Standard deviation II For each lettered part, a through c, examine the two given sets of

Number of miles run by Saturday morning joggers at a park.

Hours spent by U.S. adults watching football on Thanksgiving Day.

Amount of winnings of all people playing a particular states lottery last week.

Ages of the faculty members at your school.

Last digit of phone numbers on your campus.

More shapes Would you expect distributions of these variables to be uniform, unimodal, or bimodal? Symmetric or skewed? Explain why. a) Ages of people at a Little League game. b) Number of siblings of people in your class. c) Pulse rates of college-age males. d) Number of times each face of a die shows in 100 tosses

Sugar in cereals The histogram displays the sugar content (as a percent of weight) of 49 brands of breakfast cereals. 8 6 4 2 0 8 16 24 32 40 48 56 Sugar (%) # of Cereals a) Describe this distribution. b) What do you think might account for this shape?

Singers The display shows the heights of some of the singers in a chorus, collected so that the singers could be positioned on stage with shorter ones in front and taller ones in back. 20 15 10 5 # of Singers 60 68 76 Height (in.) a) Describe the distribution. b) Can you account for the features you see here?

Vineyards The histogram shows the sizes (in acres) of 36 vineyards in the Finger Lakes region of New York. 0 120 240 5 10 15 Size (acres) # of Vineyards a) Approximately what percentage of these vineyards are under 60 acres? b) Write a brief description of this distribution (shape, center, spread, unusual features).

Run times One of the authors collected the times (in minutes) it took him to run 4 miles on various courses during a 10-year period. Here is a histogram of the times. 50 40 30 20 10 # of Days 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 4-Mile Time (min) Describe the distribution and summarize the important features. What is it about running that might account for the shape you see?

Heart attack stays The histogram shows the lengths of hospital stays (in days) for all the female patients admitted to hospitals in New York during one year with a primary diagnosis of acute myocardial infarction (heart attack). 600 400 200 # of Female Patients 0 5 10 15 20 25 30 35 40 Stay (days) a) From the histogram, would you expect the mean or median to be larger? Explain. b) Write a few sentences describing this distribution (shape, center, spread, unusual features). c) Which summary statistics would you choose to summarize the center and spread in these data? Why?

E-mails A university teacher saved every e-mail received from students in a large Introductory Statistics class during an entire term. He then counted, for each student who had sent him at least one e-mail, how many e-mails each student had sent. 1 6 11 16 21 20 40 60 80 # of Students # of E-mails a) From the histogram, would you expect the mean or the median to be larger? Explain. b) Write a few sentences describing this distribution (shape, center, spread, unusual features). c) Which summary statistics would you choose to summarize the center and spread in these data? Why?

Super Bowl points How many points do football teams score in the Super Bowl? Here are the total numbers of points scored by both teams in each of the first 46 Super Bowl games: 45, 47, 23, 30, 29, 27, 21, 31, 22, 38, 46, 37, 66, 50, 37, 47, 44, 47, 54, 56, 59, 52, 36, 65, 39, 61, 69, 43, 75, 44, 56, 55, 53, 39, 41, 37, 69, 61, 45, 31, 46, 31, 50, 48, 56, 38 a) Find the median. b) Find the quartiles. c) Are there any outliers? d) Construct a boxplot of the data. e) Write a description of the distribution based on the boxplot and the 5-number summary.

Super Bowl wins In the Super Bowl, by how many points does the winning team outscore the losers? Here are the winning margins for the first 46 Super Bowl games: 25, 19, 9, 16, 3, 21, 7, 17, 10, 4, 18, 17, 4, 12, 17, 5, 10, 29, 22, 36, 19, 32, 4, 45, 1, 13, 35, 17, 23, 10, 14, 7, 15, 7, 27, 3, 27, 3, 3, 11, 12, 3, 4, 14, 6, 4 a) Find the median. b) Find the quartiles. c) Are there any outliers? d) Construct a boxplot of the data. e) Write a description of the distribution based on the boxplot and the 5-number summary.

