1. Graphs of Frequency Distribution Introduction to Statistics Chapter 2 Jan 21, 2010 Class #2

2. Two-dimensional graphs: Basic Set-Up

3. Commonly Used Graphs Histogram Height of bars proportional to frequency Width proportional to class boundaries Bar Chart Height proportional to frequency Width not really significant Frequency Polygon Plot points then connect with straight lines

4. Histograms

5. Simple Bar Graph

6. Grouped Bar Graph

7. Frequency Polygons

9. Shape of Frequency Distribution Symmetrical If you can draw a vertical line through the middle (so that you have a mirror image) The scores are evenly distributed Positively skewed Scores piled up on left with tail on right Negatively skewed Scores piled up on right with tail on left

10. Frequency Distribution: Different Distribution shapes

11. Be careful… See next slide for tricks researchers might use with graphs…

12. below 70 70-7980-89 90 + Players Hit Per Game 0.3 0.4 0.5 0.6 Reifman, Larrick, & Fein, 1991

13. Plotting Data: describing spread of data –A researcher is investigating short-term memory capacity: how many symbols remembered are recorded for 20 participants: 4, 6, 3, 7, 5, 7, 8, 4, 5,10 10, 6, 8, 9, 3, 5, 6, 4, 11, 6 –We can describe our data by using a Frequency Distribution. This can be presented as a table or a graph. Always presents: –The set of categories that made up the original category –The frequency of each score/category Three important characteristics: shape, central tendency, and variability

14. Frequency Distribution Tables –Highest Score is placed at top –All observed scores are listed –Gives information about distribution, variability, and centrality X = score value f = frequency fx = total value associated with frequency  f = N  X =  fX

15. Frequency Table Additions –Frequency tables can display more detailed information about distribution Percentages and proportions p = fraction of total group associated with each score (relative frequency) p = f/N As %: p(100) =100(f/N) –What does this tell about this distribution of scores?

16. Steps in Constructing a Grouped Frequency Distribution 1.Determine the Class Interval Size J Ideally, we wish to generate a frequency distribution with 10 class intervals. J We would like the size (width) of each class interval to be in units of 1, 2, 3, 5, 10, 20, 30, 50, or multiples (factor of 10) of these values.

17. Steps in Constructing a Grouped Frequency Distribution 1.Determine the Class Interval Size (continued) J To Achieve These Goals, We Employ the Following Procedure: Ø Calculate the Range (R) of the Data Set Ø Divide the Range by 10 Ø Select the Tentative Class Interval Size from the Previous Slide Closest to Your Answer in Step 2 Above (i.e. R/10). Make Certain That Your Selection Will Result in approx. 10 Class Intervals for Your Frequency Distribution.

18. Grouped Frequency Distribution Tables –Sometimes the spread of data is too wide – Grouped tables present scores as class intervals About 10 intervals An interval should be a simple round number (2, 5, 10, etc), and same width Bottom score should be a multiple of the width –Class intervals represent Continuous variable of X: E.g. 51 is bounded by real limits of 50.5- 51.5 If X is 8 and f is 3, does not mean they all have the same scores: they all fell somewhere between 7.5 and 8.5

19. Percentiles and Percentile Ranks –Percentile rank = the percentage of the sample with scores below or at the particular value –This can be represented be a cumulative frequency column –Cumulative percentage obtained by: c% = cf/N(100) –This gives information about relative position in the data distribution –X values = raw scores, without context

20. 1.Determine the Class Interval Size (continued) Example: Given the following data 100 74 84 95 95 110 99 87 100 108 85 103 99 83 91 91 84 110 113 105 100 98 100 108 100 98 100 107 79 86 123 107 87 105 88 85 99 101 93 99 Steps in Constructing a Grouped Frequency Distribution

21. 2. Determine the Starting Point (First Class Interval) of the Frequency Distribution J Start the Frequency Distribution with a Class Interval in Which the Following Guidelines Apply: u The First Number of the Class Interval is a Multiple of the Class Interval Size. u The First Interval Includes the Lowest Number or Value in the Data Set Steps in Constructing a Grouped Frequency Distribution

22. Credits http://www.statcan.ca/english/edu/power/ch8/frequency.htm http://www.le.ac.uk/pc/sk219/introtostats1.ppt#259,4,Plottin g Data: describing spread of data http://leeds- faculty.colorado.edu/luftig/Past_Course_Websites/APPM_4570 _5570/Website_without_Sound/Lecture_Slides/CHAPTER2/Ch ap_2.ppt http://leeds- faculty.colorado.edu/luftig/Past_Course_Websites/APPM_4570 _5570/Website_without_Sound/Lecture_Slides/CHAPTER2/Ch ap_2.ppt