1. + Chapter 1: Exploring Data Section 1.2 Displaying Quantitative Data with Graphs The Practice of Statistics, 4 th edition - For AP* STARNES, YATES, MOORE

2. + Chapter 1 Exploring Data Introduction: Data Analysis: Making Sense of Data 1.1Analyzing Categorical Data 1.2Displaying Quantitative Data with Graphs 1.3Describing Quantitative Data with Numbers

3. + Section 1.2 Displaying Quantitative Data with Graphs After this section, you should be able to… CONSTRUCT and INTERPRET dotplots, stemplots, and histograms DESCRIBE the shape of a distribution COMPARE distributions USE histograms wisely Learning Objectives

4. + 1)Draw a horizontal axis (a number line) and label it with the variable name. 2)Scale the axis from the minimum to the maximum value. 3)Mark a dot above the location on the horizontal axis corresponding to each data value. Displaying Quantitative Data Dotplots One of the simplest graphs to construct and interpret is adotplot. Each data value is shown as a dot above its location on a number line. How to Make a Dotplot Number of Goals Scored Per Game by the 2004 US Women’s Soccer Team 30278243511453113 33212224356155115

5. + Examining the Distribution of a Quantitative Variable The purpose of a graph is to help us understand the data. Afteryou make a graph, always ask, “What do I see?” In any graph, look for the overall pattern and for striking departures from that pattern. Describe the overall pattern of a distribution by its: Shape Center Spread Note individual values that fall outside the overall pattern. These departures are called outliers. How to Examine the Distribution of a Quantitative Variable Displaying Quantitative Data Don’t forget your SOCS!

6. + Examine this data The table and dotplot below displays the EnvironmentalProtection Agency’s estimates of highway gas mileage in milesper gallon (MPG) for a sample of 24 model year 2009 midsizecars. Displaying Quantitative Data Describe the shape, center, and spread of the distribution. Are there any outliers? Example, page 28

7. + Examine this data Here is the estimated battery life for each of 9 different smartphones (in minutes) according to http://cell- phones.toptenreviews.com/smartphones/. http://cell- phones.toptenreviews.com/smartphones/ Displaying Quantitative Data Describe the shape, center, and spread of the distribution. Are there any outliers? Solution: Shape: There is a peak at 300 and the distribution has a long tail to the right (skewed to the right). Center: The middle value is 330 minutes. Spread: The range is 460 – 300 = 160 minutes. Outliers: There is one phone with an unusually long battery life, the HTC Droid at 460 minutes. Alternate Example Smart PhoneBattery Life (minutes) Apple iPhone300 Motorola Droid385 Palm Pre300 Blackberry Bold360 Blackberry Storm330 Motorola Cliq360 Samsung Moment330 Blackberry Tour300 HTC Droid460

8. + Displaying Quantitative Data Describing Shape When you describe a distribution’s shape, concentrate onthe main features. Look for rough symmetry or clear skewness. Definitions: A distribution is roughly symmetric if the right and left sides of the graph are approximately mirror images of each other. A distribution is skewed to the right (right-skewed) if the right side of the graph (containing the half of the observations with larger values) is much longer than the left side. It is skewed to the left (left-skewed) if the left side of the graph is much longer than the right side. Symmetric Skewed-left Skewed-right

9. + Displaying Quantitative Data Alternate Example Here are dotplots for two different sets of quantitative data. The first dotplot shows the outcomes of 100 rolls of a 10-sided die. This distribution is roughly symmetric with no obvious modes. Don’t worry about the small differences in the number of dots for each die roll—this is bound to happen just by chance even if the frequencies should be the same. A distribution with this shape can be called “approximately uniform.” The second dotplot shows the number of calories in one serving of whole wheat or multigrain bread. This distribution is skewed to the left with a peak at 220 calories.

10. + Displaying Quantitative Data Comparing Distributions Some of the most interesting statistics questionsinvolve comparing two or more groups. Always discuss shape, center, spread, andpossible outliers whenever you comparedistributions of a quantitative variable. Example, page 32 Compare the distributions of household size for these two countries. Don’t forget your SOCS! Place U.K South Africa

11. + Displaying Quantitative Data Alternate Example How do the annual energy costs (in dollars) compare for refrigeratorswith top freezers and refrigerators with bottom freezers? The databelow is from the May 2010 issue of Consumer Reports. Compare the distributions of annual energy costs for these two types of refrigerators. Don’t forget your SOCS! Solution: Shape: The distribution for bottom freezers looks skewed to the right and possibly bimodal, with modes near $58 and $70 per year. The distribution for top freezers looks roughly symmetric with its main peak centered around $55. Center: The typical energy cost for the bottom freezers is higher than the typical cost for the top freezers (median of $69 vs. median of $55). Spread: There is much more variability in the energy costs for bottom freezers, since the range is $101 compared to $17 for the top freezers. Outliers: There are a couple of bottom freezers with unusually high energy costs (over $140 per year). There are no outliers for the top freezers.

