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Random number tables are tables containing a completely unordered sequence of digits from 0 to 9; that is, it is a long sequence of numbers that does not follow any pattern and any rule . For this reason, you cannot determine or calculate which number comes after another, even if you know the value and position of all the other digits in the table.
This type of table is frequently used in inferential statistics, especially during random sampling processes to select the elements of the population that will make up the sample. One of the most important conditions for the sample to be truly representative of the population under study is that the elements of a sample are selected completely at random. In addition, it is also one of the essential conditions to be able to draw valid conclusions from an inferential statistical study, such as a point estimate, a confidence interval or a hypothesis test.
That being said, in this article we will show how random number tables are built, some of their most important features, and how they are used for the sample selection process.
How are tables of random numbers generated?
There are many ways to generate tables of random numbers, however, today the most common is to generate them using computer programs designed for that purpose. Most statistical software packages have some form of random number generator. In addition, almost all the programs that are used to carry out simulations of different natural phenomena in science also use these generators.
A very easy way to generate an acceptable table of random numbers is by using a spreadsheet like Excel or Google Sheets. These sheets contain a function that allows you to generate a random number in each cell each time the sheet is updated.
Characteristics of tables of random numbers: are they really random?
The main characteristic of a table of random numbers is the fact that the numbers do not follow any pattern. However, they must also meet some other conditions to be statistically useful.
- All the figures or digits that make up the table, that is, the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 , must have the same probability of appearance in the table. This helps to avoid bias in the construction of the table.
- Each of the digits must be completely independent of all the others. That is, the fact that the first number in the table is a 7, for example, should not affect the probability that any number will appear in the next box.
This seems to be simple in theory, but in practice it is very difficult to achieve. In fact, most computerized random number generators actually generate the numbers following an algorithm, which means that they do follow a pattern. What happens is that the pattern can only be detected if many numbers are analyzed. Today, with the development of quantum computing, truly random number generators are being designed, but for our purposes, those that can be generated with Excel or another similar application work well.
Example of a table of random numbers
The following is an example of a table of random numbers generated in Excel. This table contains a total of 625 digits between 0 and 9 generated with the RANDOM.BETWEEN(0; 9) function of the aforementioned software , and can be used to practice selecting simple random samples.
It should be noted that, in this table, the first column is not part of the random numbers but consists of the identifier of the rows to make it easier to identify the starting points of the selection of random numbers.
Steps to use a table of random numbers for simple random sampling
Using the random number table for sampling is a simple 5-step process outlined below:
Step #1: Assign a unique number or index to each member of the population
The first step is to identify each member or data of the population from which we will obtain the sample with a unique number or index. In this way, when this number is selected in the random number table, we will know unequivocally what subject or data it is.
Generally speaking, the index assignment can be done arbitrarily, but some general rules and recommendations should be followed when writing these numbers:
- No index should be repeated.
- All numbers assigned as indexes must have the same number of digits. If there are one or more numbers with fewer digits than others, zeros must be added to the left to complete. For example, if we have a sample of 20 individuals and we want to number them from 1 to 20, then to the numbers from 1 to 9 we must add a leading zero so that they have two digits, just like the other numbers from 10 to 20 ( 01, 02, 03… 09, 10, etc.).
- It should be noted that it is not mandatory to start numbering from 0 or 1 (or from any other particular number). Nor is it mandatory that the numbers follow any sequence or pattern. However, for simplicity, it is customary to assign the indices in an orderly fashion to avoid repetition.
Step #2: Randomly select a starting position on the table
The starting point is of great importance when selecting random numbers in these tables. If we always start from the same place in the table and select numbers with the same number of digits, we will always get the same sequence of random numbers, which is not desirable if we later have to take a second sample. For this reason, we must choose the starting point at random and we must also try not to repeat it later.
Step #3: Group the numbers in the table into groups that have the same number of digits as the population indices
Once the starting point is selected in the random numbers table, all the numbers that have the same number of digits as the population indices will begin to be extracted, starting at the first digit that we selected in the previous step. It must be remembered that the indices were assigned in such a way that they all had the same number of figures. The idea of doing that was just to ensure that all indices had a chance to be selected.
