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Chapter. 3. Numerically Summarizing Data. Section. 3.1. Measures of Central Tendency. Objectives Determine the arithmetic mean of a variable from raw data Determine the median of a variable from raw data Explain what it means for a statistic to be resistant

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  1. Chapter 3 Numerically Summarizing Data

  2. Section 3.1 Measures of Central Tendency

  3. Objectives Determine the arithmetic mean of a variable from raw data Determine the median of a variable from raw data Explain what it means for a statistic to be resistant Determine the mode of a variable from raw data

  4. Objective 1 Determine the Arithmetic Mean of a Variable from Raw Data

  5. The arithmetic mean of a variable is computed by adding all the values of the variable in the data set and dividing by the number of observations.

  6. The population arithmetic mean, μ (pronounced “mew”), is computed using all the individuals in a population. The population mean is a parameter.

  7. The sample arithmetic mean, (pronounced “x-bar”), is computed using sample data. The sample mean is a statistic.

  8. If x1, x2, …, xN are the N observations of a variable from a population, then the population mean, µ, is

  9. If x1, x2, …, xn are the n observations of a variable from a sample, then the sample mean, , is

  10. EXAMPLEComputing a Population Mean and a Sample Mean • The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company. • 23, 36, 23, 18, 5, 26, 43 • Compute the population mean of this data. • Then take a simple random sample of n = 3 employees. Compute the sample mean. Obtain a second simple random sample of n = 3 employees. Again compute the sample mean.

  11. EXAMPLEComputing a Population Mean and a Sample Mean (a)

  12. EXAMPLEComputing a Population Mean and a Sample Mean (b) Obtain a simple random sample of size n = 3 from the population of seven employees. Use this simple random sample to determine a sample mean. Find a second simple random sample and determine the sample mean. 1 2 3 4 5 6 7 23, 36, 23, 18, 5, 26, 43

  13. Objective 2 Determine the Median of a Variable from Raw Data

  14. The median of a variable is the value that lies in the middle of the data when arranged in ascending order. We use M to represent the median.

  15. Steps in Finding the Median of a Data Set Step 1 Arrange the data in ascending order. Step 2 Determine the number of observations, n. Step 3 Determine the observation in the middle of the data set.

  16. Steps in Finding the Median of a Data Set • If the number of observations is odd, then the median is the data value exactly in the middle of the data set. That is, the median is the observation that lies in then (n + 1)/2 position. • If the number of observations is even, then the median is the mean of the two middle observations in the data set. That is, the median is the mean of the observations that lie in the n/2 position and the n/2 + 1 position.

  17. EXAMPLEComputing a Median of a Data Set with an Odd Number of Observations The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company. 23, 36, 23, 18, 5, 26, 43 Determine the median of this data. Step 1: 5, 18, 23, 23, 26, 36, 43 Step 2: There are n = 7 observations. M = 23 Step 3: 5, 18, 23, 23, 26, 36, 43

  18. EXAMPLEComputing a Median of a Data Set with an Even Number of Observations Suppose the start-up company hires a new employee. The travel time of the new employee is 70 minutes. Determine the median of the “new” data set. 23, 36, 23, 18, 5, 26, 43, 70 Step 1: 5, 18, 23, 23, 26, 36, 43, 70 Step 2: There are n = 8 observations. Step 3: 5, 18, 23, 23, 26, 36, 43, 70

  19. Objective 3 Explain What it Means for a Statistic to Be Resistant

  20. EXAMPLEComputing a Median of a Data Set with an Even Number of Observations The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company. 23, 36, 23, 18, 5, 26, 43 Suppose a new employee is hired who has a 130 minute commute. How does this impact the value of the mean and median? Mean before new hire: 24.9 minutesMedian before new hire: 23 minutes Mean after new hire: 38 minutesMedian after new hire: 24.5 minutes

  21. A numerical summary of data is said to be resistant if extreme values (very large or small) relative to the data do not affect its value substantially.

  22. EXAMPLE Describing the Shape of the Distribution The following data represent the asking price of homes for sale in Lincoln, NE. Source: http://www.homeseekers.com

  23. Find the mean and median. Use the mean and median to identify the shape of the distribution. Verify your result by drawing a histogram of the data.

  24. Find the mean and median. Use the mean and median to identify the shape of the distribution. Verify your result by drawing a histogram of the data. The mean asking price is $168,320 and the median asking price is $148,700. Therefore, we would conjecture that the distribution is skewed right.

  25. Objective 4 Determine the Mode of a Variable from Raw Data

  26. The mode of a variable is the most frequent observation of the variable that occurs in the data set. A set of data can have no mode, one mode, or more than one mode. If no observation occurs more than once, we say the data have no mode.

  27. EXAMPLE Finding the Mode of a Data Set The data on the next slide represent the Vice Presidents of the United States and their state of birth. Find the mode.

  28. Joe Biden Pennsylvania

  29. The mode is New York.

  30. Tally data to determine most frequent observation

  31. Section 3.2 Measures of Dispersion

  32. Objectives Determine the range of a variable from raw data Determine the standard deviation of a variable from raw data Determine the variance of a variable from raw data Use the Empirical Rule to describe data that are bell shaped Use Chebyshev’s Inequality to describe any data set

  33. To order food at a McDonald’s restaurant, one must choose from multiple lines, while at Wendy’s Restaurant, one enters a single line. The following data represent the wait time (in minutes) in line for a simple random sample of 30 customers at each restaurant during the lunch hour. For each sample, answer the following: (a) What was the mean wait time? (b) Draw a histogram of each restaurant’s wait time. (c ) Which restaurant’s wait time appears more dispersed? Which line would you prefer to wait in? Why?

  34. Wait Time at Wendy’s 1.50 0.79 1.01 1.66 0.94 0.67 2.53 1.20 1.46 0.89 0.95 0.90 1.88 2.94 1.40 1.33 1.20 0.84 3.99 1.90 1.00 1.54 0.99 0.35 0.90 1.23 0.92 1.09 1.72 2.00 Wait Time at McDonald’s 3.50 0.00 0.38 0.43 1.82 3.04 0.00 0.26 0.14 0.60 2.33 2.54 1.97 0.71 2.22 4.54 0.80 0.50 0.00 0.28 0.44 1.38 0.92 1.17 3.08 2.75 0.36 3.10 2.19 0.23

  35. (a) The mean wait time in each line is 1.39 minutes.

  36. (b)

  37. Objective 1 Determine the Range of a Variable from Raw Data

  38. The range, R, of a variable is the difference between the largest data value and the smallest data values. That is, Range = R = Largest Data Value – Smallest Data Value

  39. EXAMPLE Finding the Range of a Set of Data The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company. 23, 36, 23, 18, 5, 26, 43 Find the range. Range = 43 – 5 = 38 minutes

  40. Objective 2 Determine the Standard Deviation of a Variable from Raw Data

  41. The population standard deviation of a variable is the square root of the sum of squared deviations about the population mean divided by the number of observations in the population, N. That is, it is the square root of the mean of the squared deviations about the population mean. The population standard deviation is symbolically represented by σ (lowercase Greek sigma).

  42. where x1, x2, . . . , xN are the N observations in the population and μ is the population mean.

  43. A formula that is equivalent to the one on the previous slide, called the computational formula, for determining the population standard deviation is

  44. EXAMPLEComputing a Population Standard Deviation The following data represent the travel times (in minutes) to work for all seven employees of a start-up web development company. 23, 36, 23, 18, 5, 26, 43 Compute the population standard deviation of this data.

  45. Using the computational formula, yields the same result.

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