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CHAPTER SEVEN: Samples Previously, it was stated that there are two types of statistics: inferential and descriptive. The preferred would be descriptive, because it is based upon a population. However, this is seldom possible. Most statistical analyses are inferential, based upon a sample. It is imperative, then, to understand the assumptions and principles of samples. We shall begin with some crucial definitions. TERMS: The more fundamental principles that govern the behavior of samples are the Law of Large Numbers, the Central Limit Theorem and the features of The Sampling Distribution of the Mean. A Sampling Distribution- A theoretical probability distribution which would be comprised of all the potential calculations of a statistic, from an infinite variety of samples. If comprised of the infinite variety of all possible means from all possible samples of a population, that would be the sampling distribution of the mean (u_). X The Law of Large Numbers- As the sample size increases, the error of estimate reduces; because the standard error (SE) of the sampling distribution of the mean is (SE) = / n. Therefore, as N ↑, SE ↓ for the sampling distribution of the mean (u_). If u = 500 and = 100: X when n = 4, SE = 100/ 4 = 100/2 = 50 when n = 25, SE = 100/ 25 = 100/5 = 20 when n = 100, SE = 100/ 100 = 100/10 = 10 The Central Limit Theorem (CLT) - As the sample size increases,the error reduces and the sampling distribution of the mean quickly approaches normal. This is true regardless of the shape of the original population from which it was drawn. For samples with N > 30, the mean of the sampling distribution of the mean is the best unbiased estimate of the population true mean (u); and the SE of the sampling distribution of the mean is the best unbiased estimate of the SE and has the formula: SE = Sigma (σ) / the square root of N The value of these principles, is the assurance that means can be compared. In fact, Z-scores for a mean of a sample can be calculated, with a minor adjustment in the formula. For a mean Z = X - u / SE Caution! It would not be appropriate to compare means based upon the binomial approximation to the mean. TYPES OF SAMPLING: Random Samples-A subset of the referent population,in which all members of the population have an equal and constant likelihood of being selected; that is, sampling with replacement. In this way, all potential confounders may be distributed across groups in a non-systematic way, determined by chance factors alone. Block Randomization-A type of random sampling which collects subjects in blocks or proportions according to some factor or demographic characteristic. Random Assignment-The distribution of subjects, who were randomly selected, across experimental conditions to assure a pattern of assignment determined by chance factors alone.