Chapter 2. Describing, Exploring, and Comparing Data

Transcription

1 Chapter 2 Describing, Exploring, and Comparing Data

2 Important Characteristics of Data Describes the overall pattern of a distribution: Center Spread Shape Outlier Divides the data in half Differences between the data Skewness of the data Data that falls outside of the pattern Data Distributions Graphs displays distribution Numbers describe the distribution

3 Lesson 2-2 Frequency Distribution

4 Frequency Distribution A frequency distribution lists the number of occurrences for each category of data. Lower Class Limits Upper Class Limits Grades Frequency A (100 90) 5 B ( 89 80) 8 C ( 79 70) 4 D ( 69 60) 5 F (59 50) 3

5 Example Page 44, #2 Identify the class width, class midpoints, and class boundaries for the given frequency distribution. Systolic Blood Pressure of Women Frequency

6 Example Page 44, #2 Find the class width Blood Pressure Frequency

7 Example Page 44, # Blood Pressure Class Midpoints Class Boundaries

8 Relative Frequency Distribution The relative frequency is the proportion or percent of observations within a category and is found using the formula Relative Frequency = Class Frequency Sum of all Frequencies Reasons for Constructing Frequency Distributions Large data sets can be summarized. Can gain some insight into the nature of data. Have a basis for constructing graphs.

9 Example Page 44, #6 Construct the relative frequency distribution #2 Blood Pressure Frequency Relative Frequency Total % % 12.5% 2.5% 0.0% 2.5% 100%

10 Cumulative Frequency Distribution Discrete Data It displays the total number of observation less than or equal to the category. Continuous Data It displays the total number of observation less than equal to the upper class limit of a class.

11 Example Page 44, #10 Construct the cumulative frequency distribution #2 Frequency Relative Frequency Cumulative % 60.0% 12.5% 2.5% 0.0% 2.5% Frequency

12 Example Page 45, #16 In Tobacco and Alcohol Use in G-Rated Children s Animated Films, by Goldstein, Sobel, and Newman (Journal of American Medical Association, Vol 281, No. 12), the length (in seconds) of scenes showing tobacco use and alcohol use were recorded for animated children s movies. Refer to Data set 7 in Appendix B. Construct a separate frequency distribution for the lengths of time for tobacco use and alcohol use. In both cases, uses the classes of 0 99, , and so on. Compare the results and determine whether there appears to be a significant difference.

13 Example Page 45, #16 STAT 2 nd STAT

14 Example Page 45, #16 Time (Sec) Tobacco Alcohol

15 Example Page 45, #16 There does not appear to be significant difference

16 Lesson 2-3 Visualizing Data

17 Displaying Distributions Categorical Data (Qualitative) Bar Graphs Pie Charts Measurement Data (Quantitative) Histograms Dotplots Stem-and-leaf plots Ogive Frequency Polygon

18 Pie Chart When to use: The categorical data has a small number of possible categories. Are most useful for illustrating proportions of the whole data set for various categories. What to look for: Categories that form large or small proportions of the data set. Don t forget to title the graph, label the categories and include all categories that make up the whole.

19 Example Pie Chart Education of People 25 to 34 Years Old, 2000 Number of Persons (thousands) Relative Frequency Less than High School 4, % High School Graduate 11, % Some College 10, % Bachelor s Degree 8, % Advanced Degree 2, % Total 37, %

20 Example Pie Charts Education of People 25 to 34 Years Old, 2000 HS Grad Not HS Grad Some College Bachelor's Degree Advanced Degree 22.7% 6.6% 30.6% % 11.8%

21 Bar Graph When to use: The categorical data has a large number of possible categories. What to look for: Frequently or infrequently occurirng categories. Don t forget to include labels for the axes as well as a title for the graph.

22 Percent Example Bar Graphs Edcucation of People 25 to 34 Years Old, Not HS Grad HS Grad Some College Bachelor's Degree Education Advanced Degree

23 Dot Plot When to use: Numerical data sets with small number of observations. What to look for: Conveys information about a typical value in the data set. Extent in which the data values are spread out. The nature of the distribution of values along the number line. The presence of unusual values in the data set. Don t forget to title the graph and label the axis.

24 Example Dotplot Here are the numbers of home runs that Babe Ruth hit in his 15 years with the New York Yankees, 1920 to

25 Stem Plot When to use: Numerical data sets with a small to moderate number of observations What to look for: Conveys information about a typical value in the data set. Extent in which the data values are spread out. The presence of any gaps in the data. The symmetry in the distribution of values The number and location of peaks. The presence of unusual (outlier) values in the data set. Don t forget to title the graph

26 Example Stem Plot (Babe Ruth) , 5 4, 5 1, 1, 6, 6, 6, 7, 9 4, 4, 9 0

27 Displaying Distributions Categorical Data Bar Graphs Pie Charts Quantitative Data Dotplots Stem-and-leaf plots Histograms Ogive Frequency Polygon

28 Histogram When to use: Continuous numerical data sets with a moderate to large number of observations What to look for: Conveys information about a typical value in the data set. Extent in which the data values are spread out. The general shape, location and number of peaks The presence of gaps. The presence of unusual (outlier) values in the data set. Don t forget to title the graph and label axes.

