IntroductionSo far in this course we have covered the concepts of data collection: what to collect and the meaning behind each collection.Now we enter a new field of statistics: data reduction. It starts from a practical limitation: we can almost never handle the entire population of available data. For economic, time or physical reasons, we analyze only samples and, within them, keep only the information that is relevant to the question we want to answer.Population and sample“A set of entities that share at least one common characteristic is called a statistical population or statistical universe.”- Definition of populationIn any statistical study we want to investigate one or more characteristics of the elements of a population, and that characteristic must be perfectly defined. This happens when, given any element, we can say without ambiguity whether or not it belongs to the population. We therefore need a criterion for building the population that is valid for anyone, at any time or place. For example, if we want to survey elementary school students, we must define who makes up the universe: only the students enrolled today, or also those who have already left the school? The answer depends on each case.Most of the time, because it is impossible or economically or practically unfeasible, we limit the observations of a study to only part of the population. This part of the population under study is called a sample.“A sample is a finite subset of a population”- Your teacher LeonThe sample must be representative of the population, that is, it must have the same basic characteristics as the population with respect to the phenomenon we want to study. For that, the sample (or samples) must be obtained through appropriate procedures.What is samplingSampling is the technique of collecting samples in a way that guarantees, as far as possible, that chance decides who is chosen.In random sampling, every element of the population has the same chance of being chosen, which makes the sample representative. This is very important, because our conclusions about the population will be based on the results obtained from samples of that population. When the way the data are collected favors certain elements, the sample becomes biased, as the image below shows: the people who answer a survey about liking surveys are precisely the people who like answering surveys.The problem with a biased sampleWith a good sample, we can infer or estimate population parameters. Since the true parameters of a population (mean, variance, proportion) are usually unknown, we use estimators: formulas that approximate these values from the information in the sample.Types of estimatorsThere are many estimators, depending on the parameter to be estimated. Some common examples:Sample mean as an estimator of the population mean\[ \bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i \]Where \(X_i\) are the sample values and \(n\) is the sample size. The sample mean \(\bar{X}\) estimates the population mean \(\mu\).Sample variance as an estimator of the population variance\[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2 \]The sample variance \(s^2\) estimates the population variance \(\sigma^2\). The formula divides by \(n-1\), not by \(n\), to produce an unbiased estimator. This adjustment, called Bessel's correction, exists because the deviations are measured from the sample's own mean, which always sits "in the middle" of the data; so the sum of squared deviations tends to be smaller than it would be around the true mean, and dividing by \(n\) would underestimate the variability of the population.Sample proportion as an estimator of the population proportion\[ \hat{p} = \frac{\text{number of successes in the sample}}{n} \]The sample proportion \(\hat{p}\) estimates the population proportion \(p\). For example, if 18 out of 60 respondents use public transportation, \(\hat{p} = 18/60 = 0.30\), and we estimate that 30% of the population uses public transportation.Practical exampleWe want to calculate the mean height and the variance of the heights of the students in a school with 1,000 students.PopulationIf we measure all 1,000 students, we are dealing with the population. We use the formulas for the mean \(\mu\) and the variance \(\sigma^2\) (the latter dividing by \(N\)).SampleIf we select 50 students to represent the 1,000, we use the formulas for \(\bar{x}\) and \(s^2\), with the \(n-1\) correction in the variance.Understanding this difference is crucial, because statistics calculated from the sample are used to make inferences about the population.A sample has the values 2, 4 and 6. What is the sample variance s²?Answer: 4 — the mean is 4; the squared deviations add up to 4 + 0 + 4 = 8, and we divide by n − 1 = 2, getting 4. Dividing by n = 3 would give 8/3, which is the population variance formula and would underestimate the variability.Properties of estimatorsNot all estimators are equal, and statisticians look for certain properties in a good estimator. The main ones are:UnbiasednessAn estimator is unbiased if, on average, it hits the population parameter: its expected value equals the parameter. A single sample may miss up or down, but there is no systematic error in one direction.ConsistencyAn estimator is consistent