1. Welcome to Week 04 Tues MAT135 Statistics

2. Review

3. Descriptive Statistics
Descriptive statistics – describe our sample – we’ll use this to make inferences about the population

4. Descriptive Statistics
graphs n
max
min
each observation
frequencies
mean, median, mode
range, variance, standard
deviation, quartiles, IQR

5. Statistics vs Parameters
Statistic Parameter n N x μ s2 σ2 s σ

6. Questions?

7. Exploring Data
We are using the descriptive statistics to summarize our sample (and, hopefully, our population) in just a few numbers

8. Exploring Data
The “five-number summary” is: the min Q1 the median Q3 the max

9. Exploring Data
We know how to get all of these using our calculators!

10. Boxplots
There is a graph statisticians use to show this summary: the box plot

11. Boxplots
The boxplot (a.k.a. box and whisker diagram) is a standardized way of displaying the distribution of data based on the five number summary: minimum, first quartile, median, third quartile, and maximum

12. Boxplots

13. BOXPLOTS
IN-CLASS PROBLEM
Daily high temperatures Feb 2008 for Fairbanks, Alaska: 14, 12, 17, 25, 10, -1, -8, -15, -7, 0, 5, 14, 18, 14, 16, 8, -15, -13, -17, -12, 0, 1, 9, 12, 14, 7, 6, 8 Create a Boxplot

14. What do we need for a Boxplot?
BOXPLOTS
IN-CLASS PROBLEM 1
What do we need for a Boxplot?

15. BOXPLOTS
IN-CLASS PROBLEM 2
Daily high temperatures Feb 2008 for Fairbanks, Alaska: 14, 12, 17, 25, 10, -1, -8, -15, -7, 0, 5, 14, 18, 14, 16, 8, -15, -13, -17, -12, 0, 1, 9, 12, 14, 7, 6, 8 Find the 5-number summary

16. BOXPLOTS
IN-CLASS PROBLEM 2
Min = Q1 = Median = Q3 = Max =

17. BOXPLOTS
IN-CLASS PROBLEM 2
Min = -17 Q1 = -4 Median = 7.5 Q3 = 14 Max = 25 Notice they’re all in order at the bottom of your list! YAY!

18. Min = -17 Now for the box! Q1 = -4 Median = 7.5 Q3 = 14 Max = 25
BOXPLOTS
IN-CLASS PROBLEM 3
Min = -17 Now for the box! Q1 = -4 Median = 7.5 Q3 = 14 Max = 25

19. Min = -17 Q1 = -4 Median = 7.5 Q3 = 14 Max = 25 Min!
BOXPLOTS
IN-CLASS PROBLEM 3
Min = -17 Q1 = -4 Median = 7.5 Q3 = 14 Max = 25
Min!

20. Min = -17 Q1 = -4 Median = 7.5 Q3 = 14 Max = 25 Q1!
BOXPLOTS
IN-CLASS PROBLEM 3
Min = -17 Q1 = -4 Median = 7.5 Q3 = 14 Max = 25
Q1!

21. Min = -17 Q1 = -4 Median = 7.5 Q3 = 14 Max = 25 Median!
BOXPLOTS
IN-CLASS PROBLEM 3
Min = -17 Q1 = -4 Median = 7.5 Q3 = 14 Max = 25
Median!

22. Min = -17 Q1 = -4 Median = 7.5 Q3 = 14 Max = 25 Q3!
BOXPLOTS
IN-CLASS PROBLEM 3
Min = -17 Q1 = -4 Median = 7.5 Q3 = 14 Max = 25
Q3!

23. Min = -17 Q1 = -4 Median = 7.5 Q3 = 14 Max = 25 Max!
BOXPLOTS
IN-CLASS PROBLEM 3
Min = -17 Q1 = -4 Median = 7.5 Q3 = 14 Max = 25
Max!

24. Min = -17 Q1 = -4 Median = 7.5 Q3 = 14 Max = 25 Box!
BOXPLOTS
IN-CLASS PROBLEM 3
Min = -17 Q1 = -4 Median = 7.5 Q3 = 14 Max = 25
Box!

25. Min = -17 Q1 = -4 Median = 7.5 Q3 = 14 Max = 25 Whiskers!
BOXPLOTS
IN-CLASS PROBLEM 3
Min = -17 Q1 = -4 Median = 7.5 Q3 = 14 Max = 25
Whiskers!

26. Questions?

27. Outliers
Because the min and max may be outliers, a variation on the boxplot includes “fences” to show where most of the data occurs

28. Outliers
Lower fence: Q * IQR Upper fence: Q * IQR

29. Min = -17 What is the IQR? Q1 = -4 Median = 7.5 Q3 = 14 Max = 25
OUTLIERS
IN-CLASS PROBLEM 4
Min = -17 What is the IQR? Q1 = -4 Median = 7.5 Q3 = 14 Max = 25

30. OUTLIERS
IN-CLASS PROBLEM 4
Min = -17 IQR=14-(-4)=18 Q1 = -4 What is the Median = 7.5 lower fence? Q3 = 14 Max = 25

31. OUTLIERS
IN-CLASS PROBLEM 5
Min = -17 IQR=14-(-4)=18 Q1 = -4 Lower fence = Median = 7.5 Q1-1.5*IQR Q3 = (18) Max = 25 = -31

32. OUTLIERS
IN-CLASS PROBLEM 5
Min = -17 IQR=14-(-4)=18 Q1 = -4 Lower fence=-31 Median = 7.5 What is the Q3 = 14 upper fence? Max = 25

33. OUTLIERS
IN-CLASS PROBLEM 6
Min = -17 IQR=14-(-4)=18 Q1 = -4 Lower fence=-31 Median = 7.5 Upper fence= Q3 = 14 Q3+1.5*IQR Max = (18)=41

34. OUTLIERS
IN-CLASS PROBLEM 6
Min = -17 IQR=14-(-4)=18 Q1 = -4 Lower fence=-31 Median = 7.5 Upper fence=41 Q3 = 14 So, do we have Max = 25 any outliers?

