IntroductionSo far in the subject, we have seen the concepts related to data collection: what to collect and the meanings behind each collection.We will enter a new field within statistics, which concerns data reduction. This field is concerned with the inability of human beings to deal with the entire population of available data. For economic, time-related, and/or physical reasons, we restrict ourselves to analyzing only data samples and, within these samples, reducing even further, so that we keep only the relevant data.Population and Sample> The set of entities that possess at least one common characteristic is called astatistical population or statistical universe.As in any statistical study, we intend to research one or more characteristics of the elements of some population; this characteristic must be perfectly defined. And this occurs when, considering any element, we can state, without ambiguity, whether that element belongs to the population or not. Therefore, there must be a criterion for constituting the population, valid for any person, in time or space. For this reason, when we intend to conduct research among elementary school students, we need to define which students make up the universe: those who currently occupy the school desks, or should we also include those who have already attended the school? Of course, the solution to the problem will depend on each particular case.Most of the time, due to impossibility or economic or time-related infeasibility, we limit the observations related to a given research study to only part of the population. This part drawn from the population under study is called a sample.“A sample is a finite subset of a population”- Your professor LeonIt is necessary to ensure that the sample is representative of the population; that is, the sample must have the same basic characteristics as the population with regard to the phenomenon we wish to study. Therefore, the sample or samples to be used must be obtained through appropriate processes.What Is SamplingThere is a special technique—sampling—for collecting samples, which ensures, as much as possible, randomness in the selection.In this way, each element of the population has the same chance of being chosen, which gives the sample its representative character, and this is very important because, as we have seen, our conclusions regarding the population will be based on the results obtained from samples of that population.The problem of a biased sampleThus, with a good sample, we can infer or estimate population parameters based on data samples. Since the true parameters of a population (such as the mean, variance, or proportion) are generally unknown, estimators allow us to approximate these values using information from a sample.Types of EstimatorsThere are several estimators, depending on the parameter one wishes to estimate. Some common examples are:Sample Mean as an Estimator of the Population Mean\[ \bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i \]Where 𝑋𝑖 are the sample values and n is the sample size.Sample Variance (𝑠2) as an Estimator of the Population Variance (𝜎2)\[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2 \]This formula divides by 𝑛 -1 to produce an unbiased estimator. This adjustment, called Bessel’s correction, is made because a sample tends to underestimate the real variability of the population.Sample Proportion (𝑝) as an Estimator of the Population Proportion (𝑝)\[ \hat{p} = \frac{\text{number of successes in the sample}}{n} \]Where p is the number of successes in the sample.Practical ExampleIf we want to calculate the average height and the variance of the heights of students at a school with 1,000 students:PopulationIf we analyze the data of all 1,000 students, we are dealing with the population. We will use the formulas for 𝜇 and 𝜎².SampleIf we select 50 students to represent the 1,000, we will use the formulas for x_bar and 𝑠², including the 𝑛−1 correction for variance.Understanding this difference is crucial, because the statistics calculated from the sample are used to make inferences about the population.Properties of EstimatorsNot all estimators are the same, and statisticians look for several properties in a good estimator. The main properties include:Biased (Bias)An estimator is unbiased if, on average, it provides the exact value of the population parameter. This means that the expected value of the estimator is equal to the parameter.ConsistencyAn estimator is consistent if, as the sample size increases, it tends to get closer to the true value of the population parameter. In other words, the larger the sample, the more accurate the estimate will be.EfficiencyAn estimator is efficient if it has the lowest possible variance compared with other estimators for the same parameter. This means that it tends to be the most precise among the possible estimators.SufficiencyAn estimator is sufficient if it uses all the relevant information present in the sample about the parameter being estimated.Types of SamplingSimple Random SamplingThis type of sampling is equivalent to a lottery draw.ExampleLet us obtain a representative sample for research on the height of ninety students at a school: we number the students from 01 to 90 and draw 9 of these students; in this case, they represent 10% of the population.Proportional Stratified SamplingOften, the population is divided into subpopulations—strata.Since the variable under study is likely to show heterogeneous behavior from one stratum to another and homogeneous behavior within each stratum, it is advisable that the drawing of the sample elements take these strata into account.This is exactly what we do when we use proportional stratified sampling, which, in addition to considering the existence of strata, obtains the sample elements in proportion to the number of elements in each stratum.ExampleSuppose there are ninety students, of whom 54 are boys and 36 are girls. To obtain the proportional stratified sample of 10% of the population, we must separate the strata. 10% of 54 boys is 5.4, or 5 boys, and 10% of 36 girls is 3.6, or 4 girls.Systematic SamplingWhen the elements of the population are already ordered, there is no need to construct the reference system. Examples include the medical records of a hospital, the buildings on a street, production lines, etc. In these cases, the selection of the elements that will make up the sample can be done by a system imposed by the researcher. This type of sampling is called systematic.ExampleIn the case of a production line, employees can, for every ten items produced, remove one item to belong to a sample of the daily production. In this case, we would be setting the sample size at 10% of the population.Exercises1) An elementary school has 124 students. Explain how to obtain a representative sample corresponding to 15% of the population.2) In a school there are 800 students from different grades. The classes have different sizes. Explain how to obtain a sample of 100 students.3) A population is divided into three strata, with sizes, respectively: n1= 40, n2 = 100 and n3 = 60. Knowing that, when proportional stratified sampling was carried out, twelve elements of the sample were taken from the 3rd stratum, determine the total number of elements in the sample.4) Show how it would be possible to take a sample of 32 elements from an ordered population made up of 2,432 elements. In the general ordering, which of the following elements would be chosen to belong to the sample, knowing that the element of order 1,420 belongs to it? 1,648th, 290th, 725th, 2,025th, 1,120th.Answer Key3) The sample has size 40.4) The element of order 1,648 belongs to the sample.
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