1. Describing Data Visually (Part 1) Chapter33 Visual Description Dot Plots Frequency Distributions and Histograms Line Charts Bar Charts McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

2. 3A-2 Visual Description Methods of organizing, exploring and summarizing data include: Visual - Visual (charts and graphs) provides insight into characteristics of a data set without using mathematics. Numerical - Numerical (statistics or tables) provides insight into characteristics of a data set using mathematics.

3. 3A-3 Begin with univariate data (a set of n observations on one variable) and consider the following:Begin with univariate data (a set of n observations on one variable) and consider the following: CharacteristicInterpretation MeasurementWhat are the units of measurement? Are the data integer or continuous? Any missing observations? Any concerns with accuracy or sampling methods? Visual Description Central Tendency Where are the data values concentrated? What seem to be typical or middle data values?

4. 3A-4 CharacteristicInterpretation Dispersion How much variation is there in the data? How spread out are the data values? Are there unusual values? Visual Description Shape Are the data values distributed symmetrically? Skewed? Sharply peaked? Flat? Bimodal?

5. 3A-5 P/E ratios are current stock price divided by earnings per share in the last 12 months. For example: Example: Price/Earnings Ratios Example: Price/Earnings Ratios Visual Description

6. 3A-6 Sorting Sorting Measurement Measurement Look at the data and visualize how it was collected and measured. Sort the data and then summarize in a graphical display. Here are the sorted P/E ratios: histogramA histogram graphically displays sorted data. Visual Description

7. 3A-7 Sorting Sorting Sorting allows you to observe central tendency, dispersion and shape as well as minimum, maximum and range. When the number of observations is large, a sorted list of data values is difficult to analyze. visual displayTo see broader patterns in the data, analysts often prefer a visual display of the data. Visual Description

8. 3A-8 A dot plot is the simplest graphical display of n individual values of numerical data. - Easy to understand - Not good for large samples (e.g., > 5,000).A dot plot is the simplest graphical display of n individual values of numerical data. - Easy to understand - Not good for large samples (e.g., > 5,000). Steps in Making a Dot Plot Steps in Making a Dot Plot 1. Make a scale that covers the data range 2. Mark the axes and label them 3. Plot each data value as a dot above the scale at its approximate location If more than one data value lies at about the same axis location, the dots are piled up vertically. Dot Plots

9. 3A-9 Creating a Dot Plot in MegaStat

10. 3A-10 Range of data shows dispersion.Range of data shows dispersion. Can add annotations (text boxes) to call attention to specific features. Clustering shows central tendency.Clustering shows central tendency. Dot plots do not tell much of shape of distribution.Dot plots do not tell much of shape of distribution. Dot Plots

11. 3A-11 Consider the following median home prices for nine U.S. Cities. Metropolitan Area Median Home Price (000) Akron OH 119.6 Bergen-Passaic NJ 363.0 Bradenton FL 170.4 Colorado Springs CO 181.7 Hartford CT 198.5 Milwaukee WI 186.2 Raleigh-Durham NC 173.8 San Francisco CA 560.2 Topeka KS 100.7 Dot Plots Small Sample: Home Prices Small Sample: Home Prices

12. 3A-12 Small Sample: Home Prices Small Sample: Home Prices A dot plot is useful to realtors as they discuss patterns in home selling prices within their community. Dot Plots

13. 3A-13 Comparing Groups Comparing Groups stacked dot plotA stacked dot plot compares two or more groups using a common X-axis scale. Dot Plots

14. 3A-14 Bins and Bin Limits Bins and Bin Limits frequency distributionA frequency distribution is a table formed by classifying n data values into k classes (bins). Bin limitsBin limits define the values to be included in each bin. Widths must all be the same. FrequenciesFrequencies are the number of observations within each bin. relative frequencies percentagesExpress as relative frequencies (frequency divided by the total) or percentages (relative frequency times 100). Frequency Distributions and Histograms

15. 3A-15 Constructing a Frequency Distribution Constructing a Frequency Distribution 1. Find smallest and largest data values Frequency Distributions and Histograms 2. Choose the number of bins (k) - k should be much smaller than n. - Too many bins results in sparsely populated bins, too few and dissimilar data values are lumped together.

16. 3A-16 - Herbert Sturges proposes the following rule: Sample Size (n) Suggested Number of Bins (k) 165 326 647 1288 Sample Size (n) Suggested Number of Bins (k) 2569 51210 102411 Frequency Distributions and Histograms Constructing a Frequency Distribution Constructing a Frequency Distribution

17. 3A-17 Constructing a Frequency Distribution Constructing a Frequency Distribution 3. Set the bin limits: Bin width  For example, for k = 7 bins, the approximate bin width is: Bin width  To obtain “nice” limits, we round the width to 10 and start the first bin at 0 to get bin limits: 0, 10, 20, 30, 40, 50, 60, 70 Frequency Distributions and Histograms

18. 3A-18 Constructing a Frequency Distribution Constructing a Frequency Distribution 4. Put the data values in the appropriate bin In general, the lower limit is included in the bin while the upper limit is excluded. 5. Create the table, you can include Frequencies Frequencies – counts for each bin Relative frequencies Relative frequencies – absolute frequency divided by total number of data values. Cumulative frequencies Cumulative frequencies – accumulated relative frequency values as bin limits increase. Frequency Distributions and Histograms

19. 3A-19 Frequency Distributions and Histograms

20. 3A-20 Histograms Histograms histogramA histogram is a graphical representation of a frequency distribution. histogramA histogram is a bar chart. Y-axis shows frequency within each bin. X-axis ticks shows end points of each bin. Frequency Distributions and Histograms