12. + 1)Separate each observation into a stem (all but the final digit) and a leaf (the final digit). 2)Write all possible stems from the smallest to the largest in a vertical column and draw a vertical line to the right of the column. 3)Write each leaf in the row to the right of its stem. 4)Arrange the leaves in increasing order out from the stem. 5)Provide a key that explains in context what the stems and leaves represent. Displaying Quantitative Data Stemplots (Stem-and-Leaf Plots) Another simple graphical display for small data sets is astemplot. Stemplots give us a quick picture of the distributionwhile including the actual numerical values. How to Make a Stemplot

13. + Displaying Quantitative Data Stemplots (Stem-and-Leaf Plots) These data represent the responses of 20 female APStatistics students to the question, “How many pairs ofshoes do you have?” Construct a stemplot. 5026 31571924222338 13501334233049131551 Stems 1234512345 Add leaves 1 93335 2 664233 3 1840 4 9 5 0701 Order leaves 1 33359 2 233466 3 0148 4 9 5 0017 Add a key Key: 4|9 represents a female student who reported having 49 pairs of shoes.

14. + Displaying Quantitative Data Splitting Stems and Back-to-Back Stemplots When data values are “bunched up”, we can get a better picture ofthe distribution by splitting stems. Two distributions of the same quantitative variable can becompared using a back-to-back stemplot with common stems. 5026 31571924222338 13501334233049131551 001122334455001122334455 Key: 4|9 represents a student who reported having 49 pairs of shoes. Females 1476512388710 1145227510357 Males 0 4 0 555677778 1 0000124 1 2 3 3 58 4 5 Females 333 95 4332 66 410 8 9 100 7 Males “split stems”

15. + Displaying Quantitative Data Alternate Example – Who’s Taller? Which gender is taller, males or females? A sample of 14-year-oldsfrom the United Kingdom was randomly selected using theCensusAtSchool website. Here are the heights of the students (in cm): Male: 154, 157, 187, 163, 167, 159, 169, 162, 176, 177, 151, 175,174, 165, 165, 183, 180 Female: 160, 169, 152, 167, 164, 163, 160, 163, 169, 157, 158,153, 161, 165, 165, 159, 168, 153, 166, 158, 158, 166 151479 16235579 174567 18037 1514 1579 1623 165579 174 567 1803 187 Key: 15|1 represents a male who is 151 cm tall. Here are two stemplots for Male heights, one with split stems: Here is a back-to-back stemplot comparing male and female heights: FemaleMale 3321514 988871579 4331001623 99876655165579 174 567 1803 187

16. + Displaying Quantitative Data Splitting Stems and Back-to-Back Stemplots When data values are “bunched up”, we can get a better picture ofthe distribution by splitting stems. Two distributions of the same quantitative variable can becompared using a back-to-back stemplot with common stems. 5026 31571924222338 13501334233049131551 001122334455001122334455 Key: 4|9 represents a student who reported having 49 pairs of shoes. Females 1476512388710 1145227510357 Males 0 4 0 555677778 1 0000124 1 2 3 3 58 4 5 Females 333 95 4332 66 410 8 9 100 7 Males “split stems”

17. + 1)Divide the range of data into classes of equal width. 2)Find the count (frequency) or percent (relative frequency) of individuals in each class. 3)Label and scale your axes and draw the histogram. The height of the bar equals its frequency. Adjacent bars should touch, unless a class contains no individuals. Displaying Quantitative Data Histograms Quantitative variables often take many values. A graph of thedistribution may be clearer if nearby values are groupedtogether. The most common graph of the distribution of onequantitative variable is a histogram. How to Make a Histogram

18. + Making a Histogram The table on page 35 presents data on the percent ofresidents from each state who were born outside of the U.S. Displaying Quantitative Data Example, page 35 Frequency Table ClassCount 0 to <520 5 to <1013 10 to <159 15 to <205 20 to <252 25 to <301 Total50 Percent of foreign-born residents Number of States

19. + Making a Histogram The table presents data on the average points scored pergame (PTSG) for the 30 NBA teams in the 2009-2010 regularseason.. Displaying Quantitative Data Calculator Example 101.799.295.397.5102.1102 106.594108.8102.4100.895.7 101.7102.596.597.798.292.4 100.2102.1101.5102.897.7110.2 98.1100101.4104.1104.296.2

20. + 1)Don’t confuse histograms and bar graphs. 2)Don’t use counts (in a frequency table) or percents (in a relative frequency table) as data. 3)Use percents instead of counts on the vertical axis when comparing distributions with different numbers of observations. 4)Just because a graph looks nice, it’s not necessarily a meaningful display of data. Displaying Quantitative Data Using Histograms Wisely Here are several cautions based on common mistakesstudents make when using histograms. Cautions

21. + In this section, we learned that… You can use a dotplot, stemplot, or histogram to show the distribution of a quantitative variable. When examining any graph, look for an overall pattern and for notable departures from that pattern. Describe the shape, center, spread, and any outliers. Don’t forget your SOCS! Some distributions have simple shapes, such as symmetric or skewed. The number of modes (major peaks) is another aspect of overall shape. When comparing distributions, be sure to discuss shape, center, spread, and possible outliers. Histograms are for quantitative data, bar graphs are for categorical data. Use relative frequency histograms when comparing data sets of different sizes. Summary Section 1.2 Displaying Quantitative Data with Graphs

22. + Looking Ahead… We’ll learn how to describe quantitative data with numbers. Mean and Standard Deviation Median and Interquartile Range Five-number Summary and Boxplots Identifying Outliers We’ll also learn how to calculate numerical summaries with technology and how to choose appropriate measures of center and spread. In the next Section…