29 Example Histogram (Discrete Data) The manager of Wendy s fast-food restaurant is interested in studying the typical number of customers who arrive during the lunch hour. The data in the following table represent the number of customers who arrive at Wendy s for 40 randomly selected 15-minute intervals of time during lunch Number of Arrivals at Wendy s

30 Example Histogram (Discrete Data) Number of Arrivals at Wendy s Step 1 Construct a frequency distribution table How many categories are there? 11

31 Example Histogram (Discrete Data) Number of Customers Tally Frequency Relative Frequency

32 Frequency Example Histogram (Discrete Data) Arrivals at Wendy s Number of Customers

33 Relative Frequency Example Histogram (Discrete Data) Arrivals at Wendy s Number of Customers

34 Example Histogram (Continuous Data) Suppose you are considering investing in a Roth IRA. You collect the data table, which represent the three-year rate of return (in percent) for 40 small capitalization growth mutual funds

35 Example Histogram (Continuous Data) STAT

36 Example Histogram (Continuous Data) A) Construct a frequency distribution to display these data. Record your class intervals and counts Step 1 Find the class intervals Locate the smallest number (10.8) and the largest number (47.7) Lower class limit will be 10.0 with a class width of 5

37 Example Histogram (Continuous Data) 3-yr Rate of Return Frequency

38 Example Histogram (Continuous Data) 3-yr Rate of Return Total Frequency

39 Example Histogram Step 2 Graph it using the TI

40 Example Histogram

41 Frequency Example - Histogram 12 3 Year Rate of Return of Mutual Funds % Rate of Return

42 Example Histogram B) Describe the distribution of 3 Year Rate of Return. The distribution is skewed to the right with a peak at the class So that 27.5% = (11/40) of the small-cap growth fund had a 3-year return between 15% and 19.9% There is one outlier in class the

43 Histogram Too few categories Age of Spring 1998 Stat 250 Students n=92 students Age (in years)

44 Frequency (Count) Histogram Too many categories GPAs of Spring 1998 Stat 250 Students n=92 students GPA

45 Ogive A relative cumulative frequency graph (ogive) is used to find the relative standing of an individual observation.

46 Example Relative Cumulative Frequency Suppose you are considering investing in a Roth IRA. You collect the data table, which represent the three-year rate of return (in percent) for 40 small capitalization growth mutual funds

47 Example Relative Cumulative Frequency Class Freq Relative Frequency Cumulative Frequency 7 Relative cumulative Frequency Total 40

48 Example Relative Cumulative Frequency Class Freq Rel Freq Cum Freq Rel Cum Freq of the 40 mutual funds had a 3 year rate of return of 24.9% or less 65% of the mutual funds had 3 year rate of return of 24.9% or less A mutual fund with a 3 year rate of return of 45% or higher is out performing 95% of its peers.

49 Example Relative Cumulative Frequency L3 Upper Class Limits L4 Relative Cumulative Frequency

50 Example Relative Cumulative Frequency

51 Cumulative Relative Frequency Example Relative Cumulative Frequency 3 Year Rate of Return for Small Capitalization Mutal Funds Rate of Return 80% of the mutual funds had a 3 year-year rate of return less than or equal to 29.9%

52 Example Frequency Polygon Suppose you are considering investing in a Roth IRA. You collect the data table, which represent the three-year rate of return (in percent) for 40 small capitalization growth mutual funds

53 Example Frequency Polygon Class Freq Class Midpoints Total 40

54 Example Frequency Polygon L3 Class Midpoints L4 Frequency

55 Frequency Example Frequency Polygon 3 Year Rate of Return Rate of Return

56 Lesson 2-4 Measure of Center

57 Measuring the Center Mean Median Mode Midrange

58 Mean or Arithmetic Mean Find the sum of all values and then divide by the number of values Sample Population x x n x N

59 Median Arrange the data in order. Odd number values the median is the value in the exact middle. Even number values add the two middle numbers then divide by 2.

60 Mode Value that occurs most frequently. Bimodal is when two values occur with the same greatest frequency. Multimodal is when more than two values occur with the same greatest frequency. When no value is repeated, we say there is no mode.

