Describing Data with Tables

1. Describing Data with Tables

2. General Outline Frequency Distributions for Ungrouped Data Standard Relative Cumulative Frequency Distributions for Grouped Data Standard Relative Cumulative Frequency Distributions for Qualitative Data Standard Relative Cumulative Percentile Ranks for Ungrouped vs. Grouped Data Outliers

3. New Statistical Notation The number of times a score occurs is the score�s ________________, which is symbolized by f A _______________ is the general name for any organized set of data N is the ________________ indicating the number of scores

4. Simple Frequency Distribution A simple frequency distribution shows the __________________ each score occurs in a set of data The symbol for a score�s simple frequency is simply f

5. Frequency Distributions Presents the score values and their frequency of occurrence. Scores listed in rank order with _______________________________

6. How to make a frequency table 1. Make a list of each possible value down the left edge of the page, starting from the __________________________ 2. Go one by one through the scores, making a mark for each next to its value on your list.

7. 3. Make a table showing how many times each value on your list was used. 4. Figure the ______________of scores for each value.

8. Ungrouped Data: Standard Frequency Distribution

9. Raw Scores Following is a data set of raw scores. We will use these raw scores to construct a simple frequency distribution table.

10. Your turnMake a frequency table

11. Ungrouped Data: Relative Frequency Distribution

12. Your turn Using the data set from your frequency table, construct a relative frequency table.

13. Cumulative Frequency ______________________________ is the frequency of all scores at or below a particular score The symbol for cumulative frequency is cf To compute a score�s cumulative frequency, we ____________________________ for all scores below the score with the frequency for the score

14. Ungrouped Data: Cumulative Frequency Distribution

15. Grouped Data: Standard Frequency Distribution A grouped frequency distribution is a distribution in which observations (data points) are sorted into _______________that contain more than one value. Use when you have a __________________ of possible data values.

16. Grouped Scores 1. Find the _____________ of the scores. 2. Determine the ____________ of each class interval (i) i = Range/number of class intervals 3. List the limits of each class interval, placing the interval containing the lowest score value at the bottom

17. 4. Tally the raw scores into the appropriate ___________________ 5. Add the tallies for each interval to obtain the interval frequency.

18. Table 3.4 Construction of frequency distribution for grouped scoresTable 3.4 Construction of frequency distribution for grouped scores

19. Grouped Data: Standard Frequency Distribution

20. Grouped Data: Standard Frequency Distribution First, find the range of your data: largest value � smallest value = range _____________________ Find the width (or class width) of your intervals (or classes): class width = data range/# of intervals you want class width = _________________

21. Grouped Data: Standard Frequency Distribution Round class width off to the nearest convenient width: _________________________ Decide where the lowest class should start: Choose a multiple of the class/interval width. Our lowest value is 69 and 65 is a multiple of 5, so ____________________ Figure out where the lowest class should end: Class width + lower boundary - 1 _____________________

22. Grouped Data: Standard Frequency Distribution Working upward, list as many classes as you need so that you can include the largest observation (you should have one vertical column here, but I didn�t have enough room):

23. Grouped Data: Standard Frequency Distribution Mark how many values fall into each class (I�m just using a subset of columns here to save space:

24. Grouped Data: Standard Frequency Distribution Make your table, using appropriate headings for each column:

25. Grouped Data: Relative Frequency Distribution

26. Grouped Data: Cumulative Frequency Distribution

27. Qualitative Data Standard Frequency Distribution __________________ can always be converted to a standard frequency distribution. ______________________________ Qualitative data can always be converted to a relative frequency distribution. Relative f = __________

28. Qualitative Data Cumulative Frequency Distribution Qualitative data can only be converted to a cumulative frequency distribution if observations can be ordered from least to most (e.g., ___________________)

29. Outliers Outliers are very extreme observations/data points. In the following group of numbers, which data points could be considered outliers? 3 7 2 5 52 8 11 86

30. Describing Data with Graphs

31. General Outline Graphs for Qualitative Data Histograms Frequency Polygons Stem and Leaf Displays Typical shapes of these distributions Graphs for Quantitative Data Bar Graphs

32. Steps for Making Histograms 1. Make a ___________________. 2. Put the values along the bottom of the page. 3. Make a _________________ along the left edge of the paper. 4. Make a ___________ for each value.

33. Histograms

34. Graphs for Quantitative Data: Histograms

35. Histogram of Old Faithful eruption length

36. How to make a frequency polygon 1. Make a ______________________. 2. Put the values along the bottom of the page. 3. Along the left of the page, make a scale of frequencies that goes from 0 at the bottom to the _______________ for any value.

