1. 3.1 Measures of Central Tendency

2. Ch. 3 Numerically Summarizing Data The arithmetic mean of a variable is computed by determining the sum of all the values of the variable in the data set divided by the number of observations. The population arithmetic mean is computed using all the individuals in a population. –The population mean is a parameter. –The population arithmetic mean is denoted by the symbol μ

3. Population Mean If x 1, x 2, …, x N are the N observations of a variable from a population, then the population mean, µ, is

4. Sample Mean The sample arithmetic mean is computed using sample data. The sample mean is a statistic. The sample arithmetic mean is denoted by

5. Sample Mean If x 1, x 2, …, x N are the N observations of a variable from a sample, then the sample mean is

6. Sample Problem Computing 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.

7. Median 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.

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9. 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.

10. 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 mean and median of the “new” data set. 23, 36, 23, 18, 5, 26, 43, 70 3-10

11. 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? 3-11

12. 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 minutes Median before new hire: 23 minutes Mean after new hire: 38 minutes Median after new hire: 24.5 minutes 3-12

13. Resistance 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.

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15. 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 79,995128,950149,900189,900 99,899130,950151,350203,950 105,200131,800154,900217,500 111,000132,300159,900260,000 120,000134,950163,300284,900 121,700135,500165,000299,900 125,950138,500174,850309,900 126,900147,500180,000349,900 3-15

16. Sample Problem 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.

17. 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. 3-17

18. 3-18

19. Mode The mode of a variable is the most frequent observation of the variable that occurs in the data set. If there is no observation that occurs with the most frequency, we say the data has no mode. –The data on the next slide represent the Vice Presidents of the United States and their state of birth. Find the mode.

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22. Tally data to determine most frequent observation 3-22

23. One-Variable Statistics Nspire 1.Create a list & spreadsheets page 2.Title column 3.Enter data into column 4.Create a calculator page 5.Click Menu 6.6:Statistics –1:Stat Calculations –1: One-Variable Stats

24. Practice Problem The following is a list of pulse rates for nine students enrolled in a section of statistics. Determine the mean, median, & mode of the data set using technology. 76 60 81 72 80 68 73

25. 3.2 Measures of Dispersion

26. Range 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

27. EXAMPLEFinding 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.

28. Population Variance The population variance of a variable is the sum of squared deviations about the population mean divided by the number of observations in the population, N. That is it is the mean of the sum of the squared deviations about the population mean. It let’s us know how spread out data is from the average.

29. The population variance is symbolically represented by σ 2 (lower case Greek sigma squared). Note: When using the above formula, do not round until the last computation. Use as many decimals as allowed by your calculator in order to avoid round off errors. 3-29

30. EXAMPLE Computing a Population Variance 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 variance of this data. Recall that

31. xixi μ x i – μ(x i – μ) 2 2324.85714-1.857143.44898 3624.8571411.14286124.1633 2324.85714-1.857143.44898 1824.85714-6.8571447.02041 524.85714-19.8571394.3061 2624.857141.1428571.306122 4324.8571418.14286329.1633 902.8571 minutes 2 -31

32. Sample Variance The sample variance is computed by determining the sum of squared deviations about the sample mean and then dividing this result by n – 1.

33. Note: Whenever a statistic consistently overestimates or underestimates a parameter, it is called biased. To obtain an unbiased estimate of the population variance, we divide the sum of the squared deviations about the mean by n - 1. 33

34. The population standard deviation is denoted by It is obtained by taking the square root of the population variance, so that The sample standard deviation is denoted by s It is obtained by taking the square root of the sample variance, so that 3-34

35. EXAMPLE Computing 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 and variance of this data using technology. 3-35

36. One-Variable Statistics Nspire 1.Create a list & spreadsheets page 2.Title column 3.Enter data into column 4.Create a calculator page 5.Click Menu 6.6:Statistics –1:Stat Calculations –1: One-Variable Stats

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39. EXAMPLE Using the Empirical Rule The following data represent the serum HDL cholesterol of the 54 female patients of a family doctor. 414843383537444444 627577588239855554 676969706572747474 606060616263646464 545455565656575859 454747484850525253 3-39

40. (a) Compute the population mean and standard deviation. (b) Draw a histogram to verify the data is bell- shaped. (c) Determine the percentage of patients that have serum HDL within 3 standard deviations of the mean according to the Empirical Rule. (d) Determine the percentage of patients that have serum HDL between 34 and 69.1 according to the Empirical Rule. (e) Determine the actual percentage of patients that have serum HDL between 34 and 69.1. 3-40

41. (a) Using a TI-nspire graphing calculator, we find (b) 3-41© 2010 Pearson Prentice Hall. All rights reserved

42. (c) Determine the percentage of patients that have serum HDL within 3 standard deviations of the mean according to the Empirical Rule. (d) Determine the percentage of patients that have serum HDL between 34 and 69.1 according to the Empirical Rule. (e) Determine the actual percentage of patients that have serum HDL between 34 and 69.1. (look back at original data!)

43. One measure of intelligence is the Stanford-Binet Intelligence Quotient (IQ). IQ scores have bell-shaped distribution with a mean of 100 and a standard deviation of 15 A.What percentage of people has an IQ score between 70 and 130? B.What percentage of people has an IQ score less than 70 or greater than 130? C.What percentage of people has an IQ score below 85?