61 Midrange Is the value halfway between the highest and lowest values. Midrange = Highest Value + Lowest Value 2

62 Example, Page 70, #10 Find the mean, median, mode and midrange for each of the two samples, then compare the two sets of results. regular : diet :

63 Example, Page 70, #10

64 Example, Page 70, #10 Regular Diet

65 Example, Page 70, #10 Regular Diet Mean x lb x lb Median lb lb Mode None None Midrange lb lb

66 Example, Page 70, #10 Diet appears to weigh less because it has less sugar than regular coke.

67 Mean from the a Frequency Distribution use class midpoints of classes for variable x frequency class midpoint x f f x n

68 Example, Page 71, #20 The accompany frequency distribution summarizes a sample of human body temperatures. How does the mean compare to the value of 98.6 F, which is the value assumed to be the mean by most people

69 Example, Page 71, #20 Temperature Frequency Midpoint x f x

70 Example, Page 71, #20 x f x f 106 The mean appears to be substantially lower than 98.6 F

71 Skewed To The Left (Negatively)

72 Symmetric

73 Skewed To The Right (Positively)

74 How to Choose the Best Average Choose mode if there are two or more trends in the data Two or more areas of high frequency values Report one mode for each trend Choose the median if the distribution is skewed A small number of outliers are heavily influencing the mean. Choose the mean if the distribution if fairly symmetric with one mode.

75 Lesson 2-5 Measures of Variation

76 Measuring the Spread Range Quartiles Boxplots Standard Deviation Variance

77 Range The range is the difference between the largest and smallest observation. R x x max min

78 Standard Deviation The standard deviation (s) measures the average distance of observations from their mean.

79 Variance and Standard Deviation Variance s 2 x x n 1 2 Standard Deviation s s 2 x x n 1 2

80 Notation Sample s = standard deviation s² = variance Population σ = standard deviation σ² = variance

81 Example - Variance The levels of various substances in the blood influence our health. Here are measurements of the level of phosphate in the blood of a patient, in milligrams of phosphate per deciliter of blood, made on 6 consecutive visits to a clinic

82 Example - Variance A. Find the mean. x

83 Example - Variance x 4.6 x 5.4 x

84 Example - Variance Observation Deviations Square Deviations x 2 x x x x SUM 0 2 (0.2) SUM 2.06

85 Example Variance B) Find the standard deviation (s) from its definition. s 2 x x n s s

86 Example Variance C) Use your TI-83 to find x and s. Do the result agree with part B.

87 Example Variance

88 Example Page 88, #6 Listed below are ages of motorcyclists when they were fatally injured in traffic crashes. How does the variation of these ages compare to the variation of ages of licensed drivers in the general population

89 Example Page 88, #6 s 8.7 years s years²

90 Example Page 88, #6 s 8.7 years s years² Since motorcycle drivers tend to come from a particular age group, there ages would vary less than those in the general population.

91 Standard Deviation Standard deviation (s) is the square root of the variance (s² ) Units are the original units Measures the spread about the mean and should only be used when the mean is chosen as the center If s = 0 then there is no spread. Observations are the same value As s gets larger the observations are more spread out. Highly affected by outliers

92 Variance Variance (s²) measures the average squared deviation of observations from the mean Units are squared Highly affected by outliers. Best for symmetric data

93 Example Page 89, #16 In Tobacco and Alcohol Use in G-Rated Children s Animated Films, by Goldstein, Sobel, and Newman (Journal of American Medical Association, Vol. 281, No. 12), the length (in seconds) of scenes showing tobacco use and alcohol use were recorded for animated children s movies. Refer to Data set 7 in Appendix B. Find the standard deviation for the lengths of time for tobacco use and alcohol use.

94 Example Page 89, #16

95 Example Page 89, #16 Tobacco: sec; alcohol; 66.3 sec. There was slightly more variability The times for among the lengths of tobacco usage in this particular sample, there does not appear to be a significant difference between the products

96 Standard Deviation from a Frequency Distribution s 2 2 n f x f x nn ( 1) Use the class midpoints as the x values

97 Example Page 89, #19 The given frequency distribution describes the speeds of drivers ticketed by the Town of Poughkeepsie police. These drivers were traveling through a 30 mi/hr speed zone on Creek Road.

98 Example Page 89, #19 Class Midpoints f x

99 Example Page 89, #19 f x 2

100 Example Page 89, #19 Speed Frequency Class Midpoints f x f x f x Sum

101 Example Page 89, #19 s 2 n f x f x nn ( 1) 2 s s 4.1 mi/hr

102 Empirical ( Rule) For data sets that have a distribution that is approximately bell shape, the following properties apply: 68% of all values fall within 1 standard deviation of the mean. 95% of all values fall within 2 standard deviations of the mean % of all values fall within 3 standard deviations of the mean.

