Random number tables are composed of digits from 0 to 9 with equal frequency of occurrence. These tables are used in statistics for tasks such as selecting random samples. This technique is more effective when compared to manually selecting samples. It doesn't matter how they are chosen from the table it can be by row, column, diagonal or irregularly. Testing of random numbers was first done by Kendall and Babington smith.

The most practical and inexpensive method of selecting a random sample consists in the use of random number tables which have been so constructed that each of the digits 0, 1, 2,  ......., 9 appears with approximately the same frequency and independently of each other. If we have to select a sample from a population of size $N$ ($\leq$ 99), then the numbers can be combined two by two to give pairs from 00 to 99. Similarly if $N$ $\leq$ 999  or N $\leq$ 9999  and so on, then combining the digits three by three, we get numbers from 000 to 999 or 0000 to 9999 and so on. Since each of the digits 0, 1, 2, ......., 9 occurs with approximately the same frequency and independently of each other, so does each of the pairs 00 to 99, triplets 000 to 999 or quadruplets 0000 to 9999 and so on.

The method of drawing a random sample comprises the following steps :

1) Identify $N$ units in the population with the numbers 1 to $N$.

2) Select at random, any page of the random number table and pick up the numbers in any row, column or diagonal at random.

3) The population units corresponding to the numbers selected in step 2 constitute the random sample.

A random sample is one in which each unit of population has an equal chance of being included in it.
The simplest method which is normally used is illustrated by means of an example.
Suppose we want to select '$r$' candidates out of $n$. We assign numbers 1 to $n$, one number to each candidate and write these numbers (1 to $n$) on $n$ slips which are made as homogeneous as possible in shape, size etc. These slips are then put in a bag and thoroughly shuffled and then '$r$' slips are drawn one by one. The '$r$' candidates corresponding to the numbers on the slips drawn, will constitute the random sample.
Given below are the different sets of random numbers commonly used in practice. The numbers in these tables have been subjected to various statistical tests for randomness of a series and their randomness has been well established for all practical purposes.

1) Tippet's Random Number Tables:

Tippet's random number table consists of 10,400 four digited numbers, giving in all 10,400 * 4, i.e., 41600 digits selected at random from the British census reports.

2) Fisher and Yates Tables :

Fisher and Yates table comprise 15,000 digits arranged in two's. Fisher and Yates obtained these tables by drawing numbers at random from the 10th to 19th digits of A.S. Thomson's 20 figures logarithmic tables.

3) Kendall and Babington Smiths random tables: 
Kendall and Babington Smiths random tables consist of 100,000 digits grouped into 25,000 sets of 4-digited random numbers.

4) Rand corporation random number tables :

Rand corporation random number tables consist of one million random digits consisting of 200,000 random numbers of 5 digits each.
Given below is an example of random number table.
Example: The table of ten random numbers of two digits each is provided to the field investigator.
How should he use the below table to make a random selection of 5 plots out of 40?
34  96  61 85
49
 78  50  02  27  13

Solution: In this case we shall first identify the 40 plots with the numbers 1 to 40. In the given problem there are only three numbers less than 40 i.e., 02, 13 and 34, so we will not be able to draw the desired sample of size 5 from this table.
Hence we assign more than one number to each of the sampling units (plots).

01, 01 + 1 * 40, 01 + 2 * 40, 01 + 3 * 40, ........ and so on
i.e., 1, 41, 81, 121, 161, 201, ...... and so on

Similarly the second plot will be assigned the numbers
02, 02 + 1 * 40, 02 + 2 * 40, 02 + 3 * 40, ..........
02, 42, 82, 122, 162, 202, .........., and so on

Finally the last plot can be assigned the numbers 0, 40, 80, 120, 160, ..... and so on.

Select the first number in the given problem and move row wise we get,

Number from table 
Number of the sampled plot
 34  34 
 96 = 16 + 2 * 40
 16
 61 = 21 + 40
 21
 85 = 5 + 2 * 40
 5
 49 = 9 + 40  9

Thus the plot numbers 5, 9, 16, 21 and 34 constitute the desired sample.
If we select the first number from the given problem and move column wise the desired sample consists of plot numbers :
34, 38, 16, 10 and 21
Because, 78 = 40 + 38      
96 = 2 * 40 + 16,   
50 = 40 + 10 
and 61 = 40 + 21.