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Chapter 4 Displaying and Summarizing Quantitative Data CHAPTER OBJECTIVES At the conclusion of this chapter you should be able to: n 1)Construct graphs that appropriately describe quantitative data n 2)Calculate and interpret numerical summaries of quantitative data. n 3)Combine numerical methods with graphical methods to analyze a data set. n 4)Apply graphical methods of summarizing data to choose appropriate numerical summaries. n 5)Apply software and/or calculators to automate graphical and numerical summary procedures.

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Displaying Quantitative Data Histograms Stem and Leaf Displays

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Relative Frequency Histogram of Exam Grades 0.05.10.15.20.25.30 405060708090 Grade Relative frequency 100

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Frequency Histogram

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Histograms A histogram shows three general types of information: n It provides visual indication of where the approximate center of the data is. n We can gain an understanding of the degree of spread, or variation, in the data. n We can observe the shape of the distribution.

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All 200 m Races 20.2 secs or less

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Histograms Showing Different Centers

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Histograms Showing Different Centers (football head coach salaries)

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Histograms - Same Center, Different Spread (football head coach salaries)

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Excel Example: 2012-13 NFL Salaries

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Statcrunch Example: 2012-13 NFL Salaries

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Grades on a statistics exam Data: 75 66 77 66 64 73 91 65 59 86 61 86 61 58 70 77 80 58 94 78 62 79 83 54 52 45 82 48 67 55

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Frequency Distribution of Grades Class Limits Frequency 40 up to 50 50 up to 60 60 up to 70 70 up to 80 80 up to 90 90 up to 100 Total 2 6 8 7 5 2 30

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Relative Frequency Distribution of Grades Class Limits Relative Frequency 40 up to 50 50 up to 60 60 up to 70 70 up to 80 80 up to 90 90 up to 100 2/30 =.067 6/30 =.200 8/30 =.267 7/30 =.233 5/30 =.167 2/30 =.067

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Relative Frequency Histogram of Grades 0.05.10.15.20.25.30 405060708090 Grade Relative frequency 100

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Based on the histo- gram, about what percent of the values are between 47.5 and 52.5? 1. 50% 2. 5% 3. 17% 4. 30% Countdown 10

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Stem and leaf displays n Have the following general appearance stemleaf 18 9 21 2 8 9 9 32 3 8 9 40 1 56 7 64

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Stem and Leaf Displays n Partition each no. in data into a “stem” and “leaf” n Constructing stem and leaf display 1) deter. stem and leaf partition (5-20 stems) 2) write stems in column with smallest stem at top; include all stems in range of data 3) only 1 digit in leaves; drop digits or round off 4) record leaf for each no. in corresponding stem row; ordering the leaves in each row helps

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Example: employee ages at a small company 18 21 22 19 32 33 40 41 56 57 64 28 29 29 38 39; stem: 10’s digit; leaf: 1’s digit n 18: stem=1; leaf=8; 18 = 1 | 8 stemleaf 18 9 21 2 8 9 9 32 3 8 9 40 1 56 7 64

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Suppose a 95 yr. old is hired stemleaf 18 9 21 2 8 9 9 32 3 8 9 40 1 56 7 64 7 8 95

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Number of TD passes by NFL teams: 2012-2013 season ( stems are 10’s digit) stemleaf 4343 03 247 26677789 201222233444 113467889 08

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Pulse Rates n = 138

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Advantages/Disadvantages of Stem-and-Leaf Displays n Advantages 1) each measurement displayed 2) ascending order in each stem row 3) relatively simple (data set not too large) n Disadvantages display becomes unwieldy for large data sets

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Population of 185 US cities with between 100,000 and 500,000 n Multiply stems by 100,000

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Back-to-back stem-and-leaf displays. TD passes by NFL teams: 1999-2000, 2012-13 multiply stems by 10 1999-20002012-13 2403 637 2324 665526677789 43322221100201222233444 9998887666167889 4211134 08

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Below is a stem-and-leaf display for the pulse rates of 24 women at a health clinic. How many pulses are between 67 and 77? Stems are 10’s digits 1. 4 2. 6 3. 8 4. 10 5. 12 Countdown 10

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Interpreting Graphical Displays: Shape n A distribution is symmetric if the right and left sides of the histogram are approximately mirror images of each other. Symmetric distribution Complex, multimodal distribution Not all distributions have a simple overall shape, especially when there are few observations. Skewed distribution A distribution is skewed to the right if the right side of the histogram (side with larger values) extends much farther out than the left side. It is skewed to the left if the left side of the histogram extends much farther out than the right side.

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Heights of Students in Recent Stats Class

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Shape (cont.)Female heart attack patients in New York state Age: left-skewedCost: right-skewed

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AlaskaFlorida Shape (cont.): Outliers An important kind of deviation is an outlier. Outliers are observations that lie outside the overall pattern of a distribution. Always look for outliers and try to explain them. The overall pattern is fairly symmetrical except for 2 states clearly not belonging to the main trend. Alaska and Florida have unusual representation of the elderly in their population. A large gap in the distribution is typically a sign of an outlier.

