1. Chapters 1 & 3 Graphical Methods for Describing Data

2. Mr. Potter Bachelors in Mathematics w/ Education Masters in Educational Leadership Doctoral Student, Psych of Cog/Instruction World Champion in Taekwondo Married 16 years, my son Kai is 13 in 8 th grade at McMillan, my daughter Kaitlin is 4 Turn in information sheet Tutoring times Electronics – no phones, no music- Calc’s Dress code – if I tell you to tuck, it will be documented Tardies

3. Prerequisites – successful completion of Algebra 2 or above Homework Supplies –TI-84 or TI-83 plus calculator – preferred is TI-84 AP Exam –Thursday, May 12, 2016 – 12:00 pm –Test: 40 Multiple choice questions 5 free response questions 1 investigative task –What is the passing rate?

4. What is statistics? the science of collecting, organizing, analyzing, and drawing conclusions from data

5. Why should one study statistics? 1.To be informed... a)Extract information from tables, charts and graphs b)Follow numerical arguments c)Understand the basics of how data should be gathered, summarized, and analyzed to draw statistical conclusions Can dogs help patients with heart failure by reducing stress and anxiety? When people take a vacation do they really leave work behind?

6. Why should one study statistics? (continued) 2.To make informed judgments 3.To evaluate decisions that affect your life If you choose a particular major, what are your chances of finding a job when you graduate? Many companies now require drug screening as a condition of employment. With these screening tests there is a risk of a false-positive reading. Is the risk of a false result acceptable?

7. What is variability? Suppose you went into a convenience store to purchase a soft drink. Does every can on the shelf contain exactly 12 ounces? NO – there may be a little more or less in the various cans due to the variability that is inherent in the filling process. universal! In fact, variability is almost universal! It is variability that makes life interesting!! The quality, state, or degree of being variable or changeable.

8. If the Shoe Fits... The two histograms to the right display the distribution of heights of gymnasts and the distribution of heights of female basketball players. Which is which? Why? Heights – Figure A Heights – Figure B

9. If the Shoe Fits... Suppose you found a pair of size 6 shoes left outside the locker room. Which team would you go to first to find the owner of the shoes? Why? Suppose a tall woman (5 ft 11 in) tells you see is looking for her sister who is practicing with a gym. To which team would you send her? Why?

10. The Data Analysis Process 1.Understand the nature of the problem 2.Decide what to measure and how to measure it 3.Collect data 4.Summarize data and perform preliminary analysis 5.Perform formal analysis 6.Interpret results It is important to have a clear direction before gathering data. It is important to carefully define the variables to be studied and to develop appropriate methods for determining their values. It is important to understand how data is collected because the type of analysis that is appropriate depends on how the data was collected! This initial analysis provides insight into important characteristics of the data. It is important to select and apply the appropriate inferential statistical methods This step often leads to the formulation of new research questions.

11. Suppose we wanted to know the average GPA of high school graduates in the nation this year. We could collect data from all high schools in the nation. What term would be used to describe “all high school graduates”?

12. Population The entire collection of individuals or objects about which information is desired A census is performed to gather about the entire population What do you call it when you collect data about the entire population?

13. GPA Continued: Suppose we wanted to know the average GPA of high school graduates in the nation this year. We could collect data from all high schools in the nation. Why might we not want to use a census here? If we didn’t perform a census, what would we do?

14. Sample A subset of the population, selected for study in some prescribed manner What would a sample of all high school graduates across the nation look like? High school graduates from each state (region), ethnicity, gender, etc.

15. GPA Continued: Suppose we wanted to know the average GPA of high school graduates in the nation this year. We could collect data from a sample of high schools in the nation. Once we have collected the data, what would we do with it?

16. Descriptive statistics the methods of organizing & summarizing data Create a graph If the sample of high school GPAs contained 1,000 numbers, how could the data be organized or summarized? State the range of GPAs Calculate the average GPA

17. GPA Continued: Suppose we wanted to know the average GPA of high school graduates in the nation this year. We could collect data from a sample of high schools in the nation. Could we use the data from our sample to answer this question?

18. Inferential statistics involves making generalizations from a sample to a population Based on the sample, if the average GPA for high school graduates was 3.0, what generalization could be made? The average national GPA for this year’s high school graduate is approximately 3.0. Could someone claim that the average GPA for graduates in your local school district is 3.0? No. Generalizations based on the results of a sample can only be made back to the population from which the sample came from. Be sure to sample from the population of interest!!

