+ Chapter 1: Exploring Data Section 1.2 Displaying Quantitative Data with Graphs One of the simplest graphs to construct and interpret is a dotplot. Each data value is shown as a dot above its location on a number line. How to Make a Dotplot 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. Number of Goals Scored Per Game by the 2004 US Women’s Soccer Team 3 0 2 7 8 2 4 3 5 1 1 4 5 3 1 1 3 3 3 2 1 2 2 2 4 3 5 6 1 5 5 1 1 5 Displaying Quantitative Data + Dotplots The purpose of a graph is to help us understand the data. After you make a graph, always ask, “What do I see?” How to Examine the Distribution of a Quantitative Variable 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 •Outliers •Center •Spread Don’t forget your SOCS! + Examining the Distribution of a Quantitative Variable Displaying Quantitative Data When you describe a distribution’s shape, concentrate on the 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 Displaying Quantitative Data Shape + Describing Shape In the real world, distributions of data are almost never perfectly symmetric. If distributions are close to symmetric, then describe the shape as “roughly” or “approximately” symmetric. Displaying Quantitative Data Symmetric + Describing When you describe a distribution’s shape, concentrate on the main features. Look for rough symmetry or clear skewness. Unimodal Bimodal Displaying Quantitative Data Shape + Describing + this data The table and dotplot below displays the Environmental Protection Agency’s estimates of highway gas mileage in miles per gallon (MPG) for a sample of 24 model year 2009 midsize cars. 2009 Fuel Economy Guide MODEL 2009 Fuel Economy Guide 2009 Fuel Economy Guide MPG MPG MODEL <new>MODEL MPG 1 Acura RL 922 Dodge Avenger 1630 Mercedes- Benz E350 24 2 Audi A6 Quattro 1023 Hyundai Elantra 1733 Mercury Milan 29 3 Bentley Arnage 1114 Jaguar XF 1825 Mitsubishi Galant 27 4 BMW 5281 1228 Kia Optima 1932 Nissan Maxima 26 5 Buick Lacrosse 1328 Lexus GS 350 2026 Rolls Royce Phantom 18 6 Cadillac CTS 1425 Lincolon MKZ 2128 Saturn Aura 33 7 Chevrolet Malibu 1533 Mazda 6 2229 Toyota C amry 31 8 Chrysler Sebring 1630 Mercedes-Benz E350 2324 Volkswagen Passat 29 9 Dodge Avenger 1730 Mercury Milan 2429 Volvo S80 25 <new> Describe the shape, center, and spread of the distribution. Are there any outliers? Displaying Quantitative Data Examine + Check Your Understanding page 29 U.K Place South Africa Example, page 30 Compare the distributions of household size for these two countries. Don’t forget your SOCS! Displaying Quantitative Data Distributions Some of the most interesting statistics questions involve comparing two or more groups. Always discuss shape, center, spread, and possible outliers whenever you compare distributions of a quantitative variable. + Comparing South Africa Place U.K Don’t discuss the SOCS for each distribution separately! Use “about the same as” or “is much greater than” when comparing center and spread. Displaying Quantitative Data Example, page 30 Distributions + Comparing Another simple graphical display for small data sets is a stemplot. Stemplots give us a quick picture of the distribution while including the actual numerical values. How to Make a Stemplot 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 (Stem-and-Leaf Plots) + Stemplots These data represent the responses of 20 female AP Statistics students to the question, “How many pairs of shoes do you have?” Construct a stemplot. 50 26 26 31 57 19 24 22 23 38 13 50 13 34 23 30 49 13 15 51 1 1 93335 1 33359 2 2 664233 2 233466 3 3 1840 3 0148 4 4 9 4 9 5 5 0701 5 0017 Stems Add leaves Order leaves Key: 4|9 represents a female student who reported having 49 pairs of shoes. Add a key Displaying Quantitative Data (Stem-and-Leaf Plots) + Stemplots Stems and Back-to-Back Stemplots When data values are “bunched up”, we can get a better picture of the distribution by splitting stems. Two distributions of the same quantitative variable can be compared using a back-to-back stemplot with common stems. Females Males 50 26 26 31 57 19 24 22 23 38 14 7 6 5 12 38 8 7 10 10 13 50 13 34 23 30 49 13 15 51 10 11 4 5 22 7 5 10 35 7 0 0 1 1 2 2 3 3 4 4 5 5 Females “split stems” 333 95 4332 66 410 8 9 100 7 Males 0 0 1 1 2 2 3 3 4 4 5 5 4 555677778 0000124 2 58 Key: 4|9 represents a student who reported having 49 pairs of shoes. Displaying Quantitative Data + Splitting + Check Your Understanding page 32 Quantitative variables often take many values. A graph of the distribution may be clearer if nearby values are grouped together. The most common graph of the distribution of one quantitative variable is a histogram. How to Make a Histogram 1)Divide the range of data into classes of equal width. 2)Find the count or percent 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 a Histogram The table on page 35 presents data on the percent of residents from each state who were born outside of the U.S. Class Count 0 to <5 20 5 to <10 13 10 to <15 9 15 to <20 5 20 to <25 2 25 to <30 1 Total 50 Number of States Frequency Table Percent of foreign-born residents Displaying Quantitative Data Making + Example, page 34 + Check Your Understanding page 38 Here are several cautions based on common mistakes students make when using histograms. Cautions 1)Don’t confuse histograms and bar graphs. 2)Use percents instead of counts on the vertical axis when comparing distributions with different numbers of observations. 3)LABEL YOUR AXES!!! Displaying Quantitative Data Histograms Wisely + Using + Check Your Understanding page 40 + Section 1.2 Homework Pages 41-48 40, 43, 47, 58