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Center: typical value of frozen personal pizza? ~$2.65

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Spread: fuel efficiency 4, 8 cylinders 4 cylinders: more spread8 cylinders: less spread

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Other Graphical Methods for Economic Data n Time plots plot observations in time order, with time on the horizontal axis and the vari- able on the vertical axis ** Time series measurements are taken at regular intervals (monthly unemployment, quarterly GDP, weather records, electricity demand, etc.)

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Heat Maps

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Unemployment Rate, by Educational Attainment

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Water Use During Super Bowl

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Winning Times 100 M Dash

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Numerical Summaries of Quantitative Data Numerical and More Graphical Methods to Describe Univariate Data

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2 characteristics of a data set to measure n center measures where the “middle” of the data is located n variability measures how “spread out” the data is

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The median: a measure of center Given a set of n measurements arranged in order of magnitude, Median=middle valuen odd mean of 2 middle values,n even n Ex. 2, 4, 6, 8, 10; n=5; median=6 n Ex. 2, 4, 6, 8; n=4; median=(4+6)/2=5

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Student Pulse Rates (n=62) 38, 59, 60, 60, 62, 62, 63, 63, 64, 64, 65, 67, 68, 70, 70, 70, 70, 70, 70, 70, 71, 71, 72, 72, 73, 74, 74, 75, 75, 75, 75, 76, 77, 77, 77, 77, 78, 78, 79, 79, 80, 80, 80, 84, 84, 85, 85, 87, 90, 90, 91, 92, 93, 94, 94, 95, 96, 96, 96, 98, 98, 103 Median = (75+76)/2 = 75.5

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Medians are used often n Year 2014 baseball salaries Median $1,450,000 (max=$28,000,000 Zack Greinke; min=$500,000) n Median fan age: MLB 45; NFL 43; NBA 41; NHL 39 n Median existing home sales price: May 2011 $166,500; May 2010 $174,600 n Median household income (2008 dollars) 2009 $50,221; 2008 $52,029

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The median splits the histogram into 2 halves of equal area

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Examples n Example: n = 7 17.5 2.8 3.2 13.9 14.1 25.3 45.8 n Example n = 7 (ordered): n 2.8 3.2 13.9 14.1 17.5 25.3 45.8 n Example: n = 8 17.5 2.8 3.2 13.9 14.1 25.3 35.7 45.8 n Example n =8 (ordered) 2.8 3.2 13.9 14.1 17.5 25.3 35.7 45.8 m = 14.1 m = (14.1+17.5)/2 = 15.8

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Below are the annual tuition charges at 7 public universities. What is the median tuition? 4429 4960 4971 5245 5546 7586 1. 5245 2. 4965.5 3. 4960 4. 4971 Countdown 10

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Below are the annual tuition charges at 7 public universities. What is the median tuition? 4429 4960 5245 5546 4971 5587 7586 1. 5245 2. 4965.5 3. 5546 4. 4971 Countdown 10

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Measures of Spread n The range and interquartile range

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Ways to measure variability range=largest-smallest OK sometimes; in general, too crude; sensitive to one large or small data value The range measures spread by examining the ends of the data A better way to measure spread is to examine the middle portion of the data

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m = median = 3.4 Q 1 = first quartile = 2.3 Q 3 = third quartile = 4.2 Quartiles: Measuring spread by examining the middle The first quartile, Q 1, is the value in the sample that has 25% of the data at or below it (Q 1 is the median of the lower half of the sorted data). The third quartile, Q 3, is the value in the sample that has 75% of the data at or below it (Q 3 is the median of the upper half of the sorted data).

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Quartiles and median divide data into 4 pieces Q1 M Q3 Q1 M Q3 1/4 1/41/4 1/4

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Quartiles are common measures of spread n http://oirp.ncsu.edu/ir/admit http://oirp.ncsu.edu/ir/admit n http://oirp.ncsu.edu/univ/peer http://oirp.ncsu.edu/univ/peer n University of Southern California University of Southern California n Economic Value of College Majors Economic Value of College Majors

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Rules for Calculating Quartiles Step 1: find the median of all the data (the median divides the data in half) Step 2a: find the median of the lower half; this median is Q 1 ; Step 2b: find the median of the upper half; this median is Q 3. Important: when n is odd include the overall median in both halves; when n is even do not include the overall median in either half.

