Section 3.2 Measures of Central Tendency

Transcription

1 Section 3.2 Measures of Central Tendency 1 of 149

2 Section 3.2 Objectives Determine the mean, median, and mode of a population and of a sample Determine the weighted mean of a data set and the mean of a frequency distribution Describe the shape of a distribution as symmetric, uniform, or skewed and compare the mean and median for each 2 of 149

3 Measures of Central Tendency Measure of central tendency A value that represents a typical, or central, entry of a data set. Most common measures of central tendency: Mean Median Mode 3 of 149

4 Measure of Central Tendency: Mean Mean (average) The sum of all the data entries divided by the number of entries. Sigma notation: Σx = add all of the data entries (x) in the data set. Population mean: x µ = Σ N Sample mean: x = Σx n 4 of 149

5 Example: Finding a Sample Mean The prices (in dollars) for a sample of round-trip flights from Chicago, Illinois to Cancun, Mexico are listed. What is the mean price of the flights? of 149

6 Solution: Finding a Sample Mean The sum of the flight prices is Σx = = 3695 To find the mean price, divide the sum of the prices by the number of prices in the sample x = Σx n = The mean price of the flights is about $ of 149

7 Measure of Central Tendency: Median Median The value that lies in the middle of the data when the data set is ordered. Measures the center of an ordered data set by dividing it into two equal parts. If the data set has an odd number of entries: median is the middle data entry. even number of entries: median is the mean of the two middle data entries. 7 of 149

8 Example: Finding the Median The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. Find the median of the flight prices of 149

9 Solution: Finding the Median First order the data There are seven entries (an odd number), the median is the middle, or fourth, data entry. The median price of the flights is $ of 149

10 Example: Finding the Median The flight priced at $432 is no longer available. What is the median price of the remaining flights? of 149

11 Solution: Finding the Median First order the data There are six entries (an even number), the median is the mean of the two middle entries Median = = The median price of the flights is $ of 149

12 Measure of Central Tendency: Mode Mode The data entry that occurs with the greatest frequency. A data set can have one mode, more than one mode, or no mode. If no entry is repeated the data set has no mode. If two entries occur with the same greatest frequency, each entry is a mode (bimodal). 12 of 149

13 Example: Finding the Mode The prices (in dollars) for a sample of roundtrip flights from Chicago, Illinois to Cancun, Mexico are listed. Find the mode of the flight prices of 149

14 Solution: Finding the Mode Ordering the data helps to find the mode The entry of 397 occurs twice, whereas the other data entries occur only once. The mode of the flight prices is $ of 149

15 Example: Finding the Mode At a political debate a sample of audience members was asked to name the political party to which they belong. Their responses are shown in the table. What is the mode of the responses? Political Party Frequency, f Democrat 34 Republican 56 Other 21 Did not respond 9 15 of 149

16 Solution: Finding the Mode Political Party Frequency, f Democrat 34 Republican 56 Other 21 Did not respond 9 The mode is Republican (the response occurring with the greatest frequency). In this sample there were more Republicans than people of any other single affiliation. 16 of 149

17 Comparing the Mean, Median, and Mode All three measures describe a typical entry of a data set. Advantage of using the mean: The mean is a reliable measure because it takes into account every entry of a data set. Disadvantage of using the mean: Greatly affected by outliers (a data entry that is far removed from the other entries in the data set). 17 of 149

18 Example: Comparing the Mean, Median, and Mode Find the mean, median, and mode of the sample ages of a class shown. Which measure of central tendency best describes a typical entry of this data set? Are there any outliers? Ages in a class of 149

19 Solution: Comparing the Mean, Median, and Mode Ages in a class Mean: Median: x = Σx = n = 21.5 years years Mode: 20 years (the entry occurring with the greatest frequency) 19 of 149

20 Solution: Comparing the Mean, Median, and Mode Mean 23.8 years Median = 21.5 years Mode = 20 years The mean takes every entry into account, but is influenced by the outlier of 65. The median also takes every entry into account, and it is not affected by the outlier. In this case the mode exists, but it doesn't appear to represent a typical entry. 20 of 149

21 Solution: Comparing the Mean, Median, and Mode Sometimes a graphical comparison can help you decide which measure of central tendency best represents a data set. In this case, it appears that the median best describes the data set. 21 of 149

