Elementary Theory of Sampling

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* Definition

* Uses of Sampling

* Probability based sampling methods

* Other sampling methods

* Random Number Table

* Reasons for Sampling

* Difference between the following terms: sampling error and non-sampling error

* Black and Sampled Population



1 – Population: Is one set of individuals or items that we can observe, measure a feature or attribute. Examples of population:

* The set of all university students.

* The set of smokers in a region.

2 – Sampling: Refers to the procedure used to obtain one or more samples of a population. This is done once it has established a representative sample of the population, then it proceeds to the selection of items in the sample but there are many designs of the sample.

By taking several samples of a population, the statistics calculated for each sample are not necessarily equal, it is likely to vary from sample to sample.

3 – Statistical: Are the data or measurements obtained on a sample and therefore an estimate of the parameters.

4 – Parameter: Are the measures or data obtained on the probability distribution of the population, such as the mean, variance, proportion, etc..

5 – Error: The difference between a statistic and its corresponding parameter. It is a measure of the variability of repeated samples estimates on the value of the population, gives us a clear idea of how far and with what probability estimates based on a sample away from the value that would have been obtained by a complete census. Whenever you make a mistake, but the nature of the investigation will tell us to what extent we can commit it (the results are subject to sampling error and confidence intervals that vary sample to sample). Varies calculated at the beginning or end. A statistician will be more precise about his mistake and therefore is smaller. You could say that is the deviation of the sampling distribution of a statistic and its reliability.

Sampling uses.

Sampling is used in various fields:

1 – Policy: Samples of the opinions of the voters are used to measure candidates and public support in elections.

2 – Education: Samples of the test scores of students are used to determine the efficiency of a technique or teaching program.

3 – Industry: The samples of the products of an assembly line is used to control the quality.

4 – Medicine: Samples measures blood sugar in diabetic patients proved the effectiveness of a technique or a new drug.

5 – Crop harvesting maize samples in a plot projected in producing the effects of a new fertilizer.

6 – Government: A sample of opinions of voters would be used to determine the criteria of the public on matters relating to the welfare and national security.

Probability sampling methods:

1 – Simple Random Sampling: This is the most common way to get a sample in the random selection, that is, each of the individuals in a population has an equal chance of being chosen. If this requirement is not met, we say that the sample is flawed. To ensure that the random sample is not flawed, should be used to its constitution a table of random numbers. This procedure, attractive for its simplicity, has little or no practical value when we are handling the population is very large. Example:

Suppose we want to select a random sample of 5 students in a statistical group of 20 students. 20C5 gives the total number of ways to choose a sample unordered and this result is 15.504 different ways to take the sample. If we list the 15.504 on separate pieces of paper, a tremendous task, then put them in a bowl and then stir, then we have a random sample of 5 if we select a piece of paper with five names. A simpler procedure to select a random sample would write each of the 20 names on separate pieces of paper, place in a bowl, mix them and then extract five roles simultaneously.

Another method parea obtain a random sample of 5 students in a group of 20 using a random number table. You can build the table using a calculator or computer. You can also ignore these and do the writing table ten digits from 0 to 9 on slips of paper, put them in a bowl and stir, hence, the first strip selected determines the first number in the table is returned to the container and stir again after you select determines the strip followed the second number of the table and the process continues until a random digits table with many numbers as desired.

There are many situations where simple random sampling is impractical, impossible or undesirable, although it would be desirable to use simple random samples to national opinion surveys about products or presidential elections, would be too costly or time consuming.

2 – Systematic Random Sampling: A sampling technique that requires an initial random selection of observations followed by a selection of observations obtained using a system or rule. Example:

For a sample of telephone subscribers in a big city, you first obtained a random sample of the numbers of the pages of the telephone directory, to choose the twentieth name each page would obtain systematic sampling, we can choose a name from the first page directory and then select every hundredth name of the place from the already selected. For example, we could select a number at random from the first 100, suppose the chosen is 40, then select the directory names that correspond to the numbers 40, 140, 240, 340 and so on.

3 – Stratified random sampling: A sample is stratified when elements of the sample are proportional to their presence in the population. The presence of an element in a layer in another exclude their presence. For this type of sampling, the population is divided into groups or strata in order to give representation to the different factors that make up the universe of study. For the selection of the items or representatives, using the random sampling method.

In short, requires separating the population by groups called strata, and then selecting a simple random sample in each stratum. Information simple random samples from each stratum would then constitute a bulk sample. Example:

Suppose we want to get a sample of the views of a large university professors. It can be difficult to obtain a sample with all the teachers, so I suppose we choose a random sample from each college or academic department, the strata would become schools or academic departments.

4 – Random Sampling by Area or Cluster: Requires choose a simple random sample of heterogeneous population together called clusters. Each element of the population belongs to exactly one cluster, and the elements within each cluster are usually heterogeneous or dissimilar. Example:

Suppose a company of cable television service is considering opening a branch in a large city, the company plans to conduct a study to determine the percentage of families who use their services, as it is impractical to ask in every house, the company decides select a part of the city at random, which is a conglomerate.

