1. Chapter 2 Summarizing and Graphing Data  Frequency Distributions  Histograms  Statistical Graphics such as stemplots, dotplots, boxplots, etc.  Boxplots will be introduced in chapter 3.

2. Upper Class Limits ClassFrequency, f 1 – 44 5 – 85 9 – 123 13 – 164 17 – 202 A frequency distribution is a table that shows classes or intervals of data with a count of the number in each class. The frequency f of a class is the number of data points in the class. Frequencies Lower Class Limits Frequency Distributions

3. ClassFrequency, f 1 – 44 5 – 85 9 – 123 13 – 164 17 – 202 Frequency Distributions The class width is the distance between lower (or upper) limits of consecutive classes. The class width is 4. 5 – 1 = 49 – 5 = 413 – 9 = 4 17 – 13 = 4 The range is the difference between the maximum and minimum data entries.

4. Constructing a Frequency Distribution Guidelines 1.Decide on the number of classes to include. The number of classes should be between 5 and 20; otherwise, it may be difficult to detect any patterns. 2.Find the class width as follows. Determine the range of the data, divide the range by the number of classes, and round up to the next convenient number. 3.Find the class limits. You can use the minimum entry as the lower limit of the first class. To find the remaining lower limits, add the class width to the lower limit of the preceding class. Then find the upper class limits. 4.Make a tally mark for each data entry in the row of the appropriate class. 5.Count the tally marks to find the total frequency f for each class.

5. Constructing a Frequency Distribution 182021272920 193032193419 242918373822 303932443346 5449185121 Example : The following data represents the ages of 30 students in a statistics class. Construct a frequency distribution that has five classes. Ages of Students

6. Constructing a Frequency Distribution Example continued : 1. The number of classes (5) is stated in the problem. 2. The minimum data entry is 18 and maximum entry is 54, so the range is 36. Divide the range by the number of classes to find the class width. Class width = 36 5 = 7.2 Round up to 8.

7. Constructing a Frequency Distribution Example continued : 3. The minimum data entry of 18 may be used for the lower limit of the first class. To find the lower class limits of the remaining classes, add the width (8) to each lower limit. The lower class limits are 18, 26, 34, 42, and 50. The upper class limits are 25, 33, 41, 49, and 57. 4. Make a tally mark for each data entry in the appropriate class. 5. The number of tally marks for a class is the frequency for that class.

8. Constructing a Frequency Distribution Example continued : 250 – 57 342 – 49 434 – 41 826 – 33 1318 – 25 TallyFrequency, fClass Number of students Ages Check that the sum equals the number in the sample. Ages of Students

9. Midpoint The midpoint of a class is the sum of the lower and upper limits of the class divided by two. The midpoint is sometimes called the class mark. Midpoint = (Lower class limit) + (Upper class limit) 2 Frequency, fClassMidpoint 41 – 4 Midpoint = 2.5

10. Midpoint Example : Find the midpoints for the “ Ages of Students” frequency distribution. 53.5 45.5 37.5 29.5 21.5 18 + 25 = 43 43  2 = 21.5 50 – 57 42 – 49 34 – 41 26 – 33 2 3 4 8 1318 – 25 Frequency, fClass Midpoint Ages of Students

11. Relative Frequency ClassFrequency, f Relative Frequency 1 – 44 The relative frequency of a class is the portion or percentage of the data that falls in that class. To find the relative frequency of a class, divide the frequency f by the sample size n. Relative frequency = Class frequency Sample size Relative frequency 0.222

12. Relative Frequency Example : Find the relative frequencies for the “ Ages of Students” frequency distribution. 50 – 572 3 4 8 13 42 – 49 34 – 41 26 – 33 18 – 25 Frequency, fClass Relative Frequency 0.067 0.1 0.133 0.267 0.433 Portion of students

13. Cumulative Frequency The cumulative frequency of a class is the sum of the frequency for that class and all the previous classes. 30 28 25 21 13 Total number of students + + + + 50 – 572 3 4 8 13 42 – 49 34 – 41 26 – 33 18 – 25 Frequency, fClass Cumulative Frequency Ages of Students

14. A frequency histogram is a bar graph that represents the frequency distribution of a data set. Frequency Histogram 1.The horizontal scale is quantitative and measures the data values. 2.The vertical scale measures the frequencies of the classes. 3.Consecutive bars must touch. Class boundaries are the numbers that separate the classes without forming gaps between them. The horizontal scale of a histogram can be marked with either the class boundaries or the midpoints.