19. Variable any characteristic whose value may change from one individual to another Suppose we wanted to know the average GPA of high school graduates in the nation this year. Define the variable of interest. The variable of interest is the GPA of high school graduates Is this a variable... The number of wrecks per week at the intersection outside school? YES

20. Data The values for a variable from individual observations For this variable... The number of wrecks per week at the intersection outside... What could observations be? 0, 1, 2, …

21. Two types of variables categoricalnumerical discretecontinuous

22. Categorical variables Qualitative Identifies basic differentiating characteristics of the population Can you name any categorical variables?

23. Numerical variables quantitative observations or measurements take on numerical values makes sense to average these values two types - discrete & continuous Can you name any numerical variables?

24. Discrete (numerical) Isolated points along a number line usually counts of items

25. Continuous (numerical) Variable that can be any value in a given interval usually measurements of something

26. Identify the following variables: 1.the color of cars in the teacher’s lot 2.the number of calculators owned by students at your school 3.the zip code of an individual 4.the amount of time it takes students to drive to school 5.the appraised value of homes in your city Categorical discrete numerical Discrete numerical Continuous numerical Is money a measurement or a count?

27. Classifying variables by the number of variables in a data set Suppose that the PE coach records the height of each student in his class. Univariate - data that describes a single characteristic of the population This is an example of a univariate data

28. Classifying variables by the number of variables in a data set Suppose that the PE coach records the height and weight of each student in his class. Bivariate - data that describes two characteristics of the population This is an example of a bivariate data

29. Classifying variables by the number of variables in a data set Suppose that the PE coach records the height, weight, number of sit-ups, and number of push-ups for each student in his class. Multivariate - data that describes more than two characteristics (beyond the scope of this course) This is an example of a multivariate data

30. Graphs for categorical data

31. Bar Chart When to UseCategorical data How to construct –Draw a horizontal line; write the categories or labels below the line at regularly spaced intervals –Draw a vertical line; label the scale using frequency or relative frequency –Place equal-width rectangular bars above each category label with a height determined by its frequency or relative frequency

32. Bar Chart (continued) What to Look For Frequently or infrequently occurring categories Collect the following data and then display the data in a bar chart: What is your favorite ice cream flavor? Vanilla, chocolate, strawberry, or other

33. Double Bar Charts When to UseCategorical data How to construct –Constructed like bar charts, but with two (or more) groups being compared – MUST use relative frequencies on the vertical axis – MUST include a key to denote the different bars MUST Why MUST we use relative frequencies?

34. Each year the Princeton Review conducts a survey of students applying to college and of parents of college applicants. In 2009, 12,715 high school students responded to the question “Ideally how far from home would you like the college you attend to be?” Also, 3007 parents of students applying to college responded to the question “how far from home would you like the college your child attends to be?” Data is displayed in the frequency table below. Frequency Ideal DistanceStudentsParents Less than 250 miles44501594 250 to 500 miles3942902 500 to 1000 miles2416331 More than 1000 miles1907180 Create a comparative bar chart with these data. What should you do first?

35. Relative Frequency Ideal DistanceStudentsParents Less than 250 miles.35.53 250 to 500 miles.31.30 500 to 1000 miles.19.11 More than 1000 miles.15.06 Found by dividing the frequency by the total number of students Found by dividing the frequency by the total number of parents What does this graph show about the ideal distance college should be from home?

36. Segmented (or Stacked) Bar Charts When to UseCategorical data How to construct – MUST first calculate relative frequencies – Draw a bar representing 100% of the group – Divide the bar into segments corresponding to the relative frequencies of the categories

37. Relative Frequency Ideal DistanceStudentsParents Less than 250 miles.35.53 250 to 500 miles.31.30 500 to 1000 miles.19.11 More than 1000 miles.15.06 Remember the Princeton survey... Create a segmented bar graph with these data. First draw a bar that represents 100% of the students who answered the survey.

