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COURSE: JUST 3900 INTRODUCTORY STATISTICS FOR CRIMINAL JUSTICE Chapter 7: The Distribution of Sample Means Instructor: Dr. John J. Kerbs, Associate Professor Joint Ph.D. in Social Work and Sociology Samples and Populations Samples provide an incomplete picture of the population. There are aspects of the population that may not be included within a sample. The sampling error is the natural discrepancy (i.e., the difference), or amount of error, between a sample statistic and its corresponding population parameter. The sampling error is the measure of the discrepancy (i.e., difference) between the sample and the population. A Sampling Distribution A Sampling Distribution is a distribution of statistics obtained by selecting all of the possible samples of a specific size (n) from a population. The Distribution of Sample Means The Distribution of Sample Means is defined as the set of sample means for all of the possible random samples of a particular size (n) that can be selected from a specific population. Often called the Sampling Distribution of M This distribution has well-defined (and predictable) characteristics that are specified in the Central Limit Theorem The Distribution of Sample Means The three characteristics of the Distribution of Sample Means 1. Sample means should pile up around the population mean 2. The pile of sample means should tend to form a normalshaped distribution. They should pile up in the center of the distribution (around μ) and the frequencies should taper off as the distance between M and μ increases. 3. In general, the larger the sample, the closer the sample means should be to the population mean (μ). Larger samples are more representative of the population than smaller samples Sample means obtained with large samples (i.e., a large n) should cluster relatively close to the population parameter Means obtained by small samples should be more widely scattered The Central Limit Theorem The Central Limit Theorem is defined as follows: For any population with a mean (μ) and standard deviation (σ), the distribution of sample means for sample size n will have a mean of μ and a standard deviation of σ and will approach a normal distribution as n approaches infinity (∞). The Central Limit Theorem 1. The Expected Value of M is the mean of the distribution of sample means and the Expected Value of M is always equal to the mean of the population of scores (μ). 2. The shape of the distribution of sample means tends to be normal. It is guaranteed to be normal if either a) the population from which the samples are obtained is normal, or b) the sample size n ≥ 30. 3. The standard deviation of the distribution of sample means is called the Standard Error of M (σM) and is computed by the following: The Expected Value of M If two (or more) samples are selected from the same population, the two samples probably will have different means. Although the samples will have different means, you should expect the sample mans to be close to the population mean. The mean of the distribution of the sample of means is equal to the mean of the population of scores (μ): that is the expected value of M. Standard Error of M The standard error (also known as the standard deviation of the distribution of sample means, σM) provides a measure of the average distance between M (sample mean) and μ (population mean). σM describes the distribution of sample means (variability) σM shows how much distance is expected between M and μ Law of large numbers: The larger the sample size (n), the more probable or likely it is that M is close to μ. Inverse relationship: the larger the sample size, the smaller the stander error. Small standard errors indicate that sample means are close together (large standard errors indicate that means are scattered over a large range with larger difference from one sample to another) The Standard Error of M The standard error of M is defined as the standard deviation of the distribution of sample means and measures the standard distance between a sample mean and the population mean. Thus, the Standard Error of M provides a measure of how accurately, on average, a sample mean represents its corresponding population mean. The Standard Error of M Consider the changes in Standard Error of M as n increases from 1 to 4 and then to 100 for a normal population with a mean of 80 (μ=80) and a standard deviation of 20 (σ=20) Do NOT confuse “standard deviations” with “standard errors” Difference Between Standard Deviations and Standard Errors Standard Deviation measures the distance between a score and the population mean X-μ The Standard Error measures the distance between a sample mean and the population mean M–μ The Standard Error (σM) is the same as the Standard Deviation for n = 1 Note: there is only one population mean Probability and Sample Means Because the distribution of sample means tends to be normal, the z-score value obtained for a sample mean can be used with the unit normal table to obtain probabilities. The procedures for computing z-scores and finding probabilities for sample means are essentially the same as we used for individual scores Probability and Sample Means (cont'd.) However, when you are using sample means, you must remember to consider the sample size (n) and compute the standard error (σM) before you start any other computations. Also, you must be sure that the distribution of sample means satisfies at least one of the criteria for normal shape before you can use the unit normal table: 1. the population from which the samples are obtained is normal, or 2. the sample size (n) is 30 or more. z-Scores and Location within the Distribution of Sample Means Within the distribution of sample means, the location of each sample mean can be specified by a z-score: (M – μ) z = ───── or σM z = (M – μ) ───── (σ/√n) z-Scores and Location within the Distribution of Sample Means (Continued) As always, a positive z-score indicates a sample mean that is greater than μ and a negative z-score corresponds to a sample mean that is smaller than μ. The numerical value of the z-score indicates the distance between M and μ measured in terms of the standard error. Distribution for Sample Means (n = 25, μ = 500, σ = 100) A score of 540 is two standard errors above the mean (z=+2.00), which is very unlikely (see Unit Normal Table for z = +2.00, p = 0.0228) 2.28% More Thoughts on Standard Error Standard errors are nothing more than measures of reliability. Vogt (2005, p. 274) defines reliability as follows: Freedom from measurement (random) error. In practice, this boils down to consistency or stability of a measure or test or observation from one use to the next. When repeated measures of the same thing give highly similar results, the measurement instrument is said to be reliable. Small standard errors indicate that sample means are close together and so researchers can be fairly confident that an individual sample mean can act as a reliable measure of the population mean Large standard errors indicate problems with reliability