1. Frequency Distributions Chapter 3 Homework: 1, 2, 3, 12

2. Organizing Data n Describing distribution of variables l enumeration: list raw data l cumbersome to communicate n Frequency distributions l organize ---> tables or graphs l highlight important characteristics range, most frequent value ~

3. Distributions as Tables n Frequency l # of times a value of variable occurs  f = n n Tabular frequency distributions l ordered list of all values of variable & their frequencies l except f = 0 l logical order (usually descending) ~

4. Frequency Distribution X f 191 182 163 153 145 132 126 117 103 96 85 73 62 52 50 # of presentations to be able to recall 100% 8 9 7 8 16 7 10 11 16 14 12 13 12 13 12 14 8 9 15 12 18 14 12 8 11 9 18 15 11 7 9 5 6 8 10 11 10 14 16 6 11 15 9 19 12 5 Enumeration

5. Grouped Frequency Distribution n Group by class intervals l report f for intervals l Lose information: exact values l sacrifice detail for clarity n General rules l each interval same width l consecutive & do not overlap l 6 - 20 intervals Include intervals where f = 0 ~

6. Creating Grouped Frequency Distributions 1. Find range highest - lowest score 2. Choose # of class intervals 3. Determine interval width l range divided by # intervals l round width to convenient # l adjust # intervals if necessary ~

7. 4. Determine lower limit of lowest interval l should contain lowest data point l and have convenient limits 5. Prepare list of limits, work bottom---> up l highest interval must contain high score 6. Count # of observations that occur at each interval  f = n ~

8. Distributions as graphs n Summarizes data “A picture is worth ten thousand words” n Histograms & Frequency Polygrams l Interval/ratio data l grouped frequency distributions n Bar Graphs l nominal or ordinal data l frequency distributions ~

10. Histograms n X-axis l Class intervals of variables n Y-axis l Frequencies represented as vertical bars l no spaces separating bars l labels: lower or upper limit ~

11. Creating a Histograms 1. Start with grouped frequency dist. 2. Draw & label axes l Y-axis 2/3 length of X-axis l Y label: f, add tick marks & values l X label: units for variable e.g., pounds. seconds, inches, degrees l add tick marks & value labels to X evenly spaced, convenient round # ~

12. 3. Y axis should intersect X at zero l if not put in break to indicate l same goes for Y axis 4. Draw vertical lines at edges of intervals l single line, no spaces between 5. Bar height = frequency in interval 6. Provide explanatory notes l e.g. where values on border belong l clarity guiding principle ~

13. Frequency polygons n Contains same info as histogram n Frequency represented as points l interval/ratio data, grouped f dist. n Creating a frequency polygon l substitute single point for bar l at midpoint of interval l connect points with line ~

14. Relative Frequency n frequencies represented as percentage l Large # of data points n Larger sample size ---> more intervals l narrower bars ---> smoother curve ~

15. Relative Frequency # of presentations f

16. Shapes of distributions n Shapes of curves n Unimodal distribution l single value is most frequent n Bimodal l 2 most frequently occurring values l e.g. weight of all students mode for female & mode for male l distinct splitting of population l does not imply no overlap ~

17. X f Unimodal distribution X f Bimodal distribution

18. Symmetry of distributions n Symmetric l if right side mirror-image of left l unimodal or bimodal n Skewed - asymmetric l tail on one side longer than other l Positively skewed: right tail longer l Negatively skewed: left tail longer l Asymptotic: gradually approaches X axis ~

19. Asymmetric Distributions X f 0+2+4-2-4 Positively skewed X f 0+2+4-2-4 Negatively skewed

20. The Normal Distribution n Characteristic of many distributions in nature l bell-shaped n 3 characteristics l Unimodal l symmetric l asymptotic ~

21. Normal Distribution f

22. Bar Graphs n Nominal and ordinal variables l frequency represented by separated bars n Nominal variables l Histograms & Frequency polygons not appropriate l no intermediate values ~

23. Bar Graphs: Ordinal variables n Customarily use bar graphs n Histograms & Frequency polygons used occasionally l Pro: indicates intervals not necessarily equal l Con: might imply no intermediate values n Decision: focus on clarity ~