3-1 Measures of Center 89 Important Properties of the Mean ■ Sample means drawn from the same population tend to vary less than other measures of center. ■ The mean of a data set uses every data value. ■ A disadvantage of the mean is that just one extreme value (outlier) can change the value of the mean substantially. (Using the following definition, we say that the mean is not resistant.) DEFINITION The mean (or arithmetic mean) of a set of data is the measure of center found by adding all of the data values and dividing the total by the number of data values. DEFINITION A statistic is resistant if the presence of extreme values (outliers) does not cause it to change very much. Calculation and Notation of the Mean The definition of the mean can be expressed as Formula 3-1, in which the Greek letter Σ (uppercase sigma) indicates that the data values should be added, so Σx represents the sum of all data values. The symbol n denotes the sample size, which is the number of data values. FORMULA 3-1 Mean = Σx n dsum of all data values dnumber of data values If the data are a sample from a population, the mean is denoted by x (pronounced “x-bar”); if the data are the entire population, the mean is denoted by m (lowercase Greek mu). NOTATION Hint: Sample statistics are usually represented by English letters, such as x, and population parameters are usually represented by Greek letters, such as m. Σ denotes the sum of a set of data values. x is the variable usually used to represent the individual data values. n represents the number of data values in a sample. N represents the number of data values in a population. x = Σx n is the mean of a set of sample values. m = Σx N is the mean of all values in a population. Class Size Paradox There are at least two ways to obtain the mean class size, and they can have very different results. At one college, if we take the numbers of students in 737 classes, we get a mean of 40 students. But if we were to compile a list of the class sizes for each student and use this list, we would get a mean class size of 147. This large discrepancy is because there are many students in large classes, while there are few students in small classes. Without changing the number of classes or faculty, we could reduce the mean class size experienced by students by making all classes about the same size. This would also improve attendance, which is better in smaller classes.
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