Chapter 1 Exploring Data

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

2. IntroductionData Analysis: Making Sense of Data Learning Objectives After this section, you should be able to… • DEFINE “Individuals” and “Variables” • DISTINGUISH between “Categorical” and “Quantitative” variables • DEFINE “Distribution” • DESCRIBE the idea behind “Inference”

3. Data Analysis • Statistics is the science of data. • Data Analysis is the process of organizing, displaying, summarizing, and asking questions about data. Definitions: Individuals – objects (people, animals, things) described by a set of data Variable - any characteristic of an individual Categorical Variable – places an individual into one of several groups or categories. Quantitative Variable – takes numerical values for which it makes sense to find an average.

4. Read Example : Census at school ( P-3) Now do Exercise 3 on P- 7 Answers: a) Individuals: AP Stat students who completed a questionnaire on the first day of class. b) Categorical: gender, handedness, favorite type of music Quantitative: height( inches), amount of time the students expects to spend on HW (mins) , the total value of coin(cents) c) Female, right-handed, 58 inches tall, spends 60 mins on HW, prefers alternative music, had 76 cents in her pocket.

5. Do Activity : Hiring Discrimination – It just won’t fly …… • Follow the directions on Page 5 • Perform 5 repetitions of your simulation. • Turn in your results to your teacher. (Make Groups of 4)

6. A variable generally takes on many different values. In data analysis, we are interested in how often a variable takes on each value. Definition: Distribution – tells us what values a variable takes and how often it takes those values Example Dotplot of MPG Distribution Variable of Interest: MPG

7. How to Explore Data Data Analysis Examine each variable by itself. Then study relationships among the variables. Start with a graph or graphs Add numerical summaries

8. Data Analysis From Data Analysis to Inference Population Sample Collect data from a representative Sample... Make an Inference about the Population. Perform Data Analysis, keeping probability in mind…

9. Inference: Inference is the process of making a conclusion about a population based on a sample set of data.

10. IntroductionData Analysis: Making Sense of Data Summary In this section, we learned that… • A dataset contains information on individuals. • For each individual, data give values for one or more variables. • Variables can be categorical or quantitative. • The distribution of a variable describes what values it takes and how often it takes them. • Inference is the process of making a conclusion about a population based on a sample set of data.

11. Do Notes worksheet : First page

12. Looking Ahead… In the next Section… • We’ll learn how to analyze categorical data. • Bar Graphs • Pie Charts • Two-Way Tables • Conditional Distributions • We’ll also learn how to organize a statistical problem.

13. Section 1.1Analyzing Categorical Data Learning Objectives After this section, you should be able to… • CONSTRUCT and INTERPRET bar graphs and pie charts • RECOGNIZE “good” and “bad” graphs • CONSTRUCT and INTERPRET two-way tables • DESCRIBE relationships between two categorical variables • ORGANIZE statistical problems

14. Analyzing Categorical Data • Categorical Variables place individuals into one of several groups or categories • The values of a categorical variable are labels for the different categories • The distribution of a categorical variable lists the count or percent of individuals who fall into each category. Example, page 8 Variable Values Count Percent

15. Analyzing Categorical Data • Displaying categorical data Frequency tables can be difficult to read. Sometimes is is easier to analyze a distribution by displaying it with a bar graph or pie chart.

16. Graphs: Good and Bad Analyzing Categorical Data Bar graphs compare several quantities by comparing the heights of bars that represent those quantities. Our eyes react to the area of the bars as well as height. Be sure to make your bars equally wide. Avoid the temptation to replace the bars with pictures for greater appeal…this can be misleading! Alternate Example This ad for DIRECTV has multiple problems. How many can you point out?

17. Example: Bar graph • What personal media do you own? Here are the percents of 15-18 year olds who own the following personal media devices. a) Make a well-labeled bar graph. What do you see? b) Would it be appropriate to make a pie chart for these data? Why or why not?

18. Analyzing Categorical Data • Two-Way Tables and Marginal Distributions When a dataset involves two categorical variables, we begin by examining the counts or percents in various categories for one of the variables. Definition: Two-way Table – describes two categorical variables, organizing counts according to a row variable and a column variable. Example, p. 12 What are the variables described by this two-way table? How many young adults were surveyed?

19. Analyzing Categorical Data • Two-Way Tables and Marginal Distributions Definition: The Marginal Distribution of one of the categorical variables in a two-way table of counts is the distribution of values of that variable among all individuals described by the table. Note: Percents are often more informative than counts, especially when comparing groups of different sizes. To examine a marginal distribution, • Use the data in the table to calculate the marginal distribution (in percents) of the row or column totals. • Make a graph to display the marginal distribution.

20. Analyzing Categorical Data • Two-Way Tables and Marginal Distributions Example, p. 13 Examine the marginal distribution of chance of getting rich.

21. Relationships Between Categorical Variables • Marginal distributions tell us nothing about the relationship between two variables. Definition: A Conditional Distribution of a variable describes the values of that variable among individuals who have a specific value of another variable. To examine or compare conditional distributions, • Select the row(s) or column(s) of interest. • Use the data in the table to calculate the conditional distribution (in percents) of the row(s) or column(s). • Make a graph to display the conditional distribution. • Use a side-by-side bar graph or segmented bar graph to compare distributions.

22. Two-Way Tables and Conditional Distributions Example, p. 15 Calculate the conditional distribution of opinion among males. Examine the relationship between gender and opinion.