In cluster sampling, these are formed to represent, as closely as possible, the entire population, then used a simple random sample of clusters to study. Studies of social institutions such as churches, hospitals, schools and prisons are performed, generally based on cluster sampling.

Other sampling methods:

1 – Sampling Discretionary: A researcher’s criteria are chosen elements on what he believes they can bring to the study. Example.: Sampling trials; tellers in a bank or supermarket, and so on.

2 – Double Sampling: Under this type of sampling, when the result of the study of the first sample is not critical, a second sample is drawn from the same population. The two samples are combined to analyze the results. This method allows a person principiar with a relatively small sample to save costs and time. If the first sample gives a definitive result, the second sample may not be needed. For example, to test the quality of a batch of products manufactured, if the first sample yields a very high quality, the batch is accepted, if it gives a very poor quality, the lot is rejected. Only if the first sample yields an intermediate quality, the second sample will be required.

3 – Sampling Manifold: The procedure in this method is similar to that disclosed in the double sampling, except that the number of successive samples required to reach a decision is more than two samples.

4 – Intentional or opinion-based sampling: This type of sampling is characterized by a deliberate effort to obtain samples “representative” by including in the sample of supposedly typical groups. Very often their use in pre-election polls of voting areas that have marked previous voting trends.

5 – Sampling Casual or Incidental: This is a process in which the researcher directly and intentionally selected individuals of the population. The most frequent use of this procedure as the sample to individuals who are easily accessible (university professors frequently employ their own students). A particular case is that of the volunteers.

Table of random numbers.

The random number tables contain the digits 0, 1, 2, …, 7, 8, 9. These digits can be read individually or in groups in any order, in columns down, up columns in a row, diagonally, etc.., And may consider them as random. The tables are characterized by two things that make them particularly useful for random sampling. One feature is that the digits are arranged such that the likelihood that any given point in a sequence is equal to the probability of any other. The other is that the combinations of digits are equally likely to occur than other combinations of the same number of digits. These two conditions meet the requirements for random sampling, stated above. The first condition means that in a sequence of numbers, the likelihood that any digits at any point in the sequence is 1/10. The second condition means that all combinations of two digits are equally likely, in the same way that all combinations of three digits, and so on.

There are more effective methods to generate random numbers, many of which are used calculators or other electronic devices. The tables produced by these methods are fully verified to ensure that they are really random. However, the interest is not in developing these tables, but use them.

To use a table of random numbers:

1 – Make a list of the elements of the population.

2 – consecutively number the items in the list, starting with zero (0, 00, 000, etc..).

3 – Take the numbers in a table of random numbers, so that the number of digits of each is equal to the last numbered item in your list. Thus, if the last number was 18, 56 or 72, you must take a two-digit numbers.

4 – Ignore any digit that does not match the numbers on the list or repeat selected figures in the table above. Continue until the desired number of observations.

5 – Use these random numbers to identify the elements of the list for inclusion in the sample.

Donald B. Owen, Handbook of Statistical Tables, Reading Mass: Addisson-Wesley, 1962.

