ENGR 224/STAT 224 Probability and Statistics Lecture 2

1. ENGR 224/STAT 224 Probability and StatisticsLecture 2

2. Descriptive Statistics: • used to summarize important characteristics of a known set of data • Characteristics of Data include, • Center, Variation, Shape, Outliers, Time, … • We shall now see how data can be organized and presented.

3. Tables, Charts, Diagrams Very often, tables, charts, and diagrams give extensive details about the data. E.g. Nature or Shape of data Representative values (center), such as average Measure of scattering or variation Outliers

4. Definition: Frequency Tables A frequency table lists categories (or classes) along with counts (or frequency) of the number of scores that fall into each category.

5. How to Build Frequency Tables (Qualitative Data) List all the classes. Look at the raw data and count the number of entries that fall into each of the classes.

6. Example: Hospital Records of 12 peptic ulcer patients provided the following information about their blood types. O A B O A B A A O AB O APrepare a frequency table. Frequency Table of the Blood types of 12 Peptic Ulcer Patients

7. The relative frequency of any observation is obtained by dividing the frequency of that category by the total number of observations.Exercise: Let us prepare a Relative Frequency Table of the 12 Peptic Ulcer Patients. Definition: Relative Frequency

8. Relative Frequency Table of the Blood types of 12 Peptic Ulcer Patients.

9. Graphs and Charts Graphs and charts offer a distinct advantage over raw data and over tabular data. Pictures can be assimilated (understood) very quickly and can be understood by the non-expert. Examples: Pie Charts Bar Graphs Histograms Pareto Charts

10. Definition: Pie Chart Looks like a piece of pie cut into wedge-like slices. The angle of each slice is determined by the relative frequency multiplied by 360.

11. Blood Types of 12 Peptic Ulcer Patients

12. Definition: Bar Graphs Bar graphs can be used to represent qualitative data. Classes are labeled along the horizontal axis and frequency (or relative frequency) along the vertical axis.

13. Definition: Pareto Graphs These are Bar graphs where the bars are arranged highest to lowest. Note how in example we use the relative frequency.

14. Stem and Leaf Plots A stem and leaf plot is a graphical device for quantitative data that allows one to view the distribution of the data without losing information. The technique involves separating the data into two parts, the stems (the leading digits) and the leaves. Lets do an example.

15. Example: Final Grades of Statistics students 92 90 56 54 69 64 81 45 91 50 93 79 59 89 74 28 71 38 90 80 39 62 60 94 90 66 77 66 71 97 48 71 6 61 64 78 44 70 77 95 82 64 23 62 52 63 53 79 Let us construct a stem and leaf diagram for the Final grades.

16. Example: Final gradesStem and Leaf diagram 0 6 1 2 38 3 89 4 458 5 023469 6 011223444669 7 0111477899 8 0129 9 00023457

17. Additional Notes on Constructing Stem and Leaf Diagrams If you find that the number of stems is too small you can expand the Stem and leaf Diagram. Stem and Leaf diagrams can be expanded to include more rows by dividing the stems into two sections, one with leaves from 0 to 4 inclusive, and the other from 5 to 9 inclusive. For example the last two rows of the previous diagram can be expanded to 8 012 8 9 9 000234 9 57

18. Additional Notes on Constructing Stem and Leaf Diagrams If you find that the number of stems is too large you can condense the Stem and Leaf Diagram Stem and Leaf Diagrams can be condensed by writing more than one stem on a row by using a range, in which the leaves for each value in the range of stems is separated by an asterisk. For example the last four rows of the example become 6-7 011223444669*0111477899 8-9 0129*00023457

19. Frequency Tables for Quantitative Data Decide on the number of classes (between 4 and 20). Determine class width (roundup Range/# of classes). Select a convenient lowest class limit. Add the class width to the starting point to get the next lower class limit. List the lower class limits in a vertical column and enter the upper class limits. Tally the scores. Replace Tally marks with the frequency count.

20. Definition:Lower Class Limits are the smallest numbers that can actually belong to the different classes. Definition:Upper Class Limits are the largest numbers that can actually belong to the different classes. Definition:Class Boundaries are the numbers used to separate classes but without the gap created by class limits. They are obtained by decreasing lower class limit and increasing the upper class limit by the same amount so that there are no gaps between consecutive classes. Definition:Class Marks are the midpoints of the classes. Definition:Class Width is the difference between two consecutive lower class boundaries.

21. Exercise: The following data gives the amount of time (in minutes) that a doctor spent with 18 patients. 2 3 5 7 6 4 10 12 7 5 8 4 10 11 25 6 8 5Prepare a frequency table with 4 classes. Include the class marks and the class boundaries.

22. Frequency Table of Doctor Appointments

23. Guidelines on creating Frequency Tables: Be sure the classes are mutually exclusive. Include all classes even if the frequency is zero. Try to use the same class width for all classes, although open ended intervals are impossible to avoid at times (e.g. 65 and older). Try to use between 4 and 20 classes.

24. Definition: Histogram AHistogramconsists of a horizontal scale for values of the data being represented, a vertical scale for the frequencies, and bars representing the frequency for each subdivision or class of values. The width of each bar extends from its lower class boundary to its upper class boundary.

25. Histogram of Length of Doctor Appointments

26. Example: Final Grades of Statistics students 92 90 56 54 69 64 81 45 91 50 93 79 59 89 74 28 71 38 90 80 39 62 60 94 90 66 77 66 71 97 48 71 6 61 64 78 44 70 77 95 82 64 23 62 52 63 53 79 Construct a histogram with 8 classes, starting at 5.

27. Frequency Table of Final Grades

28. Histogram of Final Grades

29. Definition: Relative Frequency Histogram If the relative frequency is plotted instead of the frequency then it is called a relative frequency histogram.

30. Features of a Histogram Symmetric, if the histogram can be folded along a vertical axis so that the two halves coincide. Uniform, if the frequency of each class is the same. Histogram is nearly a straight line. Bell shaped or Normal is when the histogram has a bell shape. Skewedwhen the histogram favors one side over the other.

31. Constructing Frequency Tables for Qualitative Data and Quantitative Data Displaying Qualitative Data Stem and leaf diagrams Describing Quantitative Data Overview

32. Read the next few sections, 2.3, 2.4, 2.5, 2.6, Homework