23. Analyzing Categorical Data • Organizing a Statistical Problem • As you learn more about statistics, you will be asked to solve more complex problems. • Here is a four-step process you can follow. How to Organize a Statistical Problem: A Four-Step Process State: What’s the question that you’re trying to answer? Plan: How will you go about answering the question? What statistical techniques does this problem call for? Do: Make graphs and carry out needed calculations. Conclude: Give your practical conclusion in the setting of the real-world problem.

24. Do : P- 25 # 26. State: Do the data support the idea that people who get angry easily tend to have more heart disease? Plan: We suspect that people with different anger levels will have different rates of CHD, so we will compare the conditional distributions of CHD for each anger level. Do: We will display the conditional distributions in a table to compare the rate of CHD occurrence for each of the 3 anger levels Conclude: The conditional distributions show that while CHD occurrence is quite small overall, the percent of the population with CHD does increase as eth anger level increases.

25. Section 1.1Analyzing Categorical Data Summary In this section, we learned that… • The distribution of a categorical variable lists the categories and gives the count or percent of individuals that fall into each category. • Pie charts and bar graphs display the distribution of a categorical variable. • A two-way table of counts organizes data about two categorical variables. • The row-totals and column-totals in a two-way table give the marginal distributions of the two individual variables. • There are two sets of conditional distributions for a two-way table.

26. Section 1.1Analyzing Categorical Data Summary, continued In this section, we learned that… • We can use a side-by-side bar graph or a segmented bar graph to display conditional distributions. • To describe the association between the row and column variables, compare an appropriate set of conditional distributions. • Even a strong association between two categorical variables can be influenced by other variables lurking in the background. • You can organize many problems using the four steps state, plan, do, and conclude.

27. Looking Ahead… In the next Section… • We’ll learn how to display quantitative data. • Dotplots • Stemplots • Histograms • We’ll also learn how to describe and compare distributions of quantitative data.

28. Section 1.2Displaying Quantitative Data with Graphs Learning Objectives 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

29. Displaying Quantitative Data • Dotplots • 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 • Draw a horizontal axis (a number line) and label it with the variable name. • Scale the axis from the minimum to the maximum value. • Mark a dot above the location on the horizontal axis corresponding to each data value.

30. Displaying Quantitative Data • Examining the Distribution of a Quantitative Variable • 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 • Center • Spread • Outliers • Note individual values that fall outside the overall pattern. These departures are called outliers. Don’t forget your SOCS!

31. Displaying Quantitative Data • Examine 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. Example, page 28 Describe the shape, center, and spread of the distribution. Are there any outliers?

32. Displaying Quantitative Data • Describing Shape • 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

33. You Do: Smart Phone Battery Life Here is the estimated battery life for each of 9 different smart phones (in minutes) according to http://cellphones.toptenreviews.com/smartphones/. • Make a dot plot. * Then describe the shape, center, and spread. of the distribution. Are there any outliers?

34. Solution: 300 300 300 330 330 360 360 385 460 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.

35. Comparing 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. Example, page 32 Compare the distributions of household size for these two countries. Don’t forget your SOCS! ( Compare each characteristic) Place U.K South Africa

36. Displaying Quantitative Data • Stemplots (Stem-and-Leaf Plots) • 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 • Separate each observation into a stem (all but the final digit) and a leaf (the final digit). • 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. • Write each leaf in the row to the right of its stem. • Arrange the leaves in increasing order out from the stem. • Provide a key that explains in context what the stems and leaves represent.

37. Displaying Quantitative Data • Stemplots (Stem-and-Leaf Plots) • 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. Key: 4|9 represents a female student who reported having 49 pairs of shoes. 1 93335 2 664233 3 1840 4 9 5 0701 1 33359 2 233466 3 0148 4 9 5 0017 1 2 3 4 5 Stems Add leaves Order leaves Add a key

38. Splitting 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 Females 333 95 4332 66 410 8 9 100 7 Males 0 4 0 555677778 1 0000124 1 2 2 2 3 3 58 4 4 5 5 0 0 1 1 2 2 3 3 4 4 5 5 “split stems” Key: 4|9 represents a student who reported having 49 pairs of shoes.

39. From both sides, compare and write about the center , spread , shape and possible outliers. Do CYU : P- 34 . • Check Your Understanding, page 34: • 1. 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 57-13=44 in comparison to the range of 38-4=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. • 2. b • 3. b • 4. b

40. Histograms • 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 • Divide the range of data into classes of equal width. • Find the count (frequency) or percent (relative frequency) of individuals in each class. • 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.

41. Example, page 35 Displaying Quantitative Data • Making 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. Number of States Percent of foreign-born residents

42. Displaying Quantitative Data • Using Histograms Wisely • Here are several cautions based on common mistakes students make when using histograms. Cautions • Don’t confuse histograms and bar graphs. • Don’t use counts (in a frequency table) or percents (in a relative frequency table) as data. ( P- 40: Read the example) • Use percents instead of counts on the vertical axis when comparing distributions with different numbers of observations. • Just because a graph looks nice, it’s not necessarily a meaningful display of data.

43. Try Check your understanding: P – 39 Many people……….. (Also put these #s in the calculator.) Solution: 2. The distribution is roughly symmetric and bell-shaped. The median IQ appears to be between 110 and 120 and the IQ’s vary from 80 to 150. There do not appear to be any outliers.

44. Section 1.2Displaying Quantitative Data with Graphs Summary 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.

45. Do # 56 P- 46

46. Looking Ahead… In the next Section… • 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.