Details Here are histograms for four manufactured sets of numbers. The histograms look rather different, but all four sets have the same 5-number summary, so the boxplots for all four sets are identical to the one shown. a) Using these plots as examples, identify some features of a distribution that a boxplot may not show. b) What does this tell you about the limitations of using a boxplot to assess the shape of a distribution? Frequency 16 18 20 22 24 26 28 30 32 34 36 38 2 4 6 8 10 Frequency 16 18 20 22 24 26 28 30 32 34 36 38

Opposites In a way, boxplots are the opposite of histograms. A histogram divides the number line into equal intervals and displays the number of data values in each interval. A boxplot divides the data into equal parts and displays the portion of the number line each part covers. These two plots display the number of incarcerated prisoners in each state as of June 2006. 4 2 6 8 Frequency of Number of Prisoners10 12 0 40 80 120 Number of Prisoners (thousands) 160 200 a) Explain how you could tell, by looking at a boxplot, where the tallest bars on the histogram would be located. b) Explain how both the boxplot and the histogram can indicate a skewed distribution. c) Identify one feature of the distribution that the histogram shows but a boxplot does not. d) Identify one feature of the distribution that the boxplot shows but the histogram does not.

Still rockin Crowd Management Strategies monitors accidents at rock concerts. In their database, they list the names and other variables of victims whose deaths were attributed to crowd crush at rock concerts. Here are the histograms and boxplot of the victims ages for the data from 1999 to 2000: 12 22 32 42 Age (yr) 5 10 15 20 25 # of Deaths 10 20 30 40 50 Age (yr) a) What features of the distribution can you see in both the histogram and the boxplot? b) What features of the distribution can you see in the histogram that you could not see in the boxplot? c) What summary statistic would you choose to summarize the center of this distribution? Why? d) What summary statistic would you choose to summarize the spread of this distribution? Why?

Slalom times The Mens Combined skiing event consists of a downhill and a slalom. Here are two displays of the slalom times in the Mens Combined at the 2006 Winter Olympics: 87.5 97.5 107.5 Slalom Times (sec) 2 4 6 8 10 # of Skiers 82.5 90.0 97.5 105.0 112.5 Slalom Times Seconds a) What features of the distribution can you see in both the histogram and the boxplot? b) What features of the distribution can you see in the histogram that you could not see in the boxplot? c) What summary statistic would you choose to summarize the center of this distribution? Why? d) What summary statistic would you choose to summarize the spread of this distribution? Why?

Camp sites Shown below are the histogram and summary statistics for the number of camp sites at public parks in Vermont. a) Which statistics would you use to identify the center and spread of this distribution? Why? b) How many parks would you classify as outliers? Explain. c) Create a boxplot for these data. d) Write a few sentences describing the distribution.

Outliers The 5-number summary for the run times in minutes of the 150 highest grossing movies of 2010 looks like this: Min QI Med Q3 Max 43 98 104.5 116 160 a) Are there any outliers in these data? How can you tell? b) Construct a boxplot. (You will be unable to mark outliers if they exist.) Based on your plot, say what you can about the shape of the distribution.

Standard deviation I For each lettered part, a through c, examine the two given sets of numbers. Without doing any calculations, decide which set has the larger standard deviation and explain why. Then check by finding the standard deviations by hand

Standard deviation II For each lettered part, a through c, examine the two given sets of numbers. Without doing any calculations, decide which set has the larger standard deviation and explain why. Then check by finding the standard deviations by hand

Pizza prices The histogram shows the distribution of the prices of plain pizza slices (in $) for 156 weeks in Dallas, TX. Frequency 30 20 10 0 2.2 2.4 2.6 2.8 3.0 Price Which summary statistics would you choose to summarize the center and spread in these data? Why?

Neck size The histogram shows the neck sizes (in inches) of 250 men recruited for a health study in Utah. No. of Men 60 50 40 30 20 10 0 12 13 14 15 16 17 Neck Size (inches) Which summary statistics would you choose to summarize the center and spread in these data? Why?

Pizza prices again Look again at the histogram of the pizza prices in Exercise 23. a) Is the mean closer to $2.40, $2.60, or $2.80? Why? b) Is the standard deviation closer to $0.15, $0.50, or $1.00? Explain.

Neck sizes again Look again at the histogram of mens neck sizes in Exercise 24. a) Is the mean closer to 14, 15, or 16 inches? Why? b) Is the standard deviation closer to 1 inch, 3 inches, or 5 inches? Explain.

