Sample

Sample size - the number of cases included in the sample.

Samples can be conditionally divided into large and small, as in mathematical statistics different approaches are used depending on the size of the sample. It is believed that samples of volume greater than 30 can be attributed to large ^[1] .

When comparing two (or more) samples, their dependence is an important parameter. If a homomorphic pair can be established (that is, when one case from the sample X corresponds to one and only one case from the sample Y and vice versa) for each case in two samples (and this basis of the relationship is important for the characteristic measured on the samples), such samples are called dependent . Examples of dependent samples:

• twin pairs
• two measurements of a trait before and after experimental exposure,
• husbands and wives
• etc.

If there is no such relationship between the samples, then these samples are considered independent , for example:

• men and women
• psychologists and mathematicians .

Accordingly, dependent samples always have the same volume, and the volume of independent ones may differ.

Comparison of samples is carried out using various statistical criteria :

• Pearson criterion ( χ ² )
• Student criterion ( t )
• Wilcoxon test ( T )
• Mann Criterion - Whitney ( U )
• Sign Criteria ( G )
• and etc.

A sample may be considered representative or unrepresentative. The sample will be representative when examining a large group of people, if within this group there are representatives of different subgroups, the only way to draw the right conclusions.

An example of a non-representative sample

In the USA, one of the most famous historical examples of a non-representative sample is considered to be the case that occurred during the presidential election in 1936 ^[2] . The Literary Digest magazine, which successfully predicted the events of several previous elections, made a mistake in its predictions by sending ten million test ballots to its subscribers, as well as to people selected from the phone books of the whole country and people from car registration lists. In 25% of the returned ballots (almost 2.5 million), the votes were distributed as follows:

• 57% preferred Republican candidate Alph Landon
• 40% elected then Democratic President Franklin Roosevelt

In actual elections, Roosevelt, as you know, won, gaining more than 60% of the vote. The Literry Digest error was this: wanting to increase the representativeness of the sample — since they knew that most of their subscribers considered themselves Republicans — they expanded the sample to include people selected from phone books and registration lists. However, they did not take into account contemporary realities and in reality scored even more Republicans: during the Great Depression, mostly middle and upper classes (that is, most Republicans, not Democrats) could afford to have phones and cars.

There are several main types of group building plan ^[3] :

1. A study with experimental and control groups that are placed in different conditions.
   • Study with experimental and control groups using a pairwise selection strategy
2. A study using only one group - experimental.
3. A study using a mixed (factorial) plan - all groups are placed in different conditions.

Samples are divided into two types:

• probabilistic
• incredible

Probabilistic Samples

1. Simple probabilistic sampling:
   • Simple re-sampling. The use of such a sample is based on the assumption that each respondent is equally likely to fall into the sample. Based on the list of the general population, cards with respondent numbers are compiled. They are placed in a deck, mixed up and a card is taken out of them at random, a number is written, then it is returned. Next, the procedure is repeated as many times as we need the sample size. Less: repetition of units of selection.

The procedure for constructing a simple random sample includes the following steps:

1) you need to get a complete list of members of the general population and number this list. We recall that such a list is called the sampling frame;

2) determine the estimated sample size, that is, the expected number of respondents;

3) to extract from the table of random numbers as many numbers as we need sample units. If the sample should contain 100 people, 100 random numbers are taken from the table. These random numbers can be generated by a computer program.

4) select from the base list those observations whose numbers correspond to the written random numbers

• Simple random sampling has obvious advantages. This method is extremely easy to understand. The results of the study can be extended to the studied population. Most statistical inference approaches involve gathering information using simple random sampling. However, the simple random sampling method has at least four significant limitations:

1) it is often difficult to create the basis of a sample observation that would allow a simple random sampling.

2) the result of applying a simple random sample can be a large population, or a population distributed over a large geographical area, which significantly increases the time and cost of data collection.