Step #4: Remove from the list all numbers that do not correspond to a member of the population
An elementary rule of using the table of random numbers is that any number that does not correspond to or is not assigned to any element of the population must be discarded. For example, if in assigning the indices to the population we chose the numbers from 50 to 90, then we must discard any random number that is less than 50 or greater than 90.
Step #5: Remove repeating numbers if necessary
Some types of sampling, such as the selection of individuals or objects, do not allow repetition of the data. If this is the case, any number that is repeated during the random number selection process must be eliminated.
On the other hand, there are some applications in which repetitions are allowed. An example of this would be generating random data for a hypothetical experiment. In these cases, it is not necessarily prohibited that the numbers be repeated, since it could be the case in which two results of the experiment were the same.
Continue this process until all the elements of the sample are obtained.
This is the basic process that must be followed to use the random number table. This same procedure of extracting numbers with a fixed number of digits, eliminating those that do not correspond to a valid index and, if necessary, repeating numbers, is continued until the size of the sample that we must take has been completed.
Example of using the table of random numbers
Suppose you are asked to select a random sample of size 10 from a population containing 100 data points. We will use the table presented above to solve this problem by following the five steps described above:
- Step 1: Since we have 100 data points in the population, we will assign them the numbers from 00 to 99. In other words, each element of the population will be identified with a unique index between 00, 01, 02…97, 98 and 99. They were not numbered from 1 to 100, since, in this case, we would have to add a 0 to all indices between 1 and 99 for all indices to have the same number of digits as 100. If you had chosen this option, a problem would have arisen, and that is that there would barely be 100 indices to assign, but there are 1000 three-digit numbers. This would have meant having to remove, on average, 9 out of 10 random numbers generated by the table.
- Step 2: For the purposes of this example, we will start from the fourth column of row 9, as indicated in the following figure:
- Step 3: Since all the numbers assigned to the data are made up of two digits, the numbers in the table are grouped into groups of two, starting at the point indicated above and moving to the right. When you reach the end of a row, continue on the next. The figure below shows the grouping made in the first row.
The result is the following set of two-digit numbers: 56, 24, 83, 08, 17, 83, 47, 44, 78, 17, 84, 63, 03, 27, 24, 83, 47, 45, 38, 46, 72, 35, 13, 57, 08, 09, 51, 84, 31, 61, 50, 56, 97, 94, 70, 55, …
- Step 4: Since the population has 100 members and occupies all two-digit numbers, none of these numbers is left out of the list to begin with.
- Step 5: In the present case, since elements of a sample are being selected and these cannot be repeated, all repeating numbers must be eliminated by going through the list from left to right.
56, 24, 83, 08, 17, 83 , 47, 44, 78, 17 , 84, 63, 03, 27, 24, 83 , 47 , 45 , 38 , 46, 72, 35, 13, 57, 08 , 09, 51, 84 , 31, 61, 50, 56, 97, 94, 70, 55 , …
Finally, let’s remember that only 10 random numbers are required and here we have many more, so we select the first 10 that are not repeated, and that’s it. Consequently, the sample must be composed of the data number 56, 24, 83, 08, 17, 47, 44, 78, 84 and 63 .
References
- Devore, JL (2011). Probability And Statistics For Engineering And Sciences . Cengage Learning Publishers SA de CV
- Ian Alfred. (2016, June 4). How to Use a Random Number Table [Video]. Youtube. https://www.youtube.com/watch?v=_xLt734Af_k
- Jennifer Ward. (2015, October 14). Taking samples from a random number table [Video]. Youtube. https://www.youtube.com/watch?v=eqaYjf5zohM
- Vwillow. (2014, July 16). What is Random? [Video]. Youtube. https://www.youtube.com/watch?v=9rIy0xY99a0
- How to generate random numbers in Excel . Total Excel. Available at https://exceltotal.com/como-generar-numeros-aleatorios-en-excel/ .