35. OUTLIERS
IN-CLASS PROBLEM 7
Min = -17 IQR=14-(-4)=18 Q1 = -4 Lower fence=-31 Median = 7.5 Upper fence=41 Q3 = 14 Max and Min are Max = 25 inside the fence!

36. Outliers
How outliers are shown in a boxplot

37. Types of Boxplots

38. Questions?

39. Boxplots
Boxplots are typically used to compare different groups

40. Data Summary Table from a Ball-bouncing Experiment
Boxplots
Data Summary Table from a Ball-bouncing Experiment
Super
Ball
Wiffle
Golf
Splash
SpongyBall
Minimum 66 38 70 7 44 Q1 71 45 75 14 58
Median 76 48 78
16.5 60 Q3 50 80 23 62
Maximum 91 90 28 67

41. Boxplots

42. Boxplots

43. BOXPLOTS
IN-CLASS PROBLEM 8
What differences?

44. Boxplots
Unfortunately it is almost impossible to get a true boxplot using Excel

45. Boxplots
Unfortunately it is almost impossible to get a true boxplot using Excel (there are several YouTube videos showing how to get one…

46. Boxplots
Unfortunately it is almost impossible to get a true boxplot using Excel (there are several YouTube videos showing how to get one… but they are all wrong…)

47. Questions?

48. Exploring Data
There actually IS a useful graph you can get out of Excel that includes both an average and a measure if dispersion

49. I use the Hi/Low/Close graph
Exploring Data
I use the Hi/Low/Close graph

50. Exploring Data

51. What does this graph show?
BOXPLOTS
IN-CLASS PROBLEM 9
What does this graph show?

52. What does this graph show?
BOXPLOTS
IN-CLASS PROBLEM 10
What does this graph show?

53. Questions?

54. Normal Probability
The most popular continuous graph in statistics is the NORMAL DISTRIBUTION

55. Empirical Rule
Two descriptive statistics completely define the shape of a normal distribution: Mean µ Standard deviation σ

57. Suppose we have a normal distribution, µ = 12 σ = 2
Empirical Rule
Suppose we have a normal distribution, µ = 12 σ = 2

58. Empirical Rule
If µ = 12 12

59. Empirical Rule
If µ = 12 σ = 2

60. Empirical Rule
More sneaky stuff about the normal distribution:

61. Empirical Rule
More sneaky stuff about the normal distribution:

62. Empirical Rule
So now you can calculate even more percentages!

63. What % of the data is between the mean and +1 SD?
EMPIRICAL RULE
IN-CLASS PROBLEM 11
What % of the data is between the mean and +1 SD?

64. What % is between the mean and -1 SD?
EMPIRICAL RULE
IN-CLASS PROBLEM 12
What % is between the mean and -1 SD?

65. What % of the data is between +1 SD and +2 SD?
EMPIRICAL RULE
IN-CLASS PROBLEM 13
What % of the data is between +1 SD and +2 SD?

66. What % is between -1 SD and -2 SD?
EMPIRICAL RULE
IN-CLASS PROBLEM 14
What % is between -1 SD and -2 SD?

67. What % of the data is between +2 SD and +3 SD?
EMPIRICAL RULE
IN-CLASS PROBLEM 15
What % of the data is between +2 SD and +3 SD?

68. What % is between -2 SD and -3 SD?
EMPIRICAL RULE
IN-CLASS PROBLEM 16
What % is between -2 SD and -3 SD?

69. What % of the data is above +3 SD?
EMPIRICAL RULE
IN-CLASS PROBLEM 17
What % of the data is above +3 SD?

70. What % of the data is below -3 SD?
EMPIRICAL RULE
IN-CLASS PROBLEM 18
What % of the data is below -3 SD?

71. Questions?

72. z-scores
For the standard normal distribution, µ = 0 σ = 1

73. The standard normal is also called “z”
z-scores
The standard normal is also called “z”

74. z-scores
z = (x - µ)/σ

75. EMPIRICAL RULE
IN-CLASS PROBLEM 19
A dataset has a normal distribution with μ = 45 and σ = 13 Find the z-score for a value of 65:

76. z-scores
With a bit of algebra, we can use z = (x - µ)/σ to solve for x given a z-score

77. z-scores
z = (x - µ)/σ x =

78. EMPIRICAL RULE
IN-CLASS PROBLEM 20
A data point from a normal distribution with μ = 45 and σ = 13 has a z-score = 2.3 What is the data value?

79. In-class Project
Turn in your homework! Don’t forget your homework due next class! See you Thursday!