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The sampling distribution of means is the frequency distribution of all possi- ble sample means that occur when an infinite number of samples of the same size N are randomly selected from one raw score population cheap naproxen 500 mg mastercard arthritis in both ring fingers. This is similar to a distribution of raw scores order naproxen with a visa arthritis pain groin area, except that here each score on the X axis is a sample mean safe naproxen 500 mg arthritis in the knee cure. To the right of are the sample means the statistician obtained that are greater than 500, and to the left of are the sample means that were less than 500. This is because most scores in the population are close to 500, so most of the time the statistician will get a sample containing scores that are close to 500, so the sample mean will be close to 500. Less frequently, the statistician will obtain a strange sample containing mainly scores that are farther below or above 500, producing means that are farther below or above 500. Once in a great while, some very unusual samples will be drawn, resulting in sam- ple means that deviate greatly from 500. The story about the bored statistician is useful because it helps you to understand what a sampling distribution is. The central limit theorem is a statistical principle that defines the mean, the standard deviation, and the shape of a sampling distribution. From the central limit theorem, we know that the sampling distribution of means always (1) forms an approximately normal distribution, (2) has a equal to the of the underlying raw score population from which the sampling distribution was created, and (3), as you’ll see shortly, has a standard deviation that is mathematically related to the standard deviation of the raw score population. The importance of the central limit theorem is that with it we can describe the sam- pling distribution from any variable without actually having to infinitely sample the population of raw scores. Then we’ll know the important characteristics of the sampling distribution of means. Remember that we took a small detour, but the original problem was to evaluate our Prunepit mean of 520. To do so, we simply determine where a mean of 520 falls on the X axis of the sampling distribution in Figure 6. But if 520 lies toward the tail of the distribution, far from 500, then it is a more infrequent and unusual sample mean (the statistician seldom found such a mean). The sampling distribution is a normal distribution, and you already know how to determine the location of any “score” on a normal distribution: We use—you guessed it—z-scores. That is, we determine how far the sample mean is from the mean of the sampling distribution when measured using the standard deviation of the distribution. This will tell us the sample mean’s relative standing among all possible means that occur in this situation. To calculate the z-score for a sample mean, we need one more piece of information: the standard deviation of the sampling distribution. The Standard Error of the Mean The standard deviation of the sampling distribution of means is called the standard error of the mean. That is, in some sampling distributions, the sample means may be very different from one another and, “on average,” deviate greatly from the average sample mean. For the moment, we’ll discuss the true standard error of the mean, as if we had actually computed it using the entire sampling distribution. The σ indicates that we are describing a population, but the subscript X indicates that we are describing a population of sample means—what we call the sampling dis- tribution of means. The central limit theorem tells us that σX can be found using the following formula: The formula for the true standard error of the mean is σX σX 5 1N Using z-Scores to Describe Sample Means 127 Notice that the formula involves σX, the true standard deviation of the underlying raw score population, and N, our sample size. This is because with more variable raw scores the statistician often gets a very different set of scores from one sample to the next, so the sample means will be very different (and σX will be larger). But, if the raw scores are not so variable, then different samples will tend to contain the same scores, and so the means will be similar (and σX will be smaller). With a very small N (say 2), it is easy for each sample to be different from the next, so the sample means will differ (and σX will be larger). How- ever, with a large N, each sample will be more like the population, so all sample means will be closer to the population mean (and σX will be smaller). This is because the bored statisti- cian will often encounter a variety of high and low scores in each sample, but they will usually balance out to produce means at or close to 500.   