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. + Ways to chart quantitative data Histogram, stemplots, dotplots, and boxplots These are summary graphs for a single variable. They are very useful to understand the pattern of variability in the data. Line graphs: time plots Use when there is a meaningful sequence, like time. The line connecting the points helps emphasize any change over time.

5. + Slide 4- 5 Dotplots A dotplot is a simple display. It just places a dot along an axis for each case in the data. The dotplot to the right shows Kentucky Derby winning times, plotting each race as its own dot. You might see a dotplot displayed horizontally or vertically.

6. + 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!

7. + 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. + 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

9. + Displaying Quantitative Data Comparing Distributions Some of the most interesting statistics questionsinvolve comparing two or more groups. Always discuss shape, center, spread, in context, and possible outliers whenever you compare distributions 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

10. + 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

11. + Constructing Stemplots 858 750735 699 620 571 516 859 454860464452435414 860 858398 395 390 367 365 861

12. + 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”

13. + BACK-TO-BACK STEMPLOTS OPEN YOUR BOOK TO PAGE 45 AND LET’S TRY # 48

14. + Stemplots are quick and dirty histograms that can easily be done by hand, therefore very convenient for smaller data sets. However, they are rarely found in scientific or laymen publications. When might you NOT want to use a Stemplot? Stemplots versus histograms

15. + Displaying quantitative data: Histograms Displays counts or percents Shows trend of data User defines number of classes Good for large data sets Does not display actual data values The bars have the same width and always touch (the edges of the bars are on class boundaries which are described below). The width of a bar represents a quantitative variable x, such as age rather than a category. The height of each bar indicates frequency. 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.

16. + 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 How to Make a Histogram To find the class width, First compute: Largest value - Smallest Value Desired number of classes Increase the value computed to the next highest whole, number even if the first value was a whole number. This will ensure the classes cover the data.

17. + How to create a histogram It is an iterative process – try and try again. What bin size should you use?  Not too many bins with either 0 or 1 counts Not overly summarized that you loose all the information Not so detailed that it is no longer a summary  rule of thumb: start with 5 to10 bins Look at the distribution and refine your bins (There isn’t a unique or “perfect” solution)

18. + Using the TI-83 to make histograms The TI-83 can be used to make histograms, and will allow you to change the location and widths of the ranges. Turn to Page 38 in your textbook and follow the directions in the Technology Corner.

19. + Using the TI-83 to make histogramsI You can change the size and location of the ranges by using the Window button Use the scale key to change the number of classes. Enter the CLASS WIDTH. Press the Graph button to see the results

20. + Histogram Tips Be sure to choose classes all the same width. Use your judgment in choosing classes to display the shape. Too few classes will give a 'skyskaper' graph; Too many will produce a 'pancake' graph.

21. + Not summarized enough Too summarized Same data set

22. + Slide 4- 22 Describing the Shape of a Histogram 1. Does the histogram have a single, central hump or several separated bumps? Humps in a histogram are called modes. A histogram with one main peak is dubbed unimodal; histograms with two peaks are bimodal; histograms with three or more peaks are called multimodal.

23. + Slide 4- 23 Humps and Bumps (cont.) A histogram that doesn’t appear to have any mode and in which all the bars are approximately the same height is called uniform:

24. + Most common distribution shapes A distribution is symmetric if the right and left sides of the histogram are approximately mirror images of each other. Symmetric distribution Complex, multimodal distribution  Not all distributions have a simple overall shape, especially when there are few observations. Skewed distribution  A distribution is skewed to the right if the right side of the histogram (side with larger values) extends much farther out than the left side. It is skewed to the left if the left side of the histogram extends much farther out than the right side.

25. + Slide 4- 25 Anything Unusual? Don’t forget to make note of any unusual features denoted in the shape of the distribution. Sometimes it’s the unusual features that tell us something interesting or exciting about the data. You should always mention any stragglers, or outliers, that stand off away from the body of the distribution. Are there any gaps in the distribution? If so, we might have data from more than one group.

26. + AlaskaFlorida Outliers Always look for outliers and try to explain them. For Example: The overall pattern is fairly symmetrical except for 2 states clearly not belonging to the main trend. Alaska and Florida have unusual representation of the elderly in their population. A large gap in the distribution is typically a sign of an outlier.

27. + Slide 4- 27 Histograms are Similar to Bar Graphs and so: A relative frequency histogram displays the percentage of cases in each bin instead of the count. Relative frequency histograms qre good for comparing distributions of unequal counts

28. + AP Statistics, Section 1.1, Part 428 Notice the shape does not change when comparing frequency and relative frequency Histograms

29. + 1)Although they are similar, 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

30. + Constructing Frequency Polygons Make a frequency table that includes class midpoints and frequencies. For each class place dots above class midpoint at the height of the class frequency. Put dots on horizontal axis one class width to left of first class midpoint, and one class width to right of of last midpoint. Connect dots with straight lines.

31. + Frequency Polygon

32. + Line graphs: time plots A trend is a rise or fall that persist over time, despite small irregularities. In a time plot, time always goes on the horizontal, x axis. We describe time series by looking for an overall pattern and for striking deviations from that pattern. In a time series: A pattern that repeats itself at regular intervals of time is called seasonal variation.

33. + RRetail price of fresh oranges over time This time plot shows a regular pattern of yearly variations. These are seasonal variations in fresh orange pricing most likely due to similar seasonal variations in the production of fresh oranges. There is also an overall upward trend in pricing over time. It could simply be reflecting inflation trends or a more fundamental change in this industry. Time is on the horizontal, x axis. The variable of interest—here “retail price of fresh oranges”— goes on the vertical, y axis.

34. + A time plot can be used to compare two or more data sets covering the same time period. The pattern over time for the number of flu diagnoses closely resembles that for the number of deaths from the flu, indicating that about 8% to 10% of the people diagnosed that year died shortly afterward from complications of the flu.

35. + A picture is worth a thousand words, BUT There is nothing like hard numbers.  Look at the scales. Scales matter How you stretch the axes and choose your scales can give a different impression.

36. + 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

37. + 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…