Movie lengths The histogram shows the running times in minutes of 150 top grossing films of 2011. 30 25 20 15 10 5 40 60 80 100 Runlength (minutes) 120 140 160 180 Frequency of Runlength a) You plan to see a movie this weekend. Based on these movies, how long do you expect a typical movie to run? b) Would you be surprised to find that your movie ran for 21 2 hours (150 minutes)? c) Which would you expect to be higher: the mean or the median run time for all movies? Why?

Golf drives 2011 The display shows the average drive distance (in yards) for 186 professional golfers on the mens PGA tour in 2011. Average Drive Length (yards) 260 270 280 290 300 310 320 20 18 16 14 Frequency of Average Drive Length 12 10 8 6 4 2 a) Describe this distribution. b) Approximately what proportion of professional male golfers drive, on average, less than 280 yards? c) Estimate the mean by examining the histogram. d) Do you expect the mean to be smaller than, approximately equal to, or larger than the median? Why?

Movie lengths II Exercise 27 looked at the running times of movies released in 2011. The standard deviation of these running times is s = 17.3 minutes, and the quartiles are Q1 = 97 minutes and Q3 = 115 minutes. a) Write a sentence or two describing the spread in running times based on i) the quartiles. ii) the standard deviation. b) Do you have any concerns about using either of these descriptions of spread? Explain.

Golf drives II 2011 Exercise 28 looked at distances PGA golfers can hit the ball. The standard deviation of these average drive distances is 8.4 yards, and the quartiles are Q1 = 285.2 yards and Q3 = 297.5 yards. a) Write a sentence or two describing the spread in distances based on i) the quartiles. ii) the standard deviation. b) Do you have any concerns about using either of these descriptions of spread? Explain.

Fuel Economy The boxplot shows the fuel economy ratings for 67 model year 2012 subcompact cars. Some summary statistics are also provided. The extreme outlier is the Mitsubishi i-MiEV, an electric car whose electricity usage is equivalent to 112 miles per gallon. 20 40 60 80 100 Combined Fuel Economy Mean StDev Min Q1 Med Q3 Max n 23.76 11.87 14 20 22 25 112 67 If that electric car is removed from the data set, how will the standard deviation be affected? The IQR?

Test scores correction After entering the test scores from her Statistics class of 25 students, the instructor calculated some statistics of the scores. Upon checking, she discovered that she had entered the top score as 46, but it should have been 56. a) When she corrects this score, how will the mean and median be affected? b) What effect will correcting the error have on the IQR and the standard deviation?

Mistake A clerk entering salary data into a company spreadsheet accidentally put an extra 0 in the bosss salary, listing it as $2,000,000 instead of $200,000. Explain how this error will affect these summary statistics for the company payroll: a) measures of center: median and mean. b) measures of spread: range, IQR, and standard deviation

Cold weather A meteorologist preparing a talk about global warming compiled a list of weekly low temperatures (in degrees Fahrenheit) he observed at his southern Florida home last year. The coldest temperature for any week was 36F, but he inadvertently recorded the Celsius value of 2. Assuming that he correctly listed all the other temperatures, explain how this error will affect these summary statistics: a) measures of center: mean and median. b) measures of spread: range, IQR, and standard deviation.

Movie earnings The histogram shows total gross earnings (in millions of dollars) of major release movies in 2011. 0 100 Gross Earnings (millions) 200 300 400 40 35 30 Frequency of Gross Earnings 25 20 15 10 5 An industry publication reports that the average movie makes $41.7 million, but a watchdog group concerned with rising ticket prices says that the average earnings is $66.9 million. What statistic do you think each group is using? Explain.

Sick days During contract negotiations, a company seeks to change the number of sick days employees may take, saying that the annual average is 7 days of absence per employee. The union negotiators counter that the average employee misses only 3 days of work each year. Explain how both sides might be correct, identifying the measure of center you think each side is using and why the difference might exist.

Payroll A small warehouse employs a supervisor at $1200 a week, an inventory manager at $700 a week, six stock boys at $400 a week, and four drivers at $500 a week. a) Find the mean and median wage. b) How many employees earn more than the mean wage? c) Which measure of center best describes a typical wage at this company: the mean or the median? d) Which measure of spread would best describe the payroll: the range, the IQR, or the standard deviation? Why?