38. Less than 250 miles 250 to 500 miles 500 to 1000 miles More than 1000 miles Relative Frequency Ideal DistanceStudentsParents Less than 250 miles.35.53 250 to 500 miles.31.30 500 to 1000 miles.19.11 More than 1000 miles.15.06 First draw a bar that represents 100% of the students who answered the survey. Relative frequency Students Next, divide the bar into segments. Do the same thing for parents – don’t forget a key denoting each category Parents Notice that this segmented bar chart displays the same relationship between the opinions of students and parents concerning the ideal distance that college is from home as the double bar chart does.

39. Pie (Circle) Chart When to Use Categorical data How to construct –Draw a circle to represent the entire data set –Calculate the size of each “slice”: Relative frequency × 360° –Using a protractor, mark off each slice To describe – comment on which category had the largest proportion or smallest proportion

40. Typos on a résumé do not make a very good impression when applying for a job. Senior executives were asked how many typos in a résumé would make them not consider a job candidate. The resulting data are summarized in the table below. Number of TyposFrequencyRelative Frequency 160.40 254.36 321.14 4 or more10.07 Don’t know5.03 Create a pie chart for these data.

41. Number of TyposFrequencyRelative Frequency 160.40 254.36 321.14 4 or more10.07 Don’t know5.03 First draw a circle to represent the entire data set. Next, calculate the size of the slice for “1 typo”.40×360º =144º Draw that slice. Repeat for each slice. Here is the completed pie chart created using Minitab. What does this pie chart tell us about the number of typos occurring in résumés before the applicant would not be considered for a job?

42. Graphs for numerical data

43. Dotplot When to UseSmall numerical data sets How to construct –Draw a horizontal line and mark it with an appropriate numerical scale –Locate each value in the data set along the scale and represent it by a dot. If there are two are more observations with the same value, stack the dots vertically

44. Dotplot (continued) What to Look For –The representative or typical value –The extent to which the data values spread out –The nature of the distribution along the number line –The presence of unusual values Collect the following data and then display the data in a dotplot: How many body piercings do you have?

45. How to describe a numerical, univariate graph

46. What strikes you as the most distinctive difference among the distributions of exam scores in classes A, B, & C ?

47. 1. Center discuss where the middle of the data falls three measures of central tendency –mean, median, & mode The mean and/or median is typically reported rather than the mode.

48. What strikes you as the most distinctive difference among the distributions of scores in classes D, E, & F?

49. 2. Spread discuss how spread out the data is refers to the variability in the data Measure of spread are –Range, standard deviation, IQR Remember, Range = maximum value – minimum value Standard deviation & IQR will be discussed in Chapter 4

50. What strikes you as the most distinctive difference among the distributions of exam scores in classes G, H, & I ?

51. 3. Shape refers to the overall shape of the distribution The following slides will discuss these shapes.

52. Symmetrical refers to data in which both sides are (more or less) the same when the graph is folded vertically down the middle bell-shaped is a special type –has a center mound with two sloping tails 1. Collect data by rolling two dice and recording the sum of the two dice. Repeat three times. 2. Plot your sums on the dotplot on the board. 3. What shape does this distribution have?

53. Uniform refers to data in which every class has equal or approximately equal frequency 1. Collect data by rolling a single die and recording the number rolled. Repeat five times. 2. Plot your numbers on the dotplot on the board. 3. What shape does this distribution have? To help remember the name for this shape, picture soldier standing in straight lines. What are they wearing?

54. Skewed refers to data in which one side (tail) is longer than the other side the direction of skewness is on the side of the longer tail 1. Collect data finding the age of five coins in circulation (current year minus year of coin) and record 2. Plot the ages on the dotplot on the board. 3. What shape does this distribution have? The directions are right skewed or left skewed. Name a variable with a distribution that is skewed left.

55. Bimodal (multi-modal) refers to the number of peaks in the shape of the distribution Bimodal would have two peaks Multi-modal would have more than two peaks Bimodal distributions can occur when the data set consist of observations from two different kinds of individuals or objects. Suppose collect data on the time it takes to drive from San Luis Obispo, California to Monterey, California. Some people may take the inland route (approximately 2.5 hours) while others may take the coastal route (between 3.5 and 4 hours). What shape would this distribution have? What would a distribution be called if it had ONLY one peak? Unimodal

56. 3. Shape refers to the overall shape of the distribution symmetrical, uniform, skewed, or bimodal

57. What strikes you as the most distinctive difference among the distributions of exam scores in class J ?

58. 4. Unusual occurrences Outlier - value that lies away from the rest of the data Gaps Clusters

59. 5. In context You must write your answer in reference to the context in the problem, using correct statistical vocabulary and using complete sentences!