44. 3.4 Measures of Position and Outliers

45. The Z-Score

46. EXAMPLE Using Z-Scores The mean height of males 20 years or older is 69.1 inches with a standard deviation of 2.8 inches. The mean height of females 20 years or older is 63.7 inches with a standard deviation of 2.7 inches. Data based on information obtained from National Health and Examination Survey. Who is relatively taller? Kevin Garnett whose height is 83 inches or Candace Parker whose height is 76 inches 3-46

47. Sample Problem Score on ACT was 26 with a mean of 22 and sd of 3. Score on SAT was 950 with mean of 925 and sd of 25. Which score is "better"?

48. Quartiles divide data sets into fourths, or four equal parts. The 1 st quartile, denoted Q 1, divides the bottom 25% the data from the top 75%. Therefore, the 1 st quartile is equivalent to the 25 th percentile. The 2 nd quartile divides the bottom 50% of the data from the top 50% of the data, so that the 2 nd quartile is equivalent to the 50 th percentile, which is equivalent to the median. The 3 rd quartile divides the bottom 75% of the data from the top 25% of the data, so that the 3 rd quartile is equivalent to the 75 th percentile. 3-48

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50. A group of Brigham Young University—Idaho students (Matthew Herring, Nathan Spencer, Mark Walker, and Mark Steiner) collected data on the speed of vehicles traveling through a construction zone on a state highway, where the posted speed was 25 mph. The recorded speed of 14 randomly selected vehicles is given below: 20, 24, 27, 28, 29, 30, 32, 33, 34, 36, 38, 39, 40, 40 Find and interpret the quartiles for speed in the construction zone. In addition find the mean, median, and standard deviation. (using technology) EXAMPLE Finding and Interpreting Quartiles 3-50

51. Interpretation: 25% of the speeds are less than or equal to the first quartile, 28 miles per hour, and 75% of the speeds are greater than 28 miles per hour. 50% of the speeds are less than or equal to the second quartile, 32.5 miles per hour, and 50% of the speeds are greater than 32.5 miles per hour. 75% of the speeds are less than or equal to the third quartile, 38 miles per hour, and 25% of the speeds are greater than 38 miles per hour. 3-51

52. 3-52 Interquartile Range

53. EXAMPLE Determining and Interpreting the Interquartile Range Determine and interpret the interquartile range of the speed data. Q 1 = 28 Q 3 = 38 The range of the middle 50% of the speed of cars traveling through the construction zone is 10 miles per hour. 3-53

54. Suppose a 15 th car travels through the construction zone at 100 miles per hour. How does this value impact the mean, median, standard deviation, and interquartile range? 3-54 With Out 15 th CarWith 15 th Car Mean Median Standard Deviation IQR Which measures should we report now? When we add the 15th car which changes less – the mean or median (measures of center)? When we add the 15th car which changes les – the standard deviation or the IQR (measures of dispersion)?

55. Suppose a 15 th car travels through the construction zone at 100 miles per hour. How does this value impact the mean, median, standard deviation, and interquartile range? Without 15 th carWith 15 th car Mean32.1 mph36.7 mph Median32.5 mph33 mph Standard deviation6.2 mph18.5 mph IQR10 mph11 mph 3-55

56. Outliers

57. EXAMPLE Determining and Interpreting the Interquartile Range Check the speed data for outliers. Step 1: The first and third quartiles are Q 1 = 28 mph and Q 3 = 38 mph. Step 2: The interquartile range is 10 mph. Step 3: The fences are Lower Fence = Q 1 – 1.5(IQR) Upper Fence = Q 3 + 1.5(IQR) = 28 – 1.5(10) = 38 + 1.5(10) = 13 mph = 53 mph Step 4: There are no values less than 13 mph or greater than 53 mph. Therefore, there are no outliers. 3-57

58. Sample Problem For the following data of rainfall for Chicago, IL determine the following: 1.The Quartiles & Median 2.The IQR 3.Determine if there are any outliers..972.473.944.115.79 1.142.783.974.776.14 1.853.414.005.226.28 2.343.484.025.507.69

59. 3.5 5-Number Summary and BoxPlots

60. 5-Number Summary

61. EXAMPLEConstructing a Boxplot Every six months, the United States Federal Reserve Board conducts a survey of credit card plans in the U.S. The following data are the interest rates charged by 10 credit card issuers randomly selected for the July 2005 survey. Draw a boxplot of the data. InstitutionRate Pulaski Bank and Trust Company6.5% Rainier Pacific Savings Bank12.0% Wells Fargo Bank NA14.4% Firstbank of Colorado14.4% Lafayette Ambassador Bank14.3% Infibank13.0% United Bank, Inc.13.3% First National Bank of The Mid-Cities13.9% Bank of Louisiana9.9% Bar Harbor Bank and Trust Company14.5% Source: http://www.federalreserve.gov/pubs/SHOP/survey.htm 3-61

62. Step 1: The interquartile range (IQR) is 14.4% - 12% = 2.4%. The lower and upper fences are: Lower Fence = Q 1 – 1.5(IQR) Upper Fence = Q 3 + 1.5(IQR) = 12 – 1.5(2.4) = 14.4 + 1.5(2.4) = 8.4% = 18.0% Step 2: [ ] * 3-62

63. The interest rate boxplot indicates that the distribution is skewed left. 3-63

64. TI-nspire – Creating a BoxPlot See handout Use the Nspire calculator to create a boxplot of the rainfall data from 3.4 data..972.473.944.115.79 1.142.783.974.776.14 1.853.414.005.226.28 2.343.484.025.507.69