1. Continuous Statistical Distributions: A Practical Guide for Detection, Description and Sense Making
Unit 3

2. Continuous Statistical Distribution
Describes behavior of a continuous random variable
The probability that the c.r. variable has any value is described by a probability density function (pdf), the probability that the variable will take on any particular value.
Continuous pdfs can
Symmetric
Asymmetric (or skewed)

3. Goals Definition of continuous distributions
Probability density function, cumulative distribution function, descriptive statistics, histograms, probability plots, and mixture distributions.
Visualization of data structure with probability plots.

4. Continuous pdf shapes

5. Descriptive Statistics
Central Tendency
Mean (arithmetic mean or average)
Median: observation separating upper from lower half (50%) of data set
Mode: observation that occurs most frequently in a data set
Dispersion
Standard deviation

8. Examples include: Lognormal, Gamma, Chi-square, Weibull, Exponential, F and Extreme Value

10. Gaussian probability distribution and cumulative probability distribution functions, µ=10, σ= 1 (blue), 2 (green), and 3 (red)

11. Gaussian probability distribution and cumulative probability distribution functions, σ= 2; µ=10 (blue), 12 (green), and 14 (red)

12. Histogram (visualize ‘pdf of data sample’)
Gaussian data: Working with Random Samples (DATA)
Histogram (visualize ‘pdf of data sample’)

13. Empirical Cumulative Distribution Functions
Gaussian data: Working with Random samples
Empirical Cumulative Distribution Functions

14. Empirical Cumulative Distribution Functions
Gaussian data: Working with Random samples
Empirical Cumulative Distribution Functions
Bold line: ECDF for all samples,1000 observations

15. Probability Plot: Equal Percentiles re: Hypothetical Distribution
Gaussian data: Working with Random samples
Probability Plot: Equal Percentiles re: Hypothetical Distribution

16. Probability Plot: Equal Percentiles re: Hypothetical Distribution
Gaussian data: Working with Random samples
Probability Plot: Equal Percentiles re: Hypothetical Distribution

17. Plot the sorted data (x-axis) versus the y-axis points.
Normal Probability Plot: Equal Percentiles re: Normal (Gaussian) Distribution – IN EXCEL
For x-axis, sort (or rank) data sample observations in ascending order (from smallest to largest)
For y-axis, make a corresponding array of probability values, (i-0.5)/N, where N is the sample and i=1,2,3,…,N. Then make an array that is ‘NORMSINV()’ of these probability values, the expected value of each observation from a unit normal (mean=0, sd=1) distribution. ‘NORMINV()’ can also be used for other means and sd.
Plot the sorted data (x-axis) versus the y-axis points.

18. Make scatter plot of corresponding points
Normal Probability Plot: Equal Percentiles re: other distributions – IN EXCEL
For the x-axis, sort (or rank) data sample observations in ascending order (from smallest to largest)
For the y-axis, construct probability array (i-0.5)/N, where N is the sample and i=1,2,3,…,N.
Chi-square distribution: ‘CHIINV()’
Gamma distribution: ‘GAMMAINV()’
Beta distribution: ‘BETAINV()’
F distribution: ‘FINV()’
Make scatter plot of corresponding points

19. Probability Plot re: Unit Normal Distribution
Gaussian data: Working with Random samples
Probability Plot re: Unit Normal Distribution

20. Probability Plot re: Unit Normal Distribution
Gaussian data: Working with Random samples
Probability Plot re: Unit Normal Distribution
Bold line: plot for all samples,1000 observations

21. Probability Plot re: Unit Normal Distribution
Gaussian data: Working with Random samples
Probability Plot re: Unit Normal Distribution
Slope estimates 1/SD

22. Probability Plot re: Unit Normal Distribution
Gaussian data: Working with Random samples
Probability Plot re: Unit Normal Distribution

23. Histogram (visualize ‘pdf of data sample’)
Working with Random Samples (DATA)
Histogram (visualize ‘pdf of data sample’)

24. Empirical Cumulative Distribution Functions
Gaussian data: Working with Random samples
Empirical Cumulative Distribution Functions

25. Probability Plot re: Unit Normal Distribution
Working with Random samples
Probability Plot re: Unit Normal Distribution

26. For the y-axis, calculate ‘Cumulative Hazard’
Hazard Plots – IN EXCEL
For the x-axis, sort (or rank) data sample observations in ascending order (from smallest to largest)
For the y-axis, calculate ‘Cumulative Hazard’
For each observation, enter 1/(reverse rank order)
For the smallest of N observations, enter 1/N
For the second smallest, enter 1/(N-1) ….
Cumulative Hazard is the cumulative sum of these values for each observation. E.g., for the third smallest observation, the cumulative hazard is 1/N+1/(N-1)+1/(N-2)
Make scatter plot of corresponding points

27. Probability Plot re: Cumulative Hazard (unit exponential distribution)
Working with Random samples
Probability Plot re: Cumulative Hazard (unit exponential distribution)

28. Make scatter plot of corresponding probability points
Sample Probability-Probability (P-P) and Quantile-Quantile (Q-Q) Plots: Scatter Plot of Equal Percentiles or Quantiles of Two Samples– IN EXCEL
For the x-axis, sort (or rank) first data sample observations in ascending order (from smallest to largest)
For the y-axis, sort (or rank) second data sample observations in ascending order
Make scatter plot of corresponding probability points
If samples are from same distribution, the plot is linear.

29. Probability Plots: Are they identically distributed
Working with Random samples
Probability Plots: Are they identically distributed

31. Probability Plot re: Cumulative Hazard (unit exponential distribution)
Working with Random samples
Probability Plot re: Cumulative Hazard (unit exponential distribution)

32. Mixture Distributions

33. Mixture Distributions

34. Mixture Distributions

35. Mixture Distributions

36. Mixture Distributions

37. Mixture Distributions+ +
Mixture 2

38. Mixture Distributions=

39. Call Center Data: Call Frequency

40. Call Center Data: Call Frequency

41. Call Center Data: Call Frequency

42. Call Center Data: Call Frequency
Mean S,D,
10:09 hr ± 9 min
14:58 hr ± 34 min

43. Call Center Data: Call Frequency
10:09 hr ± 9 min
10:04 hr ± 11 min
14:58 hr ± 34 min
14:58 hr ± 15 min

44. Call Center Data: Call Frequency

45. Call Center Data: Interval Between Calls

46. Call Center Data: Interval Between Calls

47. Call Center Data: Interval Between Calls

48. Call Center Data: Interval Between Calls

49. Call Center Data: Call Service Times

50. Call Center Data: Call Service Times

51. Call Center Data: Call Service Times

52. Call Center Data: Call Service Times

53. Goals Definition of continuous distributions
Probability density function, cumulative distribution function, descriptive statistics, histograms, probability plots, and mixture distributions.
Visualization of data structure with probability plots.