truly random. The slips or the balls are thoroughly mixed and a slip or ball is picked up. Again the population of slips is mixed and the next unit is selected. In this manner, the number of slips equal to the sample size $n$ is selected. The units of the population which appear on the selected slips make the simple random sample. This method of selection is commonly used when the size of the population is small. For a large population there is a big heap of paper slips and it is difficult to mix the slips properly
• Using a Random Number Table. All the units of the population are numbered from 1 to $N$ or from 0 to $N – 1$. We consult the random number table to take a simple random sample. Suppose the size of the population is 80 and we have to select a random sample of 8 units. The units of the population are numbered from 01 to 80. We read two-digit numbers from the table of random numbers. We can start from any columns or rows of the table. Let us consult a random number table given in this content. Two-digit numbers are taken from the table and any number above 80 will be ignored, and if any number is repeated we shall not record it if sampling is done without replacement. Let us read the first two columns of the table. The random numbers from the table are 10, 37, 08, 12, 66, 31, 63 and 73. The two numbers 99 and 85 have not been recorded because the population does not contain these numbers. The units of the population whose numbers have been selected constitute the simple random sample.
Let us suppose that the size of the population is 100. If the units are numbered from 001 to 100, we shall have to read 3-digit random numbers. From the first 3 columns of the random number table, the random numbers are 100, 375, 084, 990 and 128 and so on. We find that most of the numbers are above 100 and we are wasting our time while reading the table. We can avoid this by numbering the units of the population from 00 to 99. In this way, we shall read 2-digit numbers from the table. Thus if N is 100, 1000 or 10000, the numbering is done from 00 to 99, 000 to 999 or 0000 to 9999.
• Using a Computer. Computers can be used for selecting a simple random sample. Computers are used for selecting a sample of prize-bond winners, a sample of Hajj applicants, and a sample of applicants for residential plots, and for various other purposes.