Singers The frequency table shows the heights (in inches) of 130 members of a choir. a) Find the median and IQR. b) Find the mean and standard deviation. c) Display these data with a histogram. d) Write a few sentences describing the distribution.

Gasoline 2011 In October 2011, 16 gas stations in eastern Wisconsin, posted these prices for a gallon of regular gasoline: 3.43 3.46 3.43 3.59 3.65 3.63 3.62 3.65 3.66 3.31 3.35 3.42 3.41 3.46 3.47 3.48 a) Make a stem-and-leaf display of these gas prices. Use split stems; for example, use two 3.3 stemsone for prices between $3.30 and $3.34 and the other for prices from $3.35 to $3.39. b) Describe the shape, center, and spread of this distribution. c) What unusual feature do you see?

The Great One During his 20 seasons in the NHL, Wayne Gretzky scored 50% more points than anyone who ever played professional hockey. He accomplished this amazing feat while playing in 280 fewer games than Gordie Howe, the previous record holder. Here are the number of games Gretzky played during each season: 79, 80, 80, 80, 74, 80, 80, 79, 64, 78, 73, 78, 74, 45, 81, 48, 80, 82, 82, 70 a) Create a stem-and-leaf display for these data, using split stems. b) Describe the shape of the distribution. c) Describe the center and spread of this distribution. d) What unusual feature do you see? What might explain this?

States The stem-and-leaf display shows populations of the 50 states and Washington, DC, in millions of people, according to the 2000 census. 3 2 2 1 1 0 0 State Populations (1 2 means 12 million) 4 1 6 0 5 1 9 1 5 1 2 5 1 2 5 1 6 1 6 1 6 1 6 1 6 1 7 1 8 1 8 1 8 122222333333344444 a) What measures of center and spread are most appropriate? b) Without doing any calculations, which must be larger: the median or the mean? Explain how you know. c) From the stem-and-leaf display, find the median and the interquartile range. d) Write a few sentences describing this distribution

Wayne Gretzky In Exercise 40, you examined the number of games played by hockey great Wayne Gretzky during his 20-year career in the NHL. a) Would you use the median or the mean to describe the center of this distribution? Why? b) Find the median. c) Without actually finding the mean, would you expect it to be higher or lower than the median? Explain

A-Rod 2010 Alex Rodriguez (known to fans as A-Rod) was the youngest player ever to hit 500 home runs. Here is a stem-and-leaf display of the number of home runs hit by A-Rod during the 19942010 seasons (www .baseballreference.com/players/r/rodrial01.shtml.). Describe the distribution, mentioning its shape and any unusual features.

Bird species 2010 The Cornell Lab of Ornithology holds an annual Christmas Bird Count (www.birdsource .org), in which bird watchers at various locations around the country see how many different species of birds they can spot. Here are some of the counts reported from sites in Texas during the 2010 event: 150 216 177 150 166 156 159 160 164 169 175 150 178 183 199 154 164 203 158 231 a) Create a stem-and-leaf display of these data. b) Write a brief description of the distribution. Be sure to discuss the overall shape as well as any unusual features.

Hurricanes 2010 The data below give the number of hurricanes classified as major hurricanes in the Atlantic Ocean each year from 1944 through 2006, as reported by NOAA (www.nhc.noaa.gov): 3, 3, 1, 2, 4, 3, 8, 5, 3, 4, 2, 6, 2, 2, 5, 2, 2, 7, 1, 2, 6, 1, 3, 1, 0, 5, 2, 1, 0, 1, 2, 3, 2, 1, 2, 2, 2, 3, 1, 1, 1, 3, 0, 1, 3, 2, 1, 2, 1, 1, 0, 5, 6, 1, 3, 5, 3, 4, 2, 3, 6, 7, 2, 2, 5, 2, 5 a) Create a dotplot of these data. b) Describe the distribution.