22 Weighted Mean Weighted Mean The mean of a data set whose entries have varying weights. x = Σ( x w) Σw where w is the weight of each entry x 22 of 149

23 Example: Finding a Weighted Mean You are taking a class in which your grade is determined from five sources: 50% from your test mean, 15% from your midterm, 20% from your final exam, 10% from your computer lab work, and 5% from your homework. Your scores are 86 (test mean), 96 (midterm), 82 (final exam), 98 (computer lab), and 100 (homework). What is the weighted mean of your scores? If the minimum average for an A is 90, did you get an A? 23 of 149

24 Solution: Finding a Weighted Mean Source Score, x Weight, w x w Test Mean (0.50)= 43.0 Midterm (0.15) = 14.4 Final Exam (0.20) = 16.4 Computer Lab (0.10) = 9.8 Homework (0.05) = 5.0 Σw = 1 Σ(x w) = 88.6 Σ( x w) 88.6 x = = = 88.6 Σw 1 Your weighted mean for the course is You did not get an A. 24 of 149

25 Mean of Grouped Data Mean of a Frequency Distribution Approximated by Σ( x f) x = n = Σf n where x and f are the midpoints and frequencies of a class, respectively 25 of 149

26 Finding the Mean of a Frequency Distribution In Words 1. Find the midpoint of each class. x = In Symbols (lower limit)+(upper limit) 2 2. Find the sum of the products of the midpoints and the frequencies. Σ( x f) 3. Find the sum of the frequencies. n = Σf 4. Find the mean of the frequency distribution. x = Σ( x f) n 26 of 149

27 Example: Find the Mean of a Frequency Distribution Use the frequency distribution to approximate the mean number of minutes that a sample of Internet subscribers spent online during their most recent session. Class Midpoint Frequency, f of 149

28 Solution: Find the Mean of a Frequency Distribution Class Midpoint, x Frequency, f (x f) = = = = = = = n = 50 Σ(x f) = x Σ( x f) 2089 = = n minutes 28 of 149

29 The Shape of Distributions Symmetric Distribution A vertical line can be drawn through the middle of a graph of the distribution and the resulting halves are approximately mirror images. 29 of 149

30 The Shape of Distributions Uniform Distribution (rectangular) All entries or classes in the distribution have equal or approximately equal frequencies. Symmetric. 30 of 149

31 The Shape of Distributions Skewed Left Distribution (negatively skewed) The tail of the graph elongates more to the left. The mean is to the left of the median. 31 of 149

32 The Shape of Distributions Skewed Right Distribution (positively skewed) The tail of the graph elongates more to the right. The mean is to the right of the median. 32 of 149

33 Section 3.3 Summary Determined the mean, median, and mode of a population and of a sample Determined the weighted mean of a data set and the mean of a frequency distribution Described the shape of a distribution as symmetric, uniform, or skewed and compared the mean and median for each 33 of 149

34 Section 3.3 Measures of Variation 34 of 149

35 Section 3.3 Objectives Determine the range of a data set Determine the variance and standard deviation of a population and of a sample Use the Empirical Rule and Chebychev s Theorem to interpret standard deviation Approximate the sample standard deviation for grouped data 35 of 149

36 Range Range The difference between the maximum and minimum data entries in the set. The data must be quantitative. Range = (Max. data entry) (Min. data entry) 36 of 149

37 Example: Finding the Range A corporation hired 10 graduates. The starting salaries for each graduate are shown. Find the range of the starting salaries. Starting salaries (1000s of dollars) of 149

38 Solution: Finding the Range Ordering the data helps to find the least and greatest salaries minimum Range = (Max. salary) (Min. salary) = = 10 maximum The range of starting salaries is 10 or $10, of 149

39 Deviation, Variance, and Standard Deviation Deviation The difference between the data entry, x, and the mean of the data set. Population data set: Deviation of x = x μ Sample data set: Deviation of x = x x 39 of 149

40 Example: Finding the Deviation A corporation hired 10 graduates. The starting salaries for each graduate are shown. Find the deviation of the starting salaries. Starting salaries (1000s of dollars) Solution: First determine the mean starting salary. x 415 µ = Σ = = 41.5 N of 149

41 Solution: Finding the Deviation Determine the deviation for each data entry. Salary ($1000s), x Deviation ($1000s) x μ = = = = = = = = = = 0.5 Σx = 415 Σ(x μ) = 0 41 of 149