60. Dotplot (continued) What to Look For –The representative or typical value –The extent to which the data values spread out –The nature of the distribution along the number line –The presence of unusual values Collect the following data and then display the data in a dotplot: How many body piercings do you have? Describe the distribution of the number of body piercings the class has.

61. Numerical Graphs Continued

62. Stem-and-Leaf Displays When to Use Univariate numerical data How to construct –Select one or more of the leading digits for the stem –List the possible stem values in a vertical column –Record the leaf for each observation beside each corresponding stem value –Indicate the units for stems and leaves in a key or legend To describe – comment on the center, spread, and shape of the distribution and if there are any unusual features Each number is split into two parts: Stem – consists of the first digit(s) Leaf - consists of the final digit(s) Use for small to moderate sized data sets. Doesn’t work well for large data sets. Be sure to list every stem from the smallest to the largest value If you have a long lists of leaves behind a few stems, you can split stems in order to spread out the distribution. Can also create comparative stem-and-leaf displays Remember the data set collected in Chapter 1 – how many piercings do you have? Would a stem-and-leaf display be a good graph for this distribution? Why or why not?

63. The following data are price per ounce for various brands of different brands of dandruff shampoo at a local grocery store. 0.320.210.290.540.170.280.360.23 Create a stem-and-leaf display with this data? StemLeaf 1 2 3 4 5 What would an appropriate stem be? List the stems vertically For the observation of “0.32”, write the 2 behind the “3” stem. 2 Continue recording each leaf with the corresponding stem 19 4 7 8 6 3 Describe this distribution. The median price per ounce for dandruff shampoo is $0.285, with a range of $0.37. The distribution is positively skewed with an outlier at $0.54.

64. The Census Bureau projects the median age in 2030 for the 50 states and Washington D.C. A stem-and-leaf display is shown below. Notice that you really cannot see a distinctive shape for this distribution due to the long list of leaves We can split the stems in order to better see the shape of the distribution. Notice that now you can see the shape of this distribution. We use L for lower leaf values (0-4) and H for higher leaf values (5-9).

65. The following is data on the percentage of primary-school-aged children who are enrolled in school for 19 countries in Northern Africa and for 23 countries in Central African. Northern Africa 54.6 34.348.977.859.688.597.492.583.9 98.891.697.896.192.294.998.686.696.9 88.9 Central Africa 58.334.635.545.438.663.853.961.969.9 43.085.063.458.461.940.973.934.874.4 97.461.066.779.6 Create a comparative stem- and-leaf display. What is an appropriate stem? Let’s truncate the leaves to the unit place. “4.6” becomes “4” Be sure to use comparative language when describing these distributions! The median percentage of primary-school-aged children enrolled in school is larger for countries in Northern Africa than in Central Africa, but the ranges are the same. The distribution for countries in Northern Africa is strongly negatively skewed, but the distribution for countries in Central Africa is approximately symmetrical.

66. Histograms When to UseUnivariate numerical data How to constructDiscrete data ―Draw a horizontal scale and mark it with the possible values for the variable ―Draw a vertical scale and mark it with frequency or relative frequency ―Above each possible value, draw a rectangle centered at that value with a height corresponding to its frequency or relative frequency To describe – comment on the center, spread, and shape of the distribution and if there are any unusual features Constructed differently for discrete versus continuous data For comparative histograms – use two separate graphs with the same scale on the horizontal axis

67. Queen honey bees mate shortly after they become adults. During a mating flight, the queen usually takes several partners, collecting sperm that she will store and use throughout the rest of her life. A study on honey bees provided the following data on the number of partners for 30 queen bees. 1224667878 11 835671019 7 6 975474678 10 Create a histogram for the number of partners of the queen bees.

68. First draw a horizontal axis, scaled with the possible values of the variable of interest. Next draw a vertical axis, scaled with frequency or relative frequency. Suppose we use relative frequency instead of frequency on the vertical axis. Draw a rectangle above each value with a height corresponding to the frequency. What do you notice about the shapes of these two histograms?

69. Histograms When to UseUnivariate numerical data How to constructContinuous data ―Mark the boundaries of the class intervals on the horizontal axis ―Draw a vertical scale and mark it with frequency or relative frequency ―Draw a rectangle directly above each class interval with a height corresponding to its frequency or relative frequency To describe – comment on the center, spread, and shape of the distribution and if there are any unusual features This is the type of histogram that most students are familiar with.