3) the results of applying a simple random sample are often characterized by low accuracy and a larger standard error than the results of applying other probabilistic methods.

4) as a result of applying SRS, a non-representative sample may be formed. Although the samples obtained by simple random selection on average adequately represent the general population, some of them extremely incorrectly represent the studied population. The probability of this is especially high with a small sample size.

• Simple, non-repetitive sampling. The sampling procedure is the same, only cards with respondent numbers do not return back to the deck.

1. Systematic probabilistic sampling. It is a simplified version of simple probabilistic sampling. Based on a list of the general population, respondents are selected at a certain interval (K). The value of K is determined randomly. The most reliable result is achieved with a homogeneous population; otherwise, the step size and some internal cyclic patterns of the sample may coincide (sample mixing). Cons: the same as in a simple probabilistic sample.
2. Serial (nested) sampling. The selection units are statistical series (family, school, team, etc.). The selected elements are subjected to a continuous examination. The selection of statistical units can be organized as random or systematic sampling. Less: The possibility of greater uniformity than in the general population.
3. Zoned sampling. In the case of a heterogeneous population, before using a probabilistic sample with any sampling technique, it is recommended to divide the population into homogeneous parts, such a sample is called zoned. Zoning groups can be both natural formations (for example, city districts), and any sign laid down in the basis of the study. The sign on the basis of which the separation is carried out is called the sign of stratification and zoning.
4. "Convenient" selection. The “convenient” sampling procedure consists of establishing contacts with “convenient” sampling units — a group of students, a sports team, friends and neighbors. If it is necessary to obtain information about people's reactions to the new concept, such a sample is quite justified. A “convenient” sample is often used for preliminary testing of questionnaires.

The selection of groups for their participation in a psychological experiment is carried out using various strategies that are needed in order to ensure the highest possible compliance with internal and external validity ^[4] .

• Randomization (random selection)
• Pairwise selection
• Stratometric selection
• Approximate modeling
• Attracting Real Groups

Randomization

Randomization , or random selection , is used to create simple random samples. The use of such a sample is based on the assumption that each member of the population is equally likely to fall into the sample. For example, to make a random sample of 100 university students, you can put pieces of paper with the names of all university students in a hat, and then get 100 pieces of paper out of it - this will be a random selection (Goodwin J., p. 147) ......

Pairwise selection

Pairwise selection is a strategy for constructing sample groups, in which groups of subjects are composed of subjects equivalent in the side parameters significant for the experiment. This strategy is effective for experiments using experimental and control groups with the best option - using twin pairs ( mono - and dizygotic ).

Stratometric selection

Stratometric selection - randomization with the allocation of strata (or clusters ). With this method of sampling, the general population is divided into groups (strata) with certain characteristics ( gender , age , political preferences, education , income level , etc.), and subjects with appropriate characteristics are selected.

Approximate Modeling

Approximate modeling - compiling limited samples and generalizing the conclusions about this sample to a wider population. For example, with the participation in the study of students of the 2nd year of university , the data of this study apply to "people aged 17 to 21 years." The admissibility of such generalizations is extremely limited.

Approximate modeling is the formation of a model that for a clearly defined class of systems (processes) describes its behavior (or the necessary phenomena) with acceptable accuracy.

• Ilyasov F.N. Representativeness of survey results in marketing research // Sociological studies . 2011. No. 3. P. 112-116.
• Ilyasov F.N. The inverse problem of sampling and motivation in the Forex market // Social Studies. 2016. No2. S. 49-59.
• Ilyasov F. N. Algorithms for the formation of a sample of a sociological survey // Social Research. 2017. No2. S. 60-75.
• Nasledov A. D. Mathematical methods of psychological research. - SPb .: Speech, 2004.
• Ostapenko R.I. Mathematical Foundations of Psychology . - Voronezh .: Voronezh State Pedagogical University , 2010 .-- 76 p.

• Sample according to P 50-605-80-93