21. 3A-21 Consider 3 histograms for the P/E ratio data with different bin widths. What do they tell you? Frequency Distributions and Histograms Histograms Histograms

22. 3A-22 Frequency Distributions and Histograms Excel’s Histogram Excel’s Histogram

23. 3A-23 Frequency Distributions and Histograms Mega Stat's Frequency Distribution and Histograms Mega Stat's Frequency Distribution and Histograms

24. 3A-24 Frequency Distributions and Histograms MINITAB Histogram MINITAB Histogram

25. 3A-25 Modal Class Modal Class A histogram bar that is higher than those on either side. UnimodalUnimodal – a single modal class. BimodalBimodal – two modal classes. MultimodalMultimodal – more than two modal classes. Modal classes may be artifacts of the way bin limits are chosen. Frequency Distributions and Histograms

26. 3A-26 Shape Shape A histogram suggests the shape of the population. Skew nessSkew ness – indicated by the direction of the longer tail of the histogram. It is influenced by number of bins and bin limits. – (negatively skewed) a longer left tail. Left-skewed – (negatively skewed) a longer left tail. Right-skewed – (positively skewed) a longer right tail. Symmetric – both tail areas approximately the same. Frequency Distributions and Histograms

27. 3A-27 Frequency Distributions and Histograms

28. 3A-28 Frequency Distributions and Histograms Tips for Effective Frequency Distributions Tips for Effective Frequency Distributions Check Sturges’ Rule first.Check Sturges’ Rule first. Choose a nice, round bin width.Choose a nice, round bin width. Choose bin limits that are multiples of the bin width.Choose bin limits that are multiples of the bin width. Make sure that the range is covered.Make sure that the range is covered.

29. 3A-29 Frequency Polygon and Ogive

30. 3A-30 Used to display a time series or spot trends, or to compare time periods. Can display several variables at once. Line Charts Simple Line Charts Simple Line Charts

31. 3A-31 Two-scale line chart – used to compare variables that differ in magnitude or are measured in different units. Line Charts Simple Line Charts Simple Line Charts

32. 3A-32 Grid Lines Grid Lines A line graph usually has no vertical grid lines. Horizontal lines can be added to make it easier to establish the y value. Which is easier to read? Line Charts

33. 3A-33 Log Scales Log Scales Arithmetic scaleArithmetic scale – distances on the Y-axis are proportional to the magnitude of the variable being displayed. Logarithmic scaleLogarithmic scale – (ratio scale) equal distances represent equal ratios. log scaleUse a log scale for the vertical axis when data vary over a wide range, say, by more than an order of magnitude. This will reveal more detail for small data values. Line Charts

34. 3A-34 Log Scales Log Scales Log scale is only suited for positive data values. Reveals whether the quantity is growing at an increasing percent (concave upward), constant percent (straight line), or declining percent (concave downward) Line Charts

35. 3A-35 What does the log scale graph tell you about growth rate for both series? Line Charts Example: U.S. Trade Example: U.S. Trade

36. 3A-36 When to Use Log Scales When to Use Log Scales Useful for - time series data that might be expected to grow at a compound annual percentage rate (e.g., GDP, national debt, future income) - financial charts that cover long periods of time-data that grow rapidly (e.g., revenues) Line Charts

37. 3A-37 Tips for Effective Line Charts Tips for Effective Line Charts 1. Line charts are used for time series data (never for cross-sectional data). 2. Y-axis shows numerical variable while X-axis shows time units with time increasing left to right. 3. Use a zero origin on the Y-axis unless more detail is needed. Line Charts

38. 3A-38 Tips for Effective Line Charts Tips for Effective Line Charts 4. Omit numerical labels on a line chart to avoid clutter. Use gridlines if needed. 5. Use data markers (squares, triangles, circles) if they don’t clutter the graph. 6. Don’t make lines too thick. Line Charts

39. 3A-39 Most common way to display attribute data. - Bars represent categories or attributes. - Lengths of bars represent frequencies. Vertical Bar Chart Vertical Bar Chart Horizontal Bar Chart Bar Charts Plain Bar Charts Plain Bar Charts

40. 3A-40 3-D Bar Chart Pyramid Chart Bar Charts 3-D and Novelty Bar Charts 3-D and Novelty Bar Charts

41. 3A-41 Special type of bar chart used in quality management to display the frequency of defects or errors of different types.Special type of bar chart used in quality management to display the frequency of defects or errors of different types. Categories are displayed in descending order of frequency.Categories are displayed in descending order of frequency. Focus on significant few (i.e., few categories that account for most defects or errors).Focus on significant few (i.e., few categories that account for most defects or errors). Bar Charts Pareto Charts Pareto Charts

42. 3A-42 Bar height is the sum of several subtotals. Areas may be compared by color to show patterns in the subgroups and total. Bar Charts Stacked Bar Chart Stacked Bar Chart

43. 3A-43 Bar charts can be used for time series data although it may be harder to compare trends. Bar Charts Bar Charts for Time Series Data Bar Charts for Time Series Data

44. 3A-44 Tips for Effective Bar Charts Tips for Effective Bar Charts 1. Show the numerical variable of interest with vertical bars on the Y-axis, category labels on the X-axis. 2. For time series quantities, display the category labels on the horizontal X-axis with time increasing from left to right. 3. The height or length of each bar should be proportional to the quantity displayed. 4. Put numerical values at the top of each bar, except if too cluttered. Bar Charts

45. Applied Statistics in Business and Economics End of Chapter 3A 3A-45