1. + Chapter 1: Exploring Data Section 1.2 Displaying Quantitative Data with Graphs

2. + 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

3. + 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!

4. + 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?

5. + Shape: In the dotplot, we can see three clusters of values, cars that get around 25 mpg, cars that getabout 28 to 30 mpg, and cars that get around 33mpg. We can also see large gaps between theAcura RL at 22 mpg, the Rolls Royce Phantom at18 mpg, and the Bentley Arnage at 14 mpg. Displaying Quantitative Data Center: The median is 28. So a “typical” model year 2009 midsize car got about 28 mpg on the highway. Spread: The highest value is 33 mpg and the lowest value is 14 mpg. The range is 19 mpg Outliers: We see two midsize cars with unusually low gas mileage ratings – the Bentley (14 mpg) and the Rolls Royce (18 mpg). These are potential outliers

6. + 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

8. + Checkpoint Describe the shape, center, and spread of the distribution. Also, identify any potential outliers.

9. + Shape: Skewed Right. Center: The median is 1.5 and the mean is 1.75 Spread: The number of siblings varies from 0 to 6. Outliers: There are two potential outliers; those students reporting 5 and 6 siblings.

10. + Displaying Quantitative Data Comparing Distributions Some of the most interestingstatistics questions involve comparingtwo or more groups. Always discuss shape, center,spread, and possible outlierswhenever you compare distributionsof a quantitative variable.

11. + Displaying Quantitative Data Example – Comparing Distributions How do the numbers of people living in households in theUnited Kingdom and South Africa compare? To help answerthis question, 50 students from each country were chosenand asked what was the size of their household. Below is adot plot of the household sizes reported by the surveyrespondents. Compare the distributions of household size for these two countries. Don’t forget your SOCS! Place U.K South Africa

12. + Shape: The distribution of household size for the U.K. sample is roughly symmetric and unimodal, while the distribution for the South Africa sample is skewed to the right and unimodal. Center: Household sizes for the South African students tended to be larger than for the U.K. students. The median household sizes for the two groups are 6 and 4 people respectively. Spread: There is more variability (greater spread) in the household sizes for the South African students than for the U.K. students. The range for the South African data is 23 people, while the range for the U.K. data is 4 people.

13. + Outliers: There don’t appear to be any potential outliers in the U.K. distribution. The South African distribution has two potential outliers in the right tail of the distribution – students who reported living in households with 15 and 26 people. (The U.K. households with 2 people actually will be classified as outliers when we introduce the procedure for computing outliers).

14. + 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

15. + 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.

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. + Displaying Quantitative Data Males 0 4 0 555677778 1 0000124 1 2 3 3 58 4 5 Females 0 1 333 1 95 2 4332 2 66 3 410 3 8 4 4 9 5 100 5 7

18. + Checkpoint 1) Use the back to back stemplot to write a few sentences comparing the number of pairs of shoes by males and females. Be sure to address shape, center, spread and outliers. In general, it appears that females have more pairs of shoes than males. The median report for the males was 9 pairs while the female median was 26 pairs. The females also have a larger range of 44 in comparison to the range of 34 for the males. Finally, both males and females have distributions that are skewed to the right, though the distribution for the males is more heavily skewed, as evidenced by the three likely outliers at 22, 35, and 38. The females do not have any likely outliers.

19. + Checkpoint Multiple choice: Select the best answer for questions 2 through 4. Here is a stemplot of the percents of residents aged 65 and older in the 50 states and the District of Columbia. The stems are whole percents and the leaves are tenths of a percent.

20. + 2) The low outlier is Alaska. What percent of Alaska residents are 65 or older? [A] 0.68 [B] 6.8 [C] 8.8 [D] 16.8 [E] 68 3) Ignoring the outlier, the shape of the distribution is [A] skewed right [B] skewed left [C] skewed to the middle [D] roughly symmetric [E] bimodal 4) The center of the distribution is close to [A] 13.3% [B] 12.8% [C] 12.0% [D] 11.6% [E] 6.8% to 16.8%

21. + 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

22. + Example – Making a Histogram The table below presents the data for the percentage of people per state who were born outside the United States. The individuals in this data set are the states. The variable is the percent of a state’s residents who are foreign-born.

23. + Making a Histogram Displaying Quantitative Data 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

24. + Checkpoint Many people believe that the distribution of IQ scores follows a “bell curve.” But is this really how such scores are distributed? The IQ scores of 60 fifth-grade students chosen at random from one school are shown below: 1) Construct a histogram that displays the distribution of IQ scores effectively. 2) Describe what you see. Is the distribution bell-shaped? 14513912612212513096110118 10114213412411210913411381113 12394100136109131117110127124 106124115133116102127117109137 117901031141391011221059789 10210811012811411211410282101

25. + 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

26. + Checkpoint Questions 1 and 2 relate to the following setting: About 1.6 million first-year students enroll in colleges and universities each year. What do they plan to study? The graph displays the data on the percents of first-year students who plan to major in several discipline areas. 1) Is this a bar graph or a histogram? Explain. 2) Would it be correct to describe this distribution as right skewed?. Why or why not?

27. + 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