1. Slide 1 Spring, 2005 by Dr. Lianfen Qian Lecture 2 Describing and Visualizing Data 2-1 Overview 2-2 Frequency Distributions 2-3 Visualizing Data

2. Slide 2 Spring, 2005 by Dr. Lianfen Qian  Descriptive Statistics summarize or describe the important characteristics of a known set of data  Inferential Statistics use sample data to make inferences (or generalizations) about a population 2.1 Overview

3. Slide 3 Spring, 2005 by Dr. Lianfen Qian Relationship Population  : Characteristics Sample X: Sample values Sampling Method of drawing Based on probability Estimation & Inference Method of concluding Based on statistics f(X)

4. Slide 4 Spring, 2005 by Dr. Lianfen Qian 1. Center: A representative or average value that indicates where the middle of the data set is located 2. Variation: A measure of the amount that the values vary among themselves 3. Distribution: The nature or shape of the distribution of data (such as bell-shaped, uniform, or skewed) 4. Outliers: Sample values that lie very far away from the vast majority of other sample values Important Characteristics of Data

5. Slide 5 Spring, 2005 by Dr. Lianfen Qian  Frequency Distribution lists data values (either individually or by groups of intervals), along with their corresponding frequencies or counts 2.2 Frequency Distributions

6. Slide 6 Spring, 2005 by Dr. Lianfen Qian Frequency Distribution Table for ungrouped data The Frequency of a ungrouped or regular data (discrete variable) - One value per category - how often it takes these value Gender Frequency Percent (Relative Frequency) 1 (Male)20 40% 2 (Female)30 60% Total50100%

7. Slide 7 Spring, 2005 by Dr. Lianfen Qian Example: Cotinine levels in smokers

8. Slide 8 Spring, 2005 by Dr. Lianfen Qian Grouped Frequency Distribution Table

9. Slide 9 Spring, 2005 by Dr. Lianfen Qian are the smallest numbers that can actually belong to different classes Lower Class Limits Lower Class Limits

10. Slide 10 Spring, 2005 by Dr. Lianfen Qian Upper Class Limits are the largest numbers that can actually belong to different classes Upper Class Limits

11. Slide 11 Spring, 2005 by Dr. Lianfen Qian number separating classes Class Boundaries - 0.5 99.5 199.5 299.5 399.5 499.5

12. Slide 12 Spring, 2005 by Dr. Lianfen Qian Class Midpoints Class midpoints can be found by adding the lower class limit to the upper class limit and dividing the sum by two. Class Midpoints 49.5 149.5 249.5 349.5 449.5

13. Slide 13 Spring, 2005 by Dr. Lianfen Qian Class Width is the difference between two consecutive lower class limits or two consecutive lower class boundaries Class Width 100

14. Slide 14 Spring, 2005 by Dr. Lianfen Qian 1. Large data sets can be summarized. 2. Can gain some insight into the nature of data. 3.Have a basis for constructing graphs. Disadvantage: information lost Reasons for Constructing Frequency Distributions

15. Slide 15 Spring, 2005 by Dr. Lianfen Qian 3. Starting point: Begin by choosing a lower limit of the first class, 0 4. Using the lower limit of the first class and class width, proceed to list the lower class limits: 0, 100, 200, 300, 400. 5. List the lower class limits in a vertical column and proceed to enter the upper class limits: 99, 199, 299, 399, 499. 6.Go through the data set putting a frequency count in the appropriate class for each data value. 6.Add a caption for the frequency distribution table Constructing A Grouped Frequency Table 1. Decide on the number of classes (should be between 5 and 20), say C=5 for the example: cotinine levels in smokers (subjective decision) 2. Calculate (round up). W=(491-0)/5=98.2 round to 100. class width (W)  (highest value) – (lowest value) number of classes

16. Slide 16 Spring, 2005 by Dr. Lianfen Qian Relative Frequency Distribution 11/40 = 28% 12/40 = 40% etc. Total Frequency = 40 relative frequency = class frequency sum of all frequencies

17. Slide 17 Spring, 2005 by Dr. Lianfen Qian Cumulative Frequency Distribution Cumulative Frequencies

18. Slide 18 Spring, 2005 by Dr. Lianfen Qian Frequency Tables

19. Slide 19 Spring, 2005 by Dr. Lianfen Qian In this Section we have discussed  Important characteristics of data  Frequency distributions  Procedures for constructing frequency distributions  Relative frequency distributions  Cumulative frequency distributions Recap

20. Slide 20 Spring, 2005 by Dr. Lianfen Qian 2.3 Visualizing Data Depict the nature of shape or shape of the data distribution

21. Slide 21 Spring, 2005 by Dr. Lianfen Qian A bar graph in which the horizontal scale represents the classes of data values and the vertical scale represents the frequencies. Figure 2-1 Frequency Histogram

22. Slide 22 Spring, 2005 by Dr. Lianfen Qian Has the same shape and horizontal scale as a histogram, but the vertical scale is marked with relative frequencies. Figure 2-2 Relative Frequency Histogram

23. Slide 23 Spring, 2005 by Dr. Lianfen Qian Histogram and Relative Frequency Histogram Figure 2-1 Figure 2-2

24. Slide 24 Spring, 2005 by Dr. Lianfen Qian Uses line segments connected to points directly above class midpoint values Figure 2-3 Frequency Polygon

25. Slide 25 Spring, 2005 by Dr. Lianfen Qian A line graph that depicts cumulative frequencies Figure 2-4 Ogive

26. Slide 26 Spring, 2005 by Dr. Lianfen Qian Consists of a graph in which each data value is plotted as a point along a scale of values Figure 2-5 Dot Plot

27. Slide 27 Spring, 2005 by Dr. Lianfen Qian Stem-and Leaf Plot Represents data by separating each value into two parts: the stem (such as the leftmost digit) and the leaf (such as the rightmost digit)

28. Slide 28 Spring, 2005 by Dr. Lianfen Qian Pareto Chart A bar graph for qualitative data, with the bars arranged in order according to frequencies Figure 2-6

29. Slide 29 Spring, 2005 by Dr. Lianfen Qian Pie Chart A graph depicting qualitative data as slices of a pie Figure 2-7

30. Slide 30 Spring, 2005 by Dr. Lianfen Qian Scatter Diagram A plot of paired (x,y) data with a horizontal x-axis and a vertical y-axis

31. Slide 31 Spring, 2005 by Dr. Lianfen Qian Data that have been collected at different points in time Figure 2-8 Time Series Plot

32. Slide 32 Spring, 2005 by Dr. Lianfen Qian Figure 2-9 Outmost region: deaths due to preventable diseases Innermost region: deaths from wounds Middle region: Deaths from other causes Other Graphs

33. Slide 33 Spring, 2005 by Dr. Lianfen Qian In this Section we have discussed graphs that are pictures of distributions. Keep in mind that the object of this section is not just to construct graphs, but to learn something about the data sets – that is, to understand the nature of their distributions. Recap