if, as the sample size grows, it tends to get closer to the true value of the parameter. In other words, the larger the sample, the more precise the estimate tends to be.EfficiencyAmong unbiased estimators of the same parameter, the most efficient is the one with the smallest variance. This means its estimates vary less from one sample to another.SufficiencyAn estimator is sufficient if it summarizes all the information the sample contains about the parameter being estimated: once you know it, the rest of the data adds nothing about that parameter.A researcher wants to estimate a population mean and has two unbiased estimators: A, with variance 4, and B, with variance 9. Which should be chosen, and why?Answer: A, because it is more efficient — both are right on average, but A varies less from one sample to another, so each estimate tends to be closer to the true value. Being unbiased is not enough to choose: the variance matters too.Types of samplingSimple random samplingThis type of sampling is equivalent to a lottery draw: every element of the population has the same chance of being included in the sample.ExampleWe want a representative sample to study the height of the ninety students of a school: we number the students from 01 to 90 and draw 9 of them, which represents 10% of the population.Proportional stratified samplingPopulations are often divided into subpopulations, called strata.Since the variable under study is likely to behave heterogeneously from one stratum to another and homogeneously within each stratum, the draw of the sample elements should take the strata into account.That is what proportional stratified sampling does: besides considering the existence of the strata, it takes from each stratum a number of elements proportional to its size.ExampleSuppose that, of the ninety students, 54 are boys and 36 are girls. To obtain a proportional stratified sample of 10% of the population, we separate the strata: 10% of 54 boys is 5.4, that is, 5 boys; and 10% of 36 girls is 3.6, that is, 4 girls. The sample will have 5 + 4 = 9 students, drawn within each stratum.Systematic samplingWhen the elements of the population are already ordered, there is no need to build a list for the draw. Examples are a hospital's medical records, the buildings on a street, the items on a production line, etc. In these cases, the elements of the sample can be selected by a system set by the researcher: compute the interval \(k = N/n\) (population size divided by sample size), draw a starting point between 1 and \(k\) and, from there, take one element every \(k\). This type of sampling is called systematic.ExampleOn a production line, workers can take one item out of every ten produced to form the sample of the daily production. In this case, the sample size is fixed at 10% of the population.An ordered population has 600 elements and we want a systematic sample of 40. If the first element drawn was the 7th, which of these will also be in the sample?Answer: The 97th — the interval is k = 600/40 = 15, and the sample is made up of 7, 22, 37, 52, 67, 82, 97... A shortcut: the chosen elements are those that leave remainder 7 when divided by 15; 97 = 6 × 15 + 7, while 50 leaves remainder 5 and 120 and 315 leave remainder 0.A population of 200 elements is divided into three strata of 40, 100 and 60. In a proportional stratified sample of 30 elements, how many come from the second stratum?Answer: 15 — the sample corresponds to 30/200 = 15% of the population, and 15% of 100 is 15. The other strata give 6 (15% of 40) and 9 (15% of 60), for a total of 30. Splitting 30 equally among three strata (10) ignores proportionality.Exercises1) An elementary school has 124 students. Explain how to obtain a representative sample corresponding to 15% of the population.2) A school has 800 students in different grades, and the classes have different sizes. Explain how to obtain a sample of 100 students.3) A population is divided into three strata with sizes \(n_1 = 40\), \(n_2 = 100\) and \(n_3 = 60\), respectively. Knowing that, in a proportional stratified sample, twelve elements of the sample were taken from the 3rd stratum, determine the total number of elements in the sample.4) Show how a sample of 32 elements could be taken from an ordered population of 2,432 elements. In the overall ordering, which of the following elements would be chosen for the sample, given that the element in position 1,420 belongs to it? 1,648th, 290th, 725th, 2,025th, 1,120th.Answer key1) 15% of 124 is 18.6, that is, 19 students. We number the students from 1 to 124 and draw 19 numbers (simple random sampling).2) Since the classes have different sizes, we use proportional stratified sampling, with each class as a stratum: 100/800 = 12.5%, so we draw 12.5% of the students from each class (rounding so that the total is 100).3) The sample has 40 elements: 12 out of 60 is 20% of the 3rd stratum, and 20% of 200 (40 + 100 + 60) is 40.4) The interval is 2,432/32 = 76: draw a starting point between 1 and 76 and take one element every 76. Element 1,420 leaves remainder 52 when divided by 76, and the only one on the list with the same remainder is the 1,648th (1,648 = 1,420 + 3 × 76), which belongs to the sample.
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