3690 2492 7171 7720 6509 7549 2330 5733 4730

0813 6790 6858 1489 2669 3743 1901 4971 8280

6477 5289 4092 4223 6454 7632 7577 2816 9002

0772 2160 7236 0812 4195 5589 0830 8261 9232

5692 9870 3583 8997 1533 6466 8830 7271 3809

2080 3828 7880 0586 8482 7811 6807 3309 2729

1039 3382 7600 1077 4455 8806 1822 1669 7501

7227 0104 4141 1521 9104 5563 1392 8238 4882

8506 6348 4612 8252 1062 1757 0964 2983 2244

5086 0303 7423 3298 3979 2831 2257 1508 7642

0092 1629 0377 3590 2209 4839 6332 1490 3092

0935 5565 2315 8030 7651 5189 0075 9353 1921

2605 3973 8204 4143 2677 0034 8601 3340 8383

7277 9889 0390 5579 4620 5650 0210 2082 4664

5484 3900 3485 0741 9069 5920 4326 7704 6525

6905 7127 5933 1137 7583 6450 5658 7678 3444

8387 5323 3753 1859 6043 0294 5110 6340 9137

4094 4957 0163 9717 4118 4276 9465 8820 4127

4951 3781 5101 1815 7068 6379 7252 1086 8919

9047 0199 5068 7447 1664 9278 1708 3625 2864

7274 9512 0074 6677 8676 0222 3335 1976 1645

9192 4011 0255 5458 6942 8043 6201 1587 0972

0554 1690 6333 1931 9433 2661 8690 2313 6999

9231 5627 1815 7171 8036 1832 2031 6298 6073

3995 9677 7765 3194 3222 4191 2734 4469 8617

2402 6250 9362 7373 4757 1716 1942 0417 5921

5295 7385 5474 2123 7035 9983 5192 1840 6176

5177 1191 2106 3351 5057 0967 4538 1246 3374

7315 3365 7203 1231 0546 6612 1038 1425 2709

5775 7517 8974 3961 2183 5295 3096 8536 9442

5500 2276 6307 2346 1285 7000 5306 0414 3383

3251 8902 8843 2112 8567 8131 8116 5270 5994

4675 1435 2192 0874 2897 0262 5092 5541 4014

3543 6130 4247 4859 2660 7852 9096 0578 0097

3521 8772 6612 0721 3899 2999 1263 7017 8057

5573 9396 3464 1702 9204 3389 5678 2589 0288

7478 7569 7551 3380 2152 5411 2647 7242 2800

3339 2854 9691 9562 3252 9848 6030 8472 2266

5505 8474 3167 8552 5409 1556 4247 4652 2953

6381 2086 5457 7703 2758 2963 8167 6712 9820

An example of a random number table is a list of National Lottery winning numbers throughout their history, they are characterized by that each digit is equally likely to be chosen, and their choice is independent of the other extractions . One way to do it:

Suppose we have a list of random numbers k = 5 digits (00000-99999), a population of N = 600 individuals, and we want to remove a sample of n = 6 of them. In this case ordered the entire population (using any criteria) so that each of its elements appears with a number from 1 to 600. Secondly we went to the table of random numbers, starting at any point and draw a number t, and we as the first element of the sample to the population element:

The process is repeated by taking the following numbers in the table of random numbers, to obtain the sample of 10 individuals.


can be considered as one goes observations U, which follows a uniform distribution in the interval [0.1]

Reasons for sampling.

1 – We are not really interested in all the elements but only in some cases individuals or population. This type of research is not usual in fact a sample study, but an extended case study.

2 – We are equally interested in all elements of the population and would like to study them all. But for practical reasons, we have to pick the sample. Maybe we have a population of millions of objects and is impossible to cover even a majority of them. Also in those cases (with populations of, say, 10,000) in which each object could choose to study, the study sample may be a wise choice, because it saves time and can use the time saved to more carefully study the elements. All these are good cases for a sample study.

In that sampling is not always interested in the sample population but, more precisely, on the attributes of the elements of the population. When studying the elements of the example we would want to choose items that have the same attributes as the population mean. If that is the case, our sample is representative.

This is the ideal case but in practice we have no means of knowing if the elements are representative in reality calculating probabilities tells us that in most cases there will be some differences between the sample and the population. The difference is called bias, and to some extent almost always present in the sample, simply by the accidental nature of the sampling.

However, if we have reason to suspect the presence of a systematic bias in the sample, we should always try to find what it is and see if it can be removed.

Difference between the following terms: sampling error and non-sampling error.

The difference lies in the types of errors that are measured or detected in the survey results that shed. While sampling error points from poorly written questions by interviewers in surveys, unwillingness on the part of respondents and miscalculations, the Sampling Error not locate the false information supplied by the respondents.

In short sampling error usually occurs when there is conducted the comprehensive survey of the population and the non-sampling errors can occur in a comprehensive survey of the population.

White and sampled population.

The White Population: That population belonging to a city that aims to be studied.

The sampled population: Includes real study of some of the elements of a population.

Precision and Accuracy.

Accuracy: refers to identity or at least the similarity between two or more measurements of the same quantity. To some extent, the accuracy is related to the stability of the technique of the experimenter, who may need more improvements he believes. However, depending on the nature of the particular measure under consideration, may appear a lack of precision due to a faulty control temperature, a piece of glass chipped, corroded or loose in the instruments used.

Accuracy: Refers to the proximity to the true value of the measurements obtained. For a given process, the accuracy is estimated performing physical or chemical measurements of a known pattern. For example, if an investigation depends to a titration with a standardized solution of alkali is necessary to check the reliability of this reagent from time to time, by titrating a known amount determined gravimetrically or a salt of an acid.

Can you believe that in a situation in which almost surely: simple random sampling and systematic sampling.

Both systematic sampling and random sampling are methods but have enough factors to contribute significantly to the outcome of any test, study will be lost. However, many times for not having the time or sufficient monetary resources, sometimes turning them as they are, time-saving methods, in which intuition requires investment also seems to be easier or more must leave the decision to target, such as tossing a coin. It is a business practice.

You agree with the following statement: “studies show can never be as accurate as the full accounts of people”. Reason.

Sampling studies can never become accurate and precise in the full accounts of the population, and statistical errors that often occur in these studies that yield results.

To better explain this opinion, we cite an example: A company that wants to study certain attitudes in the population of a city. For convenience, we will use a phone as a basis of selection in the survey. This is precisely where the error part in the survey, due to the omission of people who do not have telephones, which should be included according to the purpose of the study.


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