Statistics: Concepts and Controversies Chapter 11 Displaying Distributions with Graphs Chapter 11 Chapter 11

Statistics: Concepts and Controversies Recall Categorical Variable: Just record group membership (using pie chart or bar graph) Quantitative Variable: takes numerical values and it may has too many values Chapter 11 Chapter 11

Comparing tuitions using a histogram Statistics: Concepts and Controversies Comparing tuitions using a histogram Chicago State University charges in-state students $6834 per year. There are 121 colleges and universities in Illinois. Their tuitions range from $1536 to $30,729. The following is a histogram of the tuitions charged by 121 Illinois colleges and universities. Chapter 11 Chapter 11

Figure 11.2 Histogram of the tuition and fees charged by 121 Illinois colleges and universities in the 2004–2005 academic year. (Data from the Web site www.collegeillinois.com/en/collegefunding/costs.htm. This figure was created using the SPSS software package.)

Comparing tuitions using a stemplot Statistics: Concepts and Controversies Comparing tuitions using a stemplot Another method for displaying the same data is using stemplots. For example, round $6834 to nearest hundreds, that is to 68. Put the thousands digit to the left of the line, and hang the hundreds one by one on these stems. Chicago State is the red 8 on the 6 stem. The stemplot shows more detail. We can see that Chicago State’s tuition is 58th (from bottom) out of 121 colleges. Chapter 11 Chapter 11

Figure 11. 7 Stemplot of the Illinois tuition and fee data Figure 11.7 Stemplot of the Illinois tuition and fee data. (Data from the Web site www.collegeillinois.com/en/collegefunding/costs.htm. This figure was created using the Minitab software package.)

Statistics: Concepts and Controversies How to make a histogram? Chapter 11 Chapter 11

Statistics: Concepts and Controversies How to make a histogram? Step1: Divide the range of the data into classes of equal width. The data in this table range from 6.3 to 17.0, so we choose 6.0 ≤ percentage over 65 < 7.0 7.0 ≤ percentage over 65 < 8.0 … 17.0 ≤ percentage over 65 < 18.0 Make sure that the classes are exclusive (no individual is in more than one class), and exhaustive (every individual appears in some class) Chapter 11 Chapter 11

Statistics: Concepts and Controversies Step 2: Count the number of individuals in each class. Class Count 6.0 to 6.9 1 7.0 to 7.9 8.0 to 8.9 9.0 to 9.9 3 10.0 to 10.9 11.0 to 11.9 7 12.0 to 12.9 14 13.0 to 13.9 15 14.0 to 14.9 5 15.0 to 15.9 2 16.0 to 16.9 17.0 to 17.9 Chapter 11 Chapter 11

Statistics: Concepts and Controversies Step 3: Draw the histogram. Mark on the horizontal axis the scale for the variable whose distribution is to be displayed. The vertical axis contains the scale of counts. There is no horizontal space between bars unless a class is empty. Chapter 11 Chapter 11

Statistics: Concepts and Controversies Interpreting Histograms Chapter 11 Chapter 11

Statistics: Concepts and Controversies Outliers Extreme values, far from the rest of the data May occur naturally May occur due to error in recording May occur due to error in measuring Chapter 11 Chapter 11

Number of Books Read for Pleasure Statistics: Concepts and Controversies Number of Books Read for Pleasure Chapter 11 Chapter 11

Statistics: Concepts and Controversies Shape of the Data Symmetric bell-shaped other symmetric shapes Asymmetric skewed to the right skewed to the left Unimodal, bimodal Chapter 11 Chapter 11

Statistics: Concepts and Controversies Chapter 11 Chapter 11

Symmetric Distributions Bell-Shaped Statistics: Concepts and Controversies Symmetric Distributions Bell-Shaped Chapter 11 Chapter 11

Symmetric Distributions: Bell-Shaped Statistics: Concepts and Controversies Symmetric Distributions: Bell-Shaped Chapter 11 Chapter 11

Symmetric Distributions Mound-Shaped Statistics: Concepts and Controversies Symmetric Distributions Mound-Shaped Chapter 11 Chapter 11

Symmetric Distributions Uniform Statistics: Concepts and Controversies Symmetric Distributions Uniform Chapter 11 Chapter 11

Asymmetric Distributions Skewed to the Left Statistics: Concepts and Controversies Asymmetric Distributions Skewed to the Left Chapter 11 Chapter 11

Asymmetric Distributions Skewed to the Right Statistics: Concepts and Controversies Asymmetric Distributions Skewed to the Right Chapter 11 Chapter 11

Asymmetric Distributions: Skewed to the Right Statistics: Concepts and Controversies Asymmetric Distributions: Skewed to the Right Chapter 11 Chapter 11

Statistics: Concepts and Controversies How to make a stemplot? Chapter 11 Chapter 11

Statistics: Concepts and Controversies Chapter 11 Chapter 11

Statistics: Concepts and Controversies Chapter 11 Chapter 11

Statistics: Concepts and Controversies Example: Weight Data Following is a list of weights of a sample of students Chapter 11 Chapter 11

Example: Weight Data; Frequency Table Statistics: Concepts and Controversies Example: Weight Data; Frequency Table * Left endpoint is included in the group, right endpoint is not. Chapter 11 Chapter 11

Example: Weight Data; Histogram Statistics: Concepts and Controversies Example: Weight Data; Histogram 100 120 140 160 180 200 220 240 260 280 Weight * Left endpoint is included in the group, right endpoint is not. Chapter 11 Chapter 11

Weight Data: Stemplot (Stem and Leaf Plot) Statistics: Concepts and Controversies 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Weight Data: Stemplot (Stem and Leaf Plot) 5 2 Key 20|3 means 203 pounds Stems = 10’s Leaves = 1’s 2 Chapter 11 Chapter 11