70. A study examined the length of hours spent watching TV per day for a sample of children age 1 and for a sample of children age 3. Below are comparative histograms. Children Age 1Children Age 3 Notice the common scale on the horizontal axis Write a few sentences comparing the distributions. The median number of hours spent watching TV per day was greater for the 1-year-olds than for the 3-year-olds. The distribution for the 3-year-olds was more strongly skewed right than the distribution for the 1-year-olds, but the two distributions had similar ranges.

71. Cumulative Relative Frequency Plot When to use - used to answer questions about percentiles. How to construct - Mark the boundaries of the intervals on the horizontal axis - Draw a vertical scale and mark it with relative frequency - Plot the point corresponding to the upper end of each interval with its cumulative relative frequency, including the beginning point - Connect the points. Percentiles are a value with a given percent of observations at or below that value.

72. The National Climatic Center has been collecting weather data for many years. The annual rainfall amounts for Albuquerque, New Mexico from 1950 to 2008 were used to create the frequency distribution below. Annual Rainfall (in inches) Relative frequency Cumulative relative frequency 4 to <50.052 5 to <60.103 6 to <70.086 7 to <80.103 8 to <90.172 9 to <100.069 10 to < 110.207 11 to <120.103 12 to <130.052 13 to <140.052 Find the cumulative relative frequency for each interval 0.052 0.155 0.241 + + Continue this pattern to complete the table

73. The National Climatic Center has been collecting weather data for many years. The annual rainfall amounts for Albuquerque, New Mexico from 1950 to 2008 were used to create the frequency distribution below. Annual Rainfall (in inches) Relative frequency Cumulative relative frequency 4 to <50.052 5 to <60.1030.155 6 to <70.0860.241 7 to <80.1030.344 8 to <90.1720.516 9 to <100.0690.585 10 to < 110.2070.792 11 to <120.1030.895 12 to <130.0520.947 13 to <140.0520.999 In the context of this problem, explain the meaning of this value. Why isn’t this value one (1)? To create a cumulative relative frequency plot, graph a point for the upper value of the interval and the cumulative relative frequency Plot a point for each interval. Plot a starting point at (4,0). Connect the points.

74. Rainfall Cumulative relative frequency What proportion of years had rainfall amounts that were 9.5 inches or less? Approximately 0.55

75. Rainfall Cumulative relative frequency Approximately 30% of the years had annual rainfall less than what amount? Approximately 7.5 inches

76. Rainfall Cumulative relative frequency Which interval of rainfall amounts had a larger proportion of years – 9 to 10 inches or 10 to 11 inches? Explain The interval 10 to 11 inches, because its slope is steeper, indicating a larger proportion occurred.

77. Displaying Bivariate Numerical Data

78. Scatterplots When to Use Bivariate numerical data How to construct - Draw a horizontal scale and mark it with appropriate values of the independent variable - Draw a vertical scale and mark it appropriate values of the dependent variable - Plot each point corresponding to the observations To describe - comment the relationship between the variables Scatterplots are discussed in much greater depth in Chapter 5.

79. Time Series Plots When to Use - measurements collected over time at regular intervals How to construct - Draw a horizontal scale and mark it with appropriate values of time - Draw a vertical scale and mark it appropriate values of the observed variable - Plot each point corresponding to the observations and connect To describe - comment on any trends or patterns over time Can be considered bivariate data where the y-variable is the variable measured and the x- variable is time

80. The accompanying time-series plot of movie box office totals (in millions of dollars) over 18 weeks in the summer for 2001 and 2002 appeared in USA Today (September 3, 2002). Describe any trends or patterns that you see.

81. Who, What, When, Where, Why, How Who? – Individual cases about whom we record some characteristics. Individuals who answer a survey are called respondents. People on whom we experiment are subjects or participants but animals, plants, and other inanimate subjects are called experimental units. What and why? –Type of variable and why you need to look at this variable. Where, When, and How? –What methods were used to collect the data? –Where and when addresses differences in years and locations which may be important.

82. Context Context – The context tells Who was measured, How the data were collected, where the data were collected, and When and Why the study was performed. Every question must be answered in context in complete sentences.