Section 3.1 Measures of Center 89 In Example 3.1, we rounded the mean to one more decimal place than the data. We will follow this practice in general. SUMMARY We will round the mean to one more decimal place than the data. Notation for the mean Computing a mean involves adding a list of numbers. It is useful to have some notation that will allow us to discuss lists of numbers in general. When we wish to write down a list of n numbers without specifying what the numbers are, we often write x1, x2, ..., xn. To indicate that we are adding these numbers, we write x. (The symbol is the uppercase Greek letter sigma.) NOTATION • A list of n numbers is denoted x1, x2, ..., xn. • x represents the sum of these numbers:x = x1 + x2 + ··· + xn Sample means and population means Often, we wish to compute the mean of values sampled from a population. If x1, x2, ..., xn is a sample, then the mean is called the sample mean and is denoted with the symbol ¯x. Sometimes we need to discuss the mean of all the values in a population. The mean of a population is called the population mean and is denoted by μ (the Greek letter mu). Explain It Again Sample size and population size: In general, we will use a lowercase n to denote a sample size and an uppercase N to denote a population size. DEFINITION If x1, ..., xn is a sample, the sample mean is ¯x = x1 + x2 + ··· + xn n = x n If x1, ..., xN is a population, the population mean is μ = x1 + x2 + ··· + xN N = x N How the mean measures the center of the data The mean is a measure of center. Figure 3.1 presents the exam scores in Example 3.1 on a number line, and shows the position of the mean. If we imagine each data value to be a weight, then the mean is the point at which the data set would balance. 68 78 83 85 92 81.2 Mean Figure 3.1 The mean is the point where the data set would balance, if each data value were represented by an equal weight. Explain It Again The mean may not be a value in the data set: The mean is not necessarily a typical value for the data. In fact, the mean may be a value that could not possibly appear in the data set. A misconception about the mean Some people believe that the mean represents a “typical” data value. In fact, this is not necessarily so. This is shown in Example 3.1, where we computed the mean of five exam scores and obtained a result of 81.2. If, like most exams, the scores are always whole numbers, then 81.2 is certainly not a “typical” data value; in fact, it could not possibly be a data value.
navidi_monk_essential_statistics_1e_ch1_3
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