42 Deviation, Variance, and Standard Deviation Population Variance σ 2 Σ( x µ ) = N 2 Sum of squares, SS x Population Standard Deviation σ 2 Σ( x µ ) = σ = N 2 42 of 149

43 Finding the Population Variance & Standard Deviation In Words 1. Find the mean of the population data set. 2. Find the deviation of each entry. 3. Square each deviation. 4. Add to get the sum of squares. In Symbols x µ = Σ N x μ (x μ) 2 SS x = Σ(x μ) 2 43 of 149

44 Finding the Population Variance & Standard Deviation In Words 5. Divide by N to get the population variance. 6. Find the square root of the variance to get the population standard deviation. 2 Σ( x µ ) σ = N σ = In Symbols Σ( x µ ) N of 149

45 Example: Finding the Population Standard Deviation A corporation hired 10 graduates. The starting salaries for each graduate are shown. Find the population variance and standard deviation of the starting salaries. Starting salaries (1000s of dollars) Recall μ = of 149

46 Solution: Finding the Population Standard Deviation Determine SS x N = 10 Salary, x Deviation: x μ Squares: (x μ) = 0.5 ( 0.5) 2 = = 3.5 ( 3.5) 2 = = 2.5 ( 2.5) 2 = = 3.5 (3.5) 2 = = 5.5 (5.5) 2 = = 0.5 ( 0.5) 2 = = 2.5 (2.5) 2 = = 0.5 ( 0.5) 2 = = 4.5 ( 4.5) 2 = = 0.5 (0.5) 2 = 0.25 Σ(x μ) = 0 SS x = of 149

47 Solution: Finding the Population Standard Deviation Population Variance σ 2 2 Σ( x µ ) 88.5 = = N Population Standard Deviation σ σ = = The population standard deviation is about 3.0, or $ of 149

48 Deviation, Variance, and Standard Deviation Sample Variance s 2 Σ( x x) = n 1 Sample Standard Deviation 2 s 2 Σ( x x) = s = n of 149

49 Finding the Sample Variance & Standard Deviation In Words 1. Find the mean of the sample data set. 2. Find the deviation of each entry. 3. Square each deviation. 4. Add to get the sum of squares. In Symbols x = Σx n SS =Σ( x x) x x x ( x x) of 149

50 Finding the Sample Variance & Standard Deviation In Words 5. Divide by n 1 to get the sample variance. 6. Find the square root of the variance to get the sample standard deviation. In Symbols 2 Σ( x x) s = n 1 s = 2 Σ( x x) n of 149

51 Example: Finding the Sample Standard Deviation The starting salaries are for the Chicago branches of a corporation. The corporation has several other branches, and you plan to use the starting salaries of the Chicago branches to estimate the starting salaries for the larger population. Find the sample standard deviation of the starting salaries. Starting salaries (1000s of dollars) of 149

52 Solution: Finding the Sample Standard Deviation Determine SS x n = 10 Salary, x Deviation: x μ Squares: (x μ) = 0.5 ( 0.5) 2 = = 3.5 ( 3.5) 2 = = 2.5 ( 2.5) 2 = = 3.5 (3.5) 2 = = 5.5 (5.5) 2 = = 0.5 ( 0.5) 2 = = 2.5 (2.5) 2 = = 0.5 ( 0.5) 2 = = 4.5 ( 4.5) 2 = = 0.5 (0.5) 2 = 0.25 Σ(x μ) = 0 SS x = of 149

53 Solution: Finding the Sample Standard Deviation Sample Variance 2 2 Σ( x x) 88.5 s = = n Sample Standard Deviation s = s = The sample standard deviation is about 3.1, or $ of 149

54 Example: Using Technology to Find the Standard Deviation Sample office rental rates (in dollars per square foot per year) for Miami s central business district are shown in the table. Use a calculator or a computer to find the mean rental rate and the sample standard deviation. (Adapted from: Cushman & Wakefield Inc.) Office Rental Rates of 149

55 Solution: Using Technology to Find the Standard Deviation Sample Mean Sample Standard Deviation 55 of 149

56 Interpreting Standard Deviation Standard deviation is a measure of the typical amount an entry deviates from the mean. The more the entries are spread out, the greater the standard deviation. 56 of 149

