1. Displaying the Observed Distribution of Quantitative Variables Histogram –Divide the range of the variable into equally spaced intervals - called bins –Determine the frequency of observations falling within each bin –Form a histogram based on the bin frequencies The x axis is the intervals with the interval midpoint depicted. The y axis is the frequency or relative frequency Draw bars the height of frequency centered at the interval midpoint.

2. Example Data frame giving the heights of singers in the New York Choral Society. Components are named height (inches) and voice.part. Cleveland, William S. (1993). Visualizing Data. Hobart Press, Summit, New Jersey.

3. Example, cont. Range 60 to 76 inches Frequency distribution

4. Height of Singers, Histogram

5. What parameters affect the histogram? Starting Point Bin width Let’s try the same example but altering these parameters.

6. Height of Singers, Histogram

7. Height of Singers, by Voice Part

8. Histogram Graphical representation of the frequency distribution. Graphical representation of the observed values of the variable of interest. Provides a summary of the observed distribution. Shape changes with the interval definitions (starting point and interval width)

9. Time Series Plots If we observe a variable over consecutive time points. –X-axis is time –Y-axis is the value of the observed variable Demonstrates the observed changes over time of the variable. Major trends –Seasonal Variation

10. Example Ozone 11 to 22 measurement sites throughout the Houston area. Hourly measurements (average of 5 minute observations for the given hour) Focus on one site at 1pm for the year, 1997. At what levels does ozone become a concern?

11. 1997 Ozone (ppm) Location - Downtown Houston Time - 1pm

12. Bivariate/Multivariate Data? Measuring more than one variable at a time. How would you graphically describe the relationships between the variables? Scatterplot 2 dimensional histogram

13. Example Measurements of daily ozone concentration (ppb), wind speed (mph), daily maximum temperature (degrees F), and solar radiation (langleys) on 111 days from May to September 1973 in New York. Cleveland, William S. (1993). Visualizing Data. Hobart Press, Summit, New Jersey.

14. Ozone, Temperature New York, May to Sept, 1973

15. Histograms of Each Variable

16. Bivariate Histogram

17. Numerical Summaries of Data Measures of Central Tendency –Mean –Median –Mode Measures of Variation –Standard Deviation –Interquartile Range –Range

18. 5 Statistic Summary 5 Number Summary –Minimum –Q1 –Q2 –Q3 –Maximum Boxplot –Box Q1 Q2 Q3 –Lines to last obs. within Lower extreme = median - 1.5 x IQR Upper extreme = median + 1.5 x IQR –Individual points Observations beyond the extremes Many variations on Boxplots.

19. Boxplot