15. Class Boundaries Example : Find the class boundaries for the “ Ages of Students” frequency distribution. 49.5  57.5 41.5  49.5 33.5  41.5 25.5  33.5 17.5  25.5 The distance from the upper limit of the first class to the lower limit of the second class is 1. Half this distance is 0.5. Class Boundaries 50 – 572 3 4 8 13 42 – 49 34 – 41 26 – 33 18 – 25 Frequency, fClass Ages of Students

16. Frequency Histogram Example : Draw a frequency histogram for the “ Ages of Students” frequency distribution. Use the class boundaries. 2 3 4 8 13 Broken axis Ages of Students 10 8 6 4 2 0 Age (in years) f 12 14 17.5 25.533.541.549.557.5

17. Relative Frequency Histogram A relative frequency histogram has the same shape and the same horizontal scale as the corresponding frequency histogram. 0.4 0.3 0.2 0.1 0.5 Ages of Students 0 Age (in years) Relative frequency (portion of students) 17.5 25.533.541.549.557.5 0.433 0.267 0.133 0.1 0.067

18. Cumulative Frequency Graph A cumulative frequency graph or ogive, is a line graph that displays the cumulative frequency of each class at its upper class boundary. 17.5 Age (in years) Ages of Students 24 18 12 6 30 0 Cumulative frequency (portion of students) 25.533.541.549.557.5 The graph ends at the upper boundary of the last class.

19. Stem - and - Leaf Plot In a stem - and - leaf plot, each number is separated into a stem (usually the entry’s leftmost digits) and a leaf (usually the rightmost digit). This is an example of exploratory data analysis. Example : The following data represents the ages of 30 students in a statistics class. Display the data in a stem - and - leaf plot. Ages of Students 182021272920 193032193419 242918373822 303932443346 5449185121

20. Stem - and - Leaf Plot Ages of Students 1234512345 8 8 8 9 9 9 0 0 1 1 1 2 4 7 9 9 0 0 2 2 3 4 7 8 9 4 6 9 1 4 Key: 1|8 = 18 This graph allows us to see the shape of the data as well as the actual values. Most of the values lie between 20 and 39.

21. Stem - and - Leaf Plot Ages of Students 11223344551122334455 8 8 8 9 9 9 0 0 1 1 1 2 4 0 0 2 2 3 4 4 1 4 Key: 1|8 = 18 From this graph, we can conclude that more than 50% of the data lie between 20 and 34. Example : Construct a stem - and - leaf plot that has two lines for each stem. 7 9 9 7 8 9 6 9

22. Dot Plot In a dot plot, each data entry is plotted, using a point, above a horizontal axis. Example : Use a dot plot to display the ages of the 30 students in the statistics class. 182021192320 19 22192019 24291820 22 301832193319 5420181921 Ages of Students

23. Dot Plot Ages of Students 151824454821513054394233362757 From this graph, we can conclude that most of the values lie between 18 and 32.

24. Pie Chart A pie chart is a circle that is divided into sectors that represent categories. The area of each sector is proportional to the frequency of each category. Accidental Deaths in the USA in 2002 TypeFrequency Motor Vehicle 43,500 Falls 12,200 Poison 6,400 Drowning4,600 Fire4,200 Ingestion of Food/Object 2,900 Firearms1,400

25. Pie Chart To create a pie chart for the data, find the relative frequency (percent) of each category. TypeFrequency Relative Frequency Motor Vehicle 43,5000.578 Falls 12,2000.162 Poison 6,4000.085 Drowning4,6000.061 Fire4,2000.056 Ingestion of Food/Object 2,9000.039 Firearms1,4000.019 n = 75,200

26. Pie Chart Next, find the central angle. To find the central angle, multiply the relative frequency by 360°. Continued. TypeFrequency Relative Frequency Angle Motor Vehicle 43,5000.578 208.2 ° Falls 12,2000.162 58.4 ° Poison 6,4000.085 30.6 ° Drowning4,6000.061 22.0 ° Fire4,2000.056 20.1 ° Ingestion of Food/Object 2,9000.039 13.9 ° Firearms1,4000.019 6.7 °

27. Pie Chart Firearms 1.9% Motor vehicles 57.8° Poison 8.5% Falls 16.2° Drowning 6.1% Fire 5.6% Ingestion 3.9%

28. Pareto Chart A Pareto chart is a vertical bar graph in which the height of each bar represents the frequency. The bars are placed in order of decreasing height, with the tallest bar to the left. Accidental Deaths in the USA in 2002 TypeFrequency Motor Vehicle 43,500 Falls 12,200 Poison 6,400 Drowning4,600 Fire4,200 Ingestion of Food/Object 2,900 Firearms1,400

29. Pareto Chart Accidental Deaths 5000 10000 35000 40000 45000 30000 25000 20000 15000 Poison Drowning Falls Motor Vehicles Fire Firearms Ingestion of Food/Object