57 Interpreting Standard Deviation: Empirical Rule ( Rule) For data with a (symmetric) bell-shaped distribution, the standard deviation has the following characteristics: About 68% of the data lie within one standard deviation of the mean. About 95% of the data lie within two standard deviations of the mean. About 99.7% of the data lie within three standard deviations of the mean. 57 of 149

58 Interpreting Standard Deviation: Empirical Rule ( Rule) 99.7% within 3 standard deviations 95% within 2 standard deviations 68% within 1 standard deviation 34% 34% 2.35% 2.35% 13.5% 13.5% x 3s x 2s x s x x + s x + 2s x + 3s 58 of 149

59 Example: Using the Empirical Rule In a survey conducted by the National Center for Health Statistics, the sample mean height of women in the United States (ages 20-29) was 64.3 inches, with a sample standard deviation of 2.62 inches. Estimate the percent of the women whose heights are between inches and 64.3 inches. 59 of 149

60 Solution: Using the Empirical Rule Because the distribution is bell-shaped, you can use the Empirical Rule. 34% % = 47.5% of women are between and 64.3 inches tall. 60 of 149

61 Chebychev s Theorem The portion of any data set lying within k standard deviations (k > 1) of the mean is at least: 1 1 k 2 k = 2: In any data set, at least = or 75% 2 4 of the data lie within 2 standard deviations of the mean. 1 8 k = 3: In any data set, at least 1 = or 88.9% of the data lie within 3 standard deviations of the mean. 61 of 149

62 Example: Using Chebychev s Theorem The age distribution for Florida is shown in the histogram. Apply Chebychev s Theorem to the data using k = 2. What can you conclude? 62 of 149

63 Solution: Using Chebychev s Theorem k = 2: μ 2σ = (24.8) = 10.4 (use 0 since age can t be negative) μ + 2σ = (24.8) = 88.8 At least 75% of the population of Florida is between 0 and 88.8 years old. 63 of 149

64 Standard Deviation for Grouped Data Sample standard deviation for a frequency distribution s = Σ( ) n 1 2 x x f where n = Σf (the number of entries in the data set) When a frequency distribution has classes, estimate the sample mean and the sample standard deviation by using the midpoint of each class. 64 of 149

65 Example: Finding the Standard Deviation for Grouped Data You collect a random sample of the number of children per household in a region. Find the sample mean and the sample standard deviation of the data set. Number of Children in 50 Households of 149

66 Solution: Finding the Standard Deviation for Grouped Data First construct a frequency distribution. Find the mean of the frequency distribution. Σxf 91 x = = 1.8 n 50 The sample mean is about 1.8 children. x f xf (10) = (19) = (7) = (7) = (2) = (1) = (4) = 24 Σf = 50 Σ(xf )= of 149

67 Solution: Finding the Standard Deviation for Grouped Data Determine the sum of squares. x f x x ( x x) = 1.8 ( 1.8) 2 = (10) = = 0.8 ( 0.8) 2 = (19) = = 0.2 (0.2) 2 = (7) = = 1.2 (1.2) 2 = (7) = = 2.2 (2.2) 2 = (2) = = 3.2 (3.2) 2 = (1) = = 4.2 (4.2) 2 = (4) = x x f ( ) Σ = 2 ( x x) f of 149

68 Solution: Finding the Standard Deviation for Grouped Data Find the sample standard deviation. s x x ( x x) 2 Σ( x x) f = = 1.7 n x x f ( ) The standard deviation is about 1.7 children. 68 of 149

69 Section 3.3 Summary Determined the range of a data set Determined the variance and standard deviation of a population and of a sample Used the Empirical Rule and Chebychev s Theorem to interpret standard deviation Approximated the sample standard deviation for grouped data 69 of 149

70 Section 3.4 Measures of Position 70 of 149

71 Section 3.4 Objectives Determine the quartiles of a data set Determine the interquartile range of a data set Create a box-and-whisker plot Interpret other fractiles such as percentiles Determine and interpret the standard score (z-score) 71 of 149

72 Quartiles Fractiles are numbers that partition (divide) an ordered data set into equal parts. Quartiles approximately divide an ordered data set into four equal parts. First quartile, Q 1 : About one quarter of the data fall on or below Q 1. Second quartile, Q 2 : About one half of the data fall on or below Q 2 (median). Third quartile, Q 3 : About three quarters of the data fall on or below Q of 149

