72 CHAPTER 2 Descriptive Statistics Class midpoint, x Frequency, f xf 172.5 3 517.5 208.5 2 417.0 244.5 5 1222.5 280.5 6 1683.0 316.5 7 2215.5 352.5 4 1410.0 388.5 3 1165.5 n = 30 Σ = 8631 For data presented in a frequency distribution, you can estimate the mean as shown in the next definition. The mean of a frequency distribution for a sample is estimated by x = Σxf n Note that n = Σf. where x and f are the midpoint and frequency of each class, respectively. DEFINITION Finding the Mean of a Frequency Distribution In Words In Symbols 1. Find the midpoint of each class. x = 1Lower limit2 + 1Upper limit2 2 2. Find the sum of the products Σxf of the midpoints and the frequencies. 3. Find the sum of the frequencies. n = Σf 4. Find the mean of the x = Σxf n frequency distribution. GUIDELINES Finding the Mean of a Frequency Distribution The frequency distribution at the left shows the cell phone screen times (in minutes) for 30 U.S. adults on a recent day. Use the frequency distribution to estimate the mean screen time. Using the sample mean formula from page 67 with the original data set (see Example 1 in Section 2.1), the mean screen time is about 285.5 minutes. Compare this with the estimated mean. SOLUTION x = Σxf n = 8631 30 = 287.7 Interpretation The mean screen time is 287.7 minutes. This value is an estimate because it is based on class midpoints instead of the original data set. Although it is not substantially different, the mean of about 285.5 minutes found using the original data set is a more accurate result. TRY IT YOURSELF 8 Use a frequency distribution to estimate the mean of the points scored by the 55 winning teams listed on page 39. (See Try It Yourself 2 on page 43.) Using the population mean formula from page 67 with the original data set, the mean is about 30.1 points. Compare this with the estimated mean. Answer: Page A36 EXAMPLE 8 Study Tip For a frequency distribution that represents a population, the mean of the frequency distribution is estimated by m = Σxf N where N = Σf.
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