1. Statistics
Fractiles

2. Cumulative Frequencies
An ogive typically forms an “s” shape

3. Questions?

4. Fractiles
Another way of describing frequency data A measure of position Based on the ogive (cumulative frequency) or ordered data

5. Fractiles
How to do it: find n order the data divide the data into the # of pieces you want, each with an equal # of members

6. Fractiles
quartile - four pieces percentile pieces

7. Step 1: Find n! FRACTILES IN-CLASS PROBLEM
Step 1: Find n!

8. n = 12 What’s next? FRACTILES IN-CLASS PROBLEM
n = 12
What’s next?

9. What if you split it into equal halves?
FRACTILES
IN-CLASS PROBLEM
Order the data!
What if you split it into equal halves?
How many observations would be in each half?

10. 6 observations in each half! This is the 50th percentile
FRACTILES
IN-CLASS PROBLEM
Poof!
6 observations in each half!
This is the 50th percentile
or the “median”

11. The 50th percentile or the “median” 33+41 2 = = 37 FRACTILES
IN-CLASS PROBLEM
The 50th percentile
or the “median”
33+41 2
= = 37

12. What if you wanted quartiles?
FRACTILES
IN-CLASS PROBLEM
What if you wanted quartiles?
How many observations would be in each quartile?
Where would the splits be?

13. 3 observations in each quartile!
FRACTILES
IN-CLASS PROBLEM
Poof!
3 observations in each quartile!

14. 1st quartile = = 23.5 30+17 2 3rd quartile = = 58 62+54 FRACTILES
IN-CLASS PROBLEM
1st quartile = = 23.5
3rd quartile = = 58
30+17 2
62+54

15. Fractiles
Quartiles and percentiles are common, others not so much The median is also common, but it is called “the median” rather than “the 50th percentile” or “2nd quartile”

16. Questions?

17. Variability
Another measure of variability:

18. Variability
Interquartile range (IQR): IQR = 3rd quartile – 1st quartile

19. Variability
The interquartile range is in the same units as the original data (like the range and standard deviation “s”)

20. What is the IQR for our data?
FRACTILES
IN-CLASS PROBLEM 14
What is the IQR for our data? 5 11 17 30 31 33 41 46
5462 78 88

21. 1st quartile = = 23.5 30+17 2 3rd quartile = = 58 62+54 So the IQR is…
FRACTILES
IN-CLASS PROBLEM 14
1st quartile = = 23.5
3rd quartile = = 58
So the IQR is…
30+17 2
62+54

22. 1st quartile = = 23.5 30+17 2 3rd quartile = = 58 62+54
FRACTILES
IN-CLASS PROBLEM 14
1st quartile = = 23.5
3rd quartile = = 58
IQR = = 34.5
30+17 2
62+54

23. Questions?

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

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

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

27. 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

28. Boxplots

29. 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

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

31. 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 Find the 5-number summary

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

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

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

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

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

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

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

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

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

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

42. Questions?

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

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

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

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

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

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

49. OUTLIERS
IN-CLASS PROBLEM
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

50. OUTLIERS
IN-CLASS PROBLEM
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?

51. OUTLIERS
IN-CLASS PROBLEM
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!

52. Outliers
How outliers are shown in a boxplot

53. Types of Boxplots

54. Questions?

55. Boxplots
Boxplots are typically used to compare different groups

56. 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

57. Boxplots

58. Boxplots

59. BOXPLOTS
IN-CLASS PROBLEM
What differences?

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

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

62. 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…)

63. Questions?