1. Variable  An item of data  Examples: –gender –test scores –weight  Value varies from one observation to another

2. Types/Classifications of Variables  Qualitative  Quantitative –Discrete –Continuous

3. Qualitative Data  Describes the quality  Non-numerical format Counts Cannot order or measure  Examples – gender – marital status – geographical region – job title….

4. Categorical data  Non-overlapping categories or characteristics  Examples: –Completes/Incompletes –Professions –Gender

5. Quantitative Data  Frequencies  Measurements

6. Discrete  Measurements are integers  Examples: –number of employees of a company –number of incorrect answers on a test –number of participants in a program…

7. Continuous  Measurements can take on any value - usually within some range  Examples: –Age –Income  Arithmetic operations such as differences and averages make sense.

8. Qualitatiave or Quantitative? Discrete or Continuous?  Score on a placement exam  Preferred restaurant  Dollar amount of a loan  Height  Salary  Length of time to complete a task  Number of applicants  Ethnic origin

9. Treatment as Ranks  Natural order  Not strictly measured  Examples: –Age group –Likert Scale data  Distinction between adjacent points on the scale is not necessarily the same

10. Analysis Qualitative Data  Frequency tables  Modes - most frequently occurring  Graphs: Bar Charts and Pie Charts

11. Analysis Quantitative Data  Any form  Create groups or categories and generate frequency tables  All descriptive statistics

12. Effective Graphs: Quantitative Data  Histograms  Stem-and-Leaf plots  Dot Plots  Box plots  XY Scatter Plots (2 variables).

13. Examples of Graphs

14. Pie Chart

16. Histogram

17. Boxplot

18. Stem and Leaf Plot

19. Analyze Ranked Data  Frequency tables  Mode, Median, Quartiles  Graphs: –Bar Charts –Dot Plots, Pie Charts –Line Charts (2 variables)

20. Data Example Suggest some ways you could analyze these items.  Score on a placement exam  Preferred restaurant  Dollar amount of a loan  Height  Salary  Length of time to complete a task  Number of applicants  Ethnic origin

21. Tables and Graphs Note Excel will create any graph that you specify Consider the type of data before selecting your graph.

22. Frequency Table/Frequency Distribution Summarize data:  categorical  nominal  Continuous data - the data set has been divided into meaningful groups

23. Frequency Distribution Count the number of observations that fall into each category. Frequency: the number associated with each category

24. Relative Frequency Distribution Proportion of observations falling in a given category Report relative frequencies or percentages

25. Example Frequency Distribution

26. Graphs Categorical/Qualitative Data

27. Pie Charts  Circle - divided proportionately  Segment - percentage of the whole that falls into each category

28. Bar Charts  Bar charts - % in various categories  Vertical scale - frequencies, relative frequencies  Horizontal scale - categories  Allows comparisons

29. Constructing Bar Charts  All boxes should have the same width  Gaps between the boxes - no connection between  Any order.  Use to represent two categorical variables simultaneously

30. Graphs: Measured Continues Quantitative Data  Histograms  Stem and Leaf  Box plots  Line Graphs  XY Scatter Charts (2 variables)

31. Histograms  Frequency distributions of continuous variables  Drawn without gaps between the bars

32. Constructing Histograms  Non-overlapping intervals  Intervals - generally the same length  Number of values in each interval -class frequency  Relative frequencies o

33. XY Scatter Chart  Two variables  Variables: quantitative and continuous.  Plot pairs - rectangular coordinate system  Examine the relationship between two variables

34. Line Chart  Similar to the scatter chart  Values of the independent variable (shown on the horizontal axis) can be ranked values (i.e.. they do not have to be continuous variables).

35. Basic Principles for Constructing All Plots  Data should stand out clearly from background  The information should be clearly labeled –title –axes, bars, pie segments, etc. - include units that are needed to interpret data –scale including starting points.

36. Principles cont.  Source  No clutter  Minimize information or data on one graph.  Try several approaches

37. Describing Data  Shape of the Distribution –Symmetry –Skewness –Modality: most frequently occurring value –Unimodal or bimodal or uniform

38. Right Skewed Left Skewed Symmetrical

39. Describing Data  Centrality  Spread  Extreme values

40. Measures of Centrality  Mean  Median  Mode

41. Mean  Most common measure  Extremely large values in a data set will increase the value of the mean  Extremely low values will decrease it.

42. Calculating the Mean T1T2T3 858585 909090 753575 9090110 340300360Sum 857590Mean

43. Median  Central point.  Half of the data has a value than the median  Half of the data has a higher value than the median  Not affected by extremely large or small values

44. Find the Median 8590759295Data 7585909295Sorted Data Median is 90.

45. Find the Median 95909285Data 85909295Sorted Data Median: (90 + 92)/2 = 91

46. Measures of Spread

47. Range  Subtract the smallest value from the largest  Report the smallest and largest values. 8590759295Scores Range: 75 to 95 or20

48. Variance/Standard Deviation  Average variation of the data values from the mean of the values  Variance.

49. The Empirical Rule  Symmetrical Data  At least: 68% of the data values are within one standard deviation of the mean 90% of the data values are within two standard deviation of the mean 99% of the data values are within three standard deviations of the mean

50. Tchybychef’s Inequality  Skewed Data  At least: 75% of the data values are within two standard deviation of the mean. 90% of the data values are within one standard deviation of the mean.

51. Measures of Relative Standing  Percentiles  Quartiles

52. Quartiles  The lower quartile is the same as the 25th percentile. –25% of the scores are lower and –75% of the scores are higher than the lower quartile.  The upper quartile is the same as the 75th percentile. –75% of the scores are lower and

53. Correlation Describes the strength of the relationship between two (or more) variables Pearson Product-moment Correlation Coefficient - assumes continuous quantitative data

54. Relationship between Variables  Positive  Negative  No relationship.

55. Interpreting Correlation Coefficients.  0.20 to 0.35- show a slight relationship (little value in practical prediction situations)  0.50 - crude group prediction (Correlations this low do not suggest a good relationship)  0.65 to 0.85 - group predictions that are good  Over 0.85 - a close relationship between the two variables.

56. Even a high correlation coefficient does not establish a cause and effect relationship!!!!!

57. Coefficient of Determination  Square root of the correlation coefficient  Gives the percent of variation in the dependent variable that is ‘explained’ by the independent variable.  Look at an XY scatter plot

58. Least Square Line  Describe the relationship between the two variables  Make predictions of the dependent variable from the independent variable

59. Positive Relationship r will be a positive number.

60. Negative Relationship r will be a negative number.