Horsepower Create a stem-and-leaf display for these horsepowers of autos reviewed by Consumer Reports one year, and describe the distribution: 155 103 130 80 65 142 125 129 71 69 125 115 138 68 78 150 133 135 90 97 68 105 88 115 110 95 85 109 115 71 97 110 65 90 75 120 80 70

A-Rod 2010 again Students were asked to make a histogram of the number of home runs hit by Alex Rodriguez from 1995 to 2010 (see Exercise 43). One student submitted the following display: 1995 1999 2003 2007 Year 20 40 60 a) Comment on this graph. b) Create your own histogram of the data

Return of the birds Students were given the assignment to make a histogram of the data on bird counts reported in Exercise 44. One student submitted the following display: 250 200 150 100 50 Site 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 a) Comment on this graph. b) Create your own histogram of the data.

Acid rain Two researchers measured the pH (a scale on which a value of 7 is neutral and values below 7 are acidic) of water collected from rain and snow over a 6-month period in Allegheny County, PA. Describe their data with a graph and a few sentences: 4.57 5.62 4.12 5.29 4.64 4.31 4.30 4.39 4.45 5.67 4.39 4.52 4.26 4.26 4.40 5.78 4.73 4.56 5.08 4.41 4.12 5.51 4.82 4.63 4.29 4.60

Marijuana 2007 In 2007 the Council of Europe published a report entitled The European School Survey Project on Alcohol and Other Drugs (www.espad.org). Among other issues, the survey investigated the percentages of 16-year-olds who had used marijuana. Shown on the next page are the results for 34 European countries. Create an appropriate graph of these data, and describe the distribution.

Final grades A professor (of something other than Statistics!) distributed the following histogram to show the distribution of grades on his 200-point final exam. Comment on the display. 50 40 30 20 10 50 100 150 200 Final Grade

Final grades revisited After receiving many complaints about his final-grade histogram from students currently taking a Statistics course, the professor from Exercise 51 distributed the following revised histogram: # of Students4 3 2 1 75.00 92.50 110.00 127.50 145.00 162.50 180.00 197.50 Final Grade a) Comment on this display. b) Describe the distribution of grades.

Zip codes Holes-R-Us, an Internet company that sells piercing jewelry, keeps transaction records on its sales. At a recent sales meeting, one of the staff presented a histogram of the zip codes of the last 500 customers, so that the staff might understand where sales are coming from. Comment on the usefulness and appropriateness of the display.

Zip codes revisited Here are some summary statistics to go with the histogram of the zip codes of 500 customers from the Holes-R-Us Internet Jewelry Salon that we saw in Exercise 53: What can these statistics tell you about the companys sales?

Math scores 2009 The National Center for Education Statistics (http://nces.ed.gov/nationsreportcard/) reported 2009 average mathematics achievement scores for eighth graders in all 50 states: a) Find the median, the IQR, the mean, and the standard deviation of these state averages. b) Which summary statistics would you report for these data? Why? c) Write a brief summary of the performance of eighth graders nationwide

Boomtowns 2011 In 2011, the website NewGeography .com listed its ranking of the best cities for job growth in the United States. Here are the magazines top 20 larger cites, along with their weighted job rating indices. T a) Make a suitable display of the job rating indices. b) Summarize the typical job rating index among these cities with a median and mean. Why do they differ? c) Given what you know about the distribution, which of the measures in b) does the better job of summarizing the job rating indices? Why? d) Summarize the spread of the job rating indices distribution with a standard deviation and with an IQR. e) Given what you know about the distribution, which of the measures in d) does the better job of summarizing the job rating indices? Why? f) Suppose we subtract from each of the preceding job rating indices the predicted U.S. average job rating index, so that we can look at how much these indices exceed the U.S. rate. How would this change the values of the summary statistics you calculated above? g) If we were to omit the Austin and New Orleans metro areas from the data, how would you expect the mean, median, standard deviation, and IQR to change? Explain your expectations for each. h) Write a brief report about all of these job rating indices.

Population growth The following data show the percentage change in population for the 50 states and the District of Columbia from 2000 to 2009. Using appropriate graphical displays and summary statistics, write a report on the percentage change in population by state.

Prisons 2006 A report from the U.S. Department of Justice (www.ojp.usdoj.gov/bjs/) reported the percent changes in federal prison populations in 21 northeastern and midwestern states during 2006. Using appropriate graphical displays and summary statistics, write a report on the changes in prison populations.