37. How to make a frequency polygon 4. Mark a point above each value with a height for the frequency of that value. 5. _____________________________.

38. Graphs for Quantitative Data:Frequency Polygons

39. Figure 3.5 Frequency polygon: Statistics exam scores of Table 3.4.Figure 3.5 Frequency polygon: Statistics exam scores of Table 3.4.

40. Your turn Using the numbers below, make a histogram & a frequency polygon. Value Frequency 5 2 4 5 3 8 2 4 1 3 Include answer, have them draw it on the board.Include answer, have them draw it on the board.

41. Stem and Leaf Plots A stem-and-leaf plot is a display that organizes data to show its _____________________________. In a stem-and-leaf plot each data value is split into a "stem" and a "leaf".�

42. Stem and Leaf Plots Figure 3.8 Stem and leaf diagram: Statistics exam scores of Table 3.1.Figure 3.8 Stem and leaf diagram: Statistics exam scores of Table 3.1.

43. The �___________� is usually the last digit of the number and the other digits to the left of the "leaf" form the �_____________".� The number 123 would be split as: _________________ __________________ To show a one-digit number (such as 9) using a stem-and-leaf plot, use a stem of 0 and a leaf of 9.

44. Graphs for Quantitative Data: Stem and Leaf Displays Given the following data: 20 58 18 17 39 11 26 35 48 25 10 13 Make a stem and leaf display:

45. Line Graphs Sometimes used to examine frequencies ___________________________ _________________________ Are good at showing specific values of data

46. Enrollment at a large university Figure 3.2b Continued from previous slideFigure 3.2b Continued from previous slide

47. Graphs for Quantitative Data: Typical Shapes Normal Bimodal Positively Skewed Negatively Skewed

48. Normal distribution Underlying assumption of many statistical tests. _________________________

49. Characteristics of Distributions Normal Distribution Lots of scores in _____________ Few scores on the positive and negative ends Can be depicted as _________________________________

50. Normal Distribution

51. Unimodal vs. Bimodal _________________________ Graph has only one high area _______________________ Has two fairly equal high points

52. Unimodal vs. Bimodal Figure 4.2 Unimodal and bimodal histograms.Figure 4.2 Unimodal and bimodal histograms.

53. Bimodal

54. Other distributions ________________________ Two or more high points __________________________ All values have about the same frequency

55. Distributions cont.. Symmetrical If you fold the graph in half, the two halves look the same. Skew ________________________ One side is long and spread out, like a tail May be skewed to the __________________

56. Skewness Positive skew Skewed to the _________________ Negative skew Skewed to the __________________

57. Positive Skewness Is the distribution centered around the middle OR Too many high scores and too few low scores POSTIVE SKEW __________________________________

58. Negative Skewness Is the distribution centered around the middle OR Too many low scores and too few high scores NEGATIVE SKEW ____________________________________

59. Negative Skew

60. Reminder�. ______________________________________________________________________________________

61. Figure 3.7 Shapes of frequency curves.Figure 3.7 Shapes of frequency curves.

62. Non-Normal Distributions ______________ They are TOO FLAT _______________ They are TOO POINTY

63. Platykurtic, Mesokurtic, and Leptokurtic Distributions Platykurtic ________________________ in the middle Too many scores on the ends _________________________ Just Right Leptokurtic Too many scores in the ________________ Too few scores on the ends

64. Why Care about Kurtosis and Skewness? [ They will be on the test ] _____________________________________ mean distributions are not normal This is a problem for inferential statistics Don�t worry about transforming data now. Don�t worry about when to discard data now.

65. Percentile ____________________ is the percent of all scores in the data that are at or below the score If the scores cumulative frequency is known, the formula for finding the percentile is

66. Percentiles Percentiles are like quartiles, except that they divide the data set into ________________ parts instead of four equal parts Give us relative standing of an individual in a population (_____________________) Several ways to do this.

67. Percentiles

68. Percentiles One definition is the fraction of the population which is less than the specified value. So, if we are talking about the 90th percentile, 90% of scores fall below that person�s score.

69. Percentiles Example: Want to compare someone who graduate 37th from a class of 250 to someone who graduate 12th from a class of 60. First, take _________________________ To get percentile calculate _____________ So, this person graduate at the ______________

70. Percentiles What about our other student? Take _________________________ Then ___________________________ Therefore, being 37th out of 250 puts one at the 85th percentile, which is better than 12th out of 60 which is only at the 80th percentile.