Measures of Central Tendency 2.3 SECTION 2.3 Measures of Central Tendency 67 Mean, Median, and Mode Weighted Mean and Mean of Grouped Data The Shapes of Distributions What You Should Learn How to find the mean, median, and mode of a population and of a sample How to find a weighted mean of a data set, and how to estimate the sample mean of grouped data How to describe the shape of a distribution as symmetric, uniform, or skewed, and how to compare the mean and median for each Mean, Median, and Mode In Sections 2.1 and 2.2, you learned about the graphical representations of quantitative data. In Sections 2.3 and 2.4, you will learn how to supplement graphical representations with numerical statistics that describe the center and variability of a data set. A measure of central tendency is a value that represents a typical, or central, entry of a data set. The three most commonly used measures of central tendency are the mean, the median, and the mode. The mean of a data set is the sum of the data entries divided by the number of entries. To find the mean of a data set, use one of these formulas. Population Mean: m = Σx N Sample Mean: x = Σx n The lowercase Greek letter m (mu) represents the population mean and x (read as “x bar”) represents the sample mean. Note that N represents the number of entries in a population and n represents the number of entries in a sample. Recall that the uppercase Greek letter Σ (sigma) indicates a summation of values. DEFINITION Finding a Sample Mean The weights (in pounds) for a sample of adults before starting a weight-loss study are listed. What is the mean weight of the adults? 274 235 223 268 290 285 235 SOLUTION The sum of the weights is Σx = 274 + 235 + 223 + 268 + 290 + 285 + 235 = 1810. There are 7 adults in the sample, so n = 7. To find the mean weight, divide the sum of the weights by the number of adults in the sample. x = Σx n = 1810 7 ≈ 258.6. So, the mean weight of the adults is about 258.6 pounds. TRY IT YOURSELF 1 The points scored by the winning teams in the Super Bowls for the National Football League’s 2005 through 2020 seasons are listed. Find the mean. 21 29 17 27 31 31 21 34 43 28 24 34 41 13 31 31 Answer: Page A36 EXAMPLE 1 Round the last calculation to one more decimal place than the original data. Study Tip Notice that the mean in Example 1 has one more decimal place than the original set of data entries. When the mean needs to be rounded, this round-off rule will be used in the text. Another important round-off rule is that rounding should not be done until the last calculation. For help with operations and summations, see Integrated Review at MyLab Statistics
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