Weight Data: Stemplot (Stem and Leaf Plot) Statistics: Concepts and Controversies 10 0166 11 009 12 0034578 13 00359 14 08 15 00257 16 555 17 000255 18 000055567 19 245 20 3 21 025 22 0 23 24 25 26 0 Weight Data: Stemplot (Stem and Leaf Plot) Key 20|3 means 203 pounds Stems = 10’s Leaves = 1’s Chapter 11 Chapter 11

Statistics: Concepts and Controversies Key Concepts Displays (Stemplots & Histograms) Graph Shapes Symmetric Skewed to the Right Skewed to the Left Outliers Chapter 11 Chapter 11

Example: Shakespeare’s words Statistics: Concepts and Controversies Example: Shakespeare’s words The following figure shows the distributions of lengths of words used in Shakespeare’s plays. Notice that the vertical scale is not the count of words but the percentage of all words that have each length. The curve is skewed to the right which is natural because short words are common and long ones are rare. Chapter 11 Chapter 11

Statistics: Concepts and Controversies Chapter 11 Chapter 11

Statistics: Concepts and Controversies Exercise 11.3 Lightning strikes. The following figure comes from a study of lightning storms in Colorado. It shows the distribution of the hour of the day during which the first lightning flash for that day occurred. Describe the shape, center, and spread of this distribution. Are there any outliers? Chapter 11 Chapter 11

Statistics: Concepts and Controversies Chapter 11 Chapter 11

Statistics: Concepts and Controversies Exercise 11.4 Where do the young live? The following figure is a stemplot of the percentage of the residents aged under 18 in each of the 50 states. The stems are whole percentages and the leaves are tenths of a percent. Chapter 11 Chapter 11

Figure 11.10 Stemplot of the percentage of each state’s residents who are under 18 years old. (This figure was created using the Minitab software package.)

Statistics: Concepts and Controversies Utah has the largest percentage of young adults. What is the percentage for this state? Ignoring Utah, describe the shape, center, and the spread of this distribution. Is the distribution for young adults more or less spread out than the distribution in the figure for older adults? Chapter 11 Chapter 11

Figure 11. 6 Making a stemplot of the data in Table 11. 1 Figure 11.6 Making a stemplot of the data in Table 11.1. Whole percents form the stems, and tenths of a percent form the leaves. (This figure was created using the Minitab software package.)

Answer for 11.4 Utah has 31.0% young adults Without Utah , the distribution is roughly symmetric, centered at about 24.2%, spread from 21.2% to 27.6% The distribution of young adult is less spread out than the distribution of older adults. Chapter 11

Statistics: Concepts and Controversies Exercise 11.5 Minority students in engineering. The following figure is a histogram of the number of minority students (black, Hispanic, Native American) who earned doctorate degrees in engineering from each of the 152 universities in the years 2000 through 2002. Briefly describe the shape, center, and spread of this distribution. Chapter 11 Chapter 11

Figure 11.11 The distribution of number of engineering doctorates earned by minority students at 152 universities, 2000 to 2002. (Data from the 2003 National Science Foundation Survey of Earned Doctorates, found at the Web site webcaspar.nsf.gov/. This figure was created using the SPSS software package.)

Statistics: Concepts and Controversies Exercise 11.6 Returns on common stocks. The total return on a stock is the change in its market price plus any dividend payments made. Total return is usually expressed as a percentage of the beginning price. The following figure is a histogram of the distribution of total returns for all 1528 common stocks listed on the New York Stock Exchange in one year. Chapter 11 Chapter 11

Figure 11.12 The distribution of total returns for all New York Stock Exchange common stocks in one year. (Based on J. K. Ford, “Diversification: how many stocks will suffice?” American Association of Individual Investors Journal, January 1990, pp. 14–16.)

Statistics: Concepts and Controversies Describe the overall shape of the distribution of total returns. What is the approximate center of this distribution? Approximately what were the smallest and largest total returns? (This describes the spread of the distribution.) A return less than zero means that owners of stock lost money. About what percentage of all stocks lost money? Chapter 11 Chapter 11

Statistics: Concepts and Controversies Answer for 11.6 The distribution is roughly symmetric The center is about 15% The smallest return was between -70% and -60%, while the largest was between 100% and 110% About (1+1+1+1+3+5+11=23%) of stocks lost money. Chapter 11 Chapter 11

Statistics: Concepts and Controversies Exercise 11.8 Automobile fuel economy. Government regulations require automakers to give the city and highway gas mileages for each model of car. The following table gives the highway mileage (miles per gallon) for 31 model year 2004 midsize cars. Make a stemplot of the highway gas mileages of these cars. What can you say about the overall shape of the distribution? Where is the center (the value such that half the cars have better gas mileage and half have worse gas mileage)? Two of these cars are subject to the “gas guzzler tax” because of their low gas mileage. Which two? Chapter 11 Chapter 11

Model MPG Acura RL 24 Lexus ES350 27 Bentley Arnage 15 Lexus GS460 BMW 535i 26 Lincoln Town Car 23 Buick Lacrosse Maybach 57 16 Cadillac CTS Maybach 62 Cadillac STS Mazda 6 Chevy Malibu 30 Benz E350 Chrysler Sebring Benz E550 22 Dodge Avenger Nissan Maxima 25 Honda Accord 31 Pontiac Grand Prix 28 Hyundai Sonata Rolls Royce Phantom 18 Infiniti G35 Saturn Aura Infiniti M35 Toyota Camry Jaguar S-Type R Volkswagen Passat Kia Optima Volvo S80 AWD Kia Spectra 32 Chapter 11