1. 3A-1

2. Describing Data Visually (Part 1) Visual Description Visual Description Dot Plots Dot Plots Frequency Distributions and Histograms Frequency Distributions and Histograms Line Charts Line Charts Bar Charts Bar Charts Chapter 3A3A McGraw-Hill/Irwin© 2008 The McGraw-Hill Companies, Inc. All rights reserved.

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

4. 3A-4 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:CharacteristicInterpretationMeasurement What 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?

5. 3A-5CharacteristicInterpretationDispersion 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?

6. 3A-6 P/E ratios are current stock price divided by earnings per share in the last 12 months. For example:P/E ratios are current stock price divided by earnings per share in the last 12 months. For example:  Example: Price/Earnings Ratios Visual Description

7. 3A-7 Look at the data and visualize how it was collected and measured.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:Sort the data and then summarize in a graphical display. Here are the sorted P/E ratios:810101013131414151516161718191920202122 23262627292934485568 A histogram graphically displays sorted data.A histogram graphically displays sorted data.  Measurement Visual Description  Sorting

8. 3A-8 Sorting allows you to observe central tendency, dispersion and shape as well as minimum, maximum and range.Sorting allows you to observe central tendency, dispersion and shape as well as minimum, maximum and range.  Sorting Visual Description What else do you observe?What else do you observe?

9. 3A-9 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). 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.  Steps in Making a Dot Plot Dot Plots

10. 3A-10 Range of data shows dispersion.Range of data shows dispersion. Can add annotations (text boxes) to call attention to specific features.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.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

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

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

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

15. 3A-15 1. Sort data in ascending order (e.g., P/E ratios) 810 13 14 15 16 171819 20 2122 2326 2729 34485568 Frequency Distributions and Histograms  Constructing a Frequency Distribution 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. 3A-15

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

17. 3A-17 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  Constructing a Frequency Distribution 3A-17

18. 3A-18 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 – counts for each bin Relative frequencies – absolute frequency divided by total number of data values. Cumulative frequencies – accumulated relative frequency values as bin limits increase. Frequency Distributions and Histograms  Constructing a Frequency Distribution 3A-18

19. 3A-19 Bin RangeFrequency Relative Frequency Cumulative Relative Frequency 0<P/E Ratio<1010.0333 10<P/E Ratio<20150.50000.5333 20<P/E Ratio<30100.33330.8666 30<P/E Ratio<4010.03330.8999 40<P/E Ratio<5010.03330.9332 50<P/E Ratio<6010.03330.9665 60<P/E Ratio<7010.03330.9998 What are the bin limits for the P/E ratio data? Frequency Distributions and Histograms 3A-19

20. 3A-20 A histogram is a graphical representation of a frequency distribution. A 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  Histograms 3A-20

21. 3A-21 A histogram bar that is higher than those on either side.A histogram bar that is higher than those on either side. Monomodal – a single modal class.Monomodal – a single modal class. Bimodal – two modal classes.Bimodal – two modal classes. Multimodal – more than two modal classes.Multimodal – more than two modal classes. Modal classes may be artifacts of the way bin limits are chosen.Modal classes may be artifacts of the way bin limits are chosen. Frequency Distributions and Histograms  Modal Class 3A-21

22. 3A-22 A histogram suggests the shape of the population.A histogram suggests the shape of the population. Skewness – indicated by the direction of the longer tail of the histogram.Skewness – indicated by the direction of the longer tail of the histogram. It is influenced by number of bins and bin limits.It is influenced by number of bins and bin limits. 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  Shape 3A-22

23. 3A-23