103 The Empirical Rule

104 The Empirical Rule

105 The Empirical Rule

106 Example, Page 90, #26 Use the weights of regular coke listed in data set 17 from Appendix B, we find that the mean is lb, the standard deviations is lb, and the distribution is approximately bell-shape. Using the empirical rule, what is the approximate percentage of cans of regular coke with weights between: A lb and lb B lb and lb

107 Example, Page 90, #26 mean = lb and the standard deviations = lb A lb and lb? B lb and lb? 1 standard deviation 68% 2 standard deviations 95% x 2s x s x x s x 2s

108 Lesson 2-6 Measure of Relative Standing

109 Z-Scores (Standard Score)

110 Z-Score (Standard Score) The standard score or z score, is the number of standard deviations that a given value x is above or below the mean. It is found using the following expressions: Sample Population z x s x z x σ μ Round z to two decimal places

111 Unusual Values/Ordinary Values Ordinary values: Unusual Values: z score between 2 and 2 sd z score < -2 or z score > 2 sd

112 Example Page 99, #2 Assume that adults have pulse rates (beats per minute) with a mean of 72.9 and a standard deviation of When this question was written, the author s pulse rate was 48. A. What is the difference between the author s pulse and the mean

113 Example Page 99, #2 mean of 72.9 and a standard deviation of the author s pulse rate was 48. B. How many standard deviations is that [the difference found in part (a).] z x μ σ C. Convert a pulse rate of 48 to a z-score. z x μ σ 12.3

114 Example Page 99, #2 mean of 72.9 and a standard deviation of the author s pulse rate was 48. D. If we considered usual pulse rates to be those that convert to z scores between -2 and 2, is a pulse rate of 48 usual or unusual? Unusual

115 Example Page 100, #8 For men ages between 18 and 24 years, serum cholesterol levels (in mg/100 ml) have a mean of and a standard deviation of Find the z score corresponding to a male, aged years, who has serum cholesterol of mg/100 ml. Is this level unusually high? x μ σ z x μ σ 40.7 No, this level is not considered unusually high since 1.99 < 2.00

116 Quartiles Quartiles divides the observation into fourths, or four equal parts. Smallest Data Value Q1 Q2 Q3 Largest Data Value 25% of the data 25% of the data 25% of the data 25% of the data

117 Percentiles Just as there are three quartiles separating a data set into four parts, there are also 99 percentiles, denoted P, P,... P which partition the data into 100 groups. Smallest Data Value Q Q 1 P25 Q2 P P Largest Data Value 25% of the data 25% of the data 25% of the data 25% of the data

118 Finding the Percentile of a Given Score Percentile of value x = number of values less than x 100 total number of values

119 Example Page 100, #14 Find the percentile corresponding to the given cotinine levels of 210. Use Table P

120 Converting from the kth Percentile to the Corresponding Data Value Notation n total number of values in the data set L = k 100 n k L percentile being used locator that gives the position of a value P k kth percentile

121 Example Page 100, #22 Find the P21 21 L th Score is 48 P

122 Example Page 100, #20 Find the Q2 50 L The mean of the 20 th and 21 st scores P

123 Converting from kth Percentile to the Corresponding Data Value

124 Lesson 2-7 Exploratory Data Analysis

125 Exploratory Data Analysis Is the process of using statistical tools (such as graphs, measures of center, and measures of variation) to investigate data sets in order to understand their important characteristics

126 Outlier An outlier can have a dramatic effect on the mean An outlier have a dramatic effect on the standard deviation An outlier can have a dramatic effect on the scale of the histogram so that the true nature of the distribution is totally obscured

127 Five Number Summary Smallest observation (minimum) Quartile 1 Quartile 2 (median) Quartile 3 Largest observation (maximum)

128 Box plots A boxplot ( or box-and-whisker-diagram) is a graph of a data set that consists of a line extending from the minimum value to the maximum value, and a box with lines drawn at the first quartile, Q 1 ; the median; and the third quartile, Q 3

129 Interquartile Range (IQR) The interquartile range (IQR) is the distance between the first and third quartiles IQR Q Q 3 1

130 Outliers Upper Cutoff Q 1.5( IQR) 3 Lower Cutoff Q 1.5( IQR) 1

131 Example Page 109, #7 In Tobacco and Alcohol Use in G-Rated Children s Animated Films, by Goldstein, Sobel, and Newman (Journal of American Medical Association, Vol. 281, No. 12), the length (in seconds) of scenes showing alcohol use was recorded for animated children s movies. Refer to Data set 7 in Appendix B. Find the 5 number summary and construct a box plot. Based on the box plots, does the distribution appear to be symmetric or is it skewed?

132 Example Page 109, #7

133 Example Page 109, #7 min Q Q Q x x max x x 39 x 414

134 Example Page 109, #7 Based on the box plot, the distribution appears to be extremely right skewed