1. Data Handbook Chapter 4 & 5

2. Data A series of readings that represents a natural population parameter A series of readings that represents a natural population parameter It provides information about the population itself It provides information about the population itself

3. Organizing Data Important prelude to describing and interpreting data Important prelude to describing and interpreting data

4. Charting Data Tables Tables –Organized by rows and columns Column 1 Column 2 Column 3 Row 1 Row 2 Row 3

5. Charting Data Graphs Graphs –Organized by horizontal (abscissa) and vertical (ordinate) axes

6. Charting Data Graphs Graphs –Proper legend –Properly labeled axes

8. Graphs Multiple graphs used for comparing data should map the same variables on the ordinate and abscissa and use the same scale for each graph. Multiple graphs used for comparing data should map the same variables on the ordinate and abscissa and use the same scale for each graph.

10. Describing data Descriptions of data indirectly describes actual population parameters Descriptions of data indirectly describes actual population parameters Describing the data distribution is a first step in this process Describing the data distribution is a first step in this process

11. Data Distributions Pattern of frequency Pattern of frequency Frequency is how often a particular value or set of values occurs in a data set Frequency is how often a particular value or set of values occurs in a data set

12. Histogram Frequency Graph

13. Types of distributions Uniform Uniform Unimodal Unimodal Bimodal Bimodal Normal Normal Skewed Skewed

14. Uniform The distribution has an equal frequency (number of occurrences) of each value or category of values The distribution has an equal frequency (number of occurrences) of each value or category of values

15. Uniform Distribution of Tree Heights

16. Uniform Frequency Graph

17. This is not a uniform distribution.

18. Unimodal The distribution has an unequal frequency (number of occurances) of each value or category of values The distribution has an unequal frequency (number of occurances) of each value or category of values The distribution has distinct central values that have a greater frequency than the others The distribution has distinct central values that have a greater frequency than the others

19. Unimodal Distribution of Tree Heights

20. Unimodal Frequency Graph

21. Skewed The distribution has distinct central values that have a greater frequency than the others The distribution has distinct central values that have a greater frequency than the others The less frequent values are not evenly distributed on either side of the high point The less frequent values are not evenly distributed on either side of the high point

22. Skewed Distribution of Tree Heights

23. Skewed Frequency Graph

24. Bimodal The distribution has two distinct values or sets of values that have greater frequencies than the others The distribution has two distinct values or sets of values that have greater frequencies than the others These values are separated from one another by less frequent values These values are separated from one another by less frequent values Often indicative of two populations Often indicative of two populations

25. Bimodal Distribution of Tree Heights

26. Bimodal Frequency Graph

27. Two Populations of Trees

28. Normal Frequencies are equally spread out on either side of a central high point Frequencies are equally spread out on either side of a central high point Bell shaped Bell shaped Most frequent type of distribution Most frequent type of distribution

29. Normal Frequency Graph

30. Interpreting Data Descriptive statistics are used to summarize data Descriptive statistics are used to summarize data Several descriptive statistics are used to describe two important aspects of data distributions: Several descriptive statistics are used to describe two important aspects of data distributions: –Central Tendency –Dispersion

31. Central Tendency Most data are spread out around a central high point Most data are spread out around a central high point The central values are the ones that occur most often and thus important to report The central values are the ones that occur most often and thus important to report

32. Measures of Central Tendency Three common measurements Three common measurements –Mean »Average value –Median »Center value –Mode »Most frequent value

33. Mean “Typical Value” “Typical Value” N Mean =  X i i=1 i=1N

34. Normal Distribution and Central Measures In a perfectly normal distribution the mean, median and mode are all the same In a perfectly normal distribution the mean, median and mode are all the same

35. Perfectly Normal Distribution

36. Dispersion The distribution of values that occur less often The distribution of values that occur less often The spread of the data around the central values is important to report The spread of the data around the central values is important to report Dispersion is about the degree of clustering of the data Dispersion is about the degree of clustering of the data

37. Measures of Dispersion Two common measurements Two common measurements –Range »Distance between the lowest and highest values –Standard Deviation »Average deviation from the mean

38. Calculate Mean and Range

39. Ranges

40. Calculating Standard Deviation

41. Normal Distribution and Dispersion 68.26% of values fall within one standard deviation on either side of the mean 68.26% of values fall within one standard deviation on either side of the mean 95.44% of values fall within two standard deviations on either side of the mean 95.44% of values fall within two standard deviations on either side of the mean 99.74% of values fall within three standard deviations on either side of the mean 99.74% of values fall within three standard deviations on either side of the mean

42. Normal Distribution and Standard Deviation

43. Graph of Dispersion

44. Accuracy & Precision

45. Accuracy

46. Error Error = Accuracy of a particular data point relative to an accepted value Error = Accuracy of a particular data point relative to an accepted value Absolute Error = I Accepted – Data I Absolute Error = I Accepted – Data I Percent Error = I Accepted – Data I x 100 Percent Error = I Accepted – Data I x 100Accepted

47. Precision Precision is a measure of how consistent the data within a data set are relative to each other Precision is a measure of how consistent the data within a data set are relative to each other One measure of precision of a data set is the standard deviation  (the mean) is the accepted value One measure of precision of a data set is the standard deviation SD provided that  (the mean) is the accepted value  + SD  + SD

48. Calculation of SD of a Data Set N  N  SD =   –  X i   i=1 i=1 N-1 N-1