73 Example: Finding Quartiles The number of nuclear power plants in the top 15 nuclear power-producing countries in the world are listed. Find the first, second, and third quartiles of the data set Solution: Q 2 divides the data set into two halves. Lower half Upper half Q 2 73 of 149

74 Solution: Finding Quartiles The first and third quartiles are the medians of the lower and upper halves of the data set. Lower half Upper half Q 1 Q 2 Q 3 About one fourth of the countries have 10 or fewer nuclear power plants; about one half have 18 or fewer; and about three fourths have 31 or fewer. 74 of 149

75 Interquartile Range Interquartile Range (IQR) The difference between the third and first quartiles. IQR = Q 3 Q 1 75 of 149

76 Example: Finding the Interquartile Range Find the interquartile range of the data set Recall Q 1 = 10, Q 2 = 18, and Q 3 = 31 Solution: IQR = Q 3 Q 1 = = 21 The number of power plants in the middle portion of the data set vary by at most of 149

77 Box-and-whisker plot Box-and-Whisker Plot Exploratory data analysis tool. Highlights important features of a data set. Requires (five-number summary): Minimum entry First quartile Q 1 Median Q 2 Third quartile Q 3 Maximum entry 77 of 149

78 Drawing a Box-and-Whisker Plot 1. Find the five-number summary of the data set. 2. Construct a horizontal scale that spans the range of the data. 3. Plot the five numbers above the horizontal scale. 4. Draw a box above the horizontal scale from Q 1 to Q 3 and draw a vertical line in the box at Q Draw whiskers from the box to the minimum and maximum entries. Whisker Box Whisker Minimum entry Median, Q 2 Q 3 Maximum entry 78 of 149

79 Example: Drawing a Box-and-Whisker Plot Draw a box-and-whisker plot that represents the data set Min = 6, Q 1 = 10, Q 2 = 18, Q 3 = 31, Max = 104, Solution: About half the data values are between 10 and 31. By looking at the length of the right whisker, you can conclude 104 is a possible outlier. 79 of 149

80 Percentiles and Other Fractiles Fractiles Summary Symbols Quartiles Divide a data set into 4 equal Q 1, Q 2, Q 3 parts Deciles Divide a data set into 10 D 1, D 2, D 3,, D 9 equal parts Percentiles Divide a data set into 100 equal parts P 1, P 2, P 3,, P of 149

81 Example: Interpreting Percentiles The ogive represents the cumulative frequency distribution for SAT test scores of college-bound students in a recent year. What test score represents the 62 nd percentile? How should you interpret this? (Source: College Board) 81 of 149

82 Solution: Interpreting Percentiles The 62 nd percentile corresponds to a test score of This means that 62% of the students had an SAT score of 1600 or less. 82 of 149

83 The Standard Score Standard Score (z-score) Represents the number of standard deviations a given value x falls from the mean μ. z = value mean standard deviation = x µ σ 83 of 149

84 Example: Comparing z-scores from Different Data Sets In 2009, Heath Ledger won the Oscar for Best Supporting Actor at age 29 for his role in the movie The Dark Knight. Penelope Cruz won the Oscar for Best Supporting Actress at age 34 for her role in Vicky Cristina Barcelona. The mean age of all Best Supporting Actor winners is 49.5, with a standard deviation of The mean age of all Best Supporting Actress winners is 39.9, with a standard deviation of Find the z-scores that correspond to the ages of Ledger and Cruz. Then compare your results. 84 of 149

85 Solution: Comparing z-scores from Different Data Sets Heath Ledger z x µ = = 1.49 σ 13.8 Penelope Cruz z x µ = = 0.42 σ standard deviations below the mean 0.42 standard deviations below the mean 85 of 149

86 Solution: Comparing z-scores from Different Data Sets Both z-scores fall between 2 and 2, so neither score would be considered unusual. Compared with other Best Supporting Actor winners, Heath Ledger was relatively younger, whereas the age of Penelope Cruz was only slightly lower than the average age of other Best Supporting Actress winners. 86 of 149

87 Section 3.4 Summary Determined the quartiles of a data set Determined the interquartile range of a data set Created a box-and-whisker plot Interpreted other fractiles such as percentiles Determined and interpreted the standard score (z-score) 87 of 149