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Example n 2 4 6 8 10 12 14 16 18 20 n = 10 n Median n m = (10+12)/2 = 22/2 = 11 n Q 1 : median of lower half 2 4 6 8 10 Q 1 = 6 n Q 3 : median of upper half 12 14 16 18 20 Q 3 = 16 11

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Quartile example: odd no. of data values n HR’s hit by Babe Ruth in each season as a Yankee 54 59 35 41 46 25 47 60 54 46 49 46 41 34 22 Ordered values: 22 25 34 35 41 41 46 46 46 47 49 54 54 59 60 Median: value in ordered position 8. median = 46 Lower half (including overall median): 22 25 34 35 41 41 46 46 Upper half (including overall median): 46 46 47 49 54 54 59 60

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Pulse Rates n = 138 Median: mean of pulses in locations 69 & 70: median= (70+70)/2=70 Q 1 : median of lower half (lower half = 69 smallest pulses); Q 1 = pulse in ordered position 35; Q 1 = 63 Q 3 median of upper half (upper half = 69 largest pulses); Q 3 = pulse in position 35 from the high end; Q 3 =78

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Below are the weights of 31 linemen on the NCSU football team. What is the value of the first quartile Q 1 ? #stemleaf 22255 42357 62426 7257 1026257 122759 (4)281567 152935599 1030333 73145 532155 2336 1340 1. 287 2. 257.5 3. 263.5 4. 262.5 Countdown 10

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Interquartile range n lower quartile Q 1 n middle quartile: median n upper quartile Q 3 n interquartile range (IQR) IQR = Q 3 – Q 1 measures spread of middle 50% of the data

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Example: beginning pulse rates n Q 3 = 78; Q 1 = 63 n IQR = 78 – 63 = 15

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Below are the weights of 31 linemen on the NCSU football team. The first quartile Q 1 is 263.5. What is the value of the IQR? #stemleaf 22255 42357 62426 7257 1026257 122759 (4)281567 152935599 1030333 73145 532155 2336 1340 1. 23.5 2. 39.5 3. 46 4. 69.5 Countdown 10

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5-number summary of data n Minimum Q 1 median Q 3 maximum n Pulse data 45 63 70 78 111

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m = median = 3.4 Q 3 = third quartile = 4.2 Q 1 = first quartile = 2.3 Largest = max = 6.1 Smallest = min = 0.6 Five-number summary: min Q 1 m Q 3 max Boxplot: display of 5-number summary BOXPLOT

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Boxplot: display of 5-number summary n Example: age of 66 “crush” victims at rock concerts 1999-2000. 5-number summary: 13 17 19 22 47

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Boxplot construction 1) construct box with ends located at Q1 and Q3; in the box mark the location of median (usually with a line or a “+”) 2) fences are determined by moving a distance 1.5(IQR) from each end of the box; 2a) upper fence is 1.5*IQR above the upper quartile 2b) lower fence is 1.5*IQR below the lower quartile Note: the fences only help with constructing the boxplot; they do not appear in the final boxplot display

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Box plot construction (cont.) 3) whiskers: draw lines from the ends of the box left and right to the most extreme data values found within the fences; 4) outliers: special symbols represent each data value beyond the fences; 4a) sometimes a different symbol is used for “far outliers” that are more than 3 IQRs from the quartiles

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Q 3 = third quartile = 4.2 Q 1 = first quartile = 2.3 Largest = max = 7.9 Boxplot: display of 5-number summary BOXPLOT 8 Interquartile range Q 3 – Q 1 = 4.2 − 2.3 = 1.9 Distance to Q 3 7.9 − 4.2 = 3.7 1.5 * IQR = 1.5*1.9=2.85. Individual #25 has a value of 7.9 years, which is 3.7 years above the third quartile. This is more than 2.85 = 1.5*IQR above Q 3. Thus, individual #25 is a suspected outlier.

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ATM Withdrawals by Day, Month, Holidays

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Beg. of class pulses (n=138) n Q 1 = 63, Q 3 = 78 n IQR=78 63=15 n 1.5(IQR)=1.5(15)=22.5 n Q 1 - 1.5(IQR): 63 – 22.5=40.5 n Q 3 + 1.5(IQR): 78 + 22.5=100.5 70 63 78 40.5 100.5 45

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Below is a box plot of the yards gained in a recent season by the 136 NFL receivers who gained at least 50 yards. What is the approximate value of Q 3 ? 0 136 273 410 547 684 821 958 1095 1232 1369 Pass Catching Yards by Receivers 1. 450 2. 750 3. 215 4. 545 Countdown 10

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Rock concert deaths: histogram and boxplot

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Automating Boxplot Construction n Excel “out of the box” does not draw boxplots. n Many add-ins are available on the internet that give Excel the capability to draw box plots. n Statcrunch (http://statcrunch.stat.ncsu.edu) draws box plots.

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Q 3 = third quartile = 4.2 Q 1 = first quartile = 2.3 Largest = max = 7.9 Statcrunch Boxplot

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Tuition 4-yr Colleges

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Statcrunch: 2012-13 NFL Salaries by Position

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College Football Head Coach Salaries by Conference

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2013 Major League Baseball Salaries by Team

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End of General Numerical Summaries. Next: Numerical Summaries of Symmetric Data

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