1. Welcome to Week 03 Thurs MAT135 Statistics

2. Review

3. Descriptive Statistics
graphs n max min each observation frequencies “averages”

4. Descriptive Statistics
And… Measures of variability!

5. Descriptive Statistics
Averages tell where the data tends to pile up

6. Descriptive Statistics
Another good way to describe data is how spread out it is

7. Variability
Measures of variability tell how close to the “average” the sample data tend to be

8. Variability
Just like measures of central tendency, there are several measures of variability

9. Variability
Range = max – min Variance = sum of (obs – x )2 n − 1 s = variance

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

11. Questions?

12. Descriptive Statistics
Other numbers and calculations can be used to summarize our data

13. Frequencies Frequency – the number in a category Number of Users 9 1815 8 1

14. Frequencies
Cumulative frequency – the number of observations that fall in that category or a previous category This can only be done if the categories can be ordered

15. Cumulative Frequencies
How many observations occur in a given category and any previous ordered categories:
Minutes Internet Usage
Number of Users
Cumulative Number of Users
1-20 9
21-40 18 27
41-60 15 42
61-80 8 50
81-100 1 51
121+ 52

16. Cumulative Frequencies
The last value is always “n” the sample size
Minutes Internet Usage
Number of Users
Cumulative Number of Users
1-20 9
21-40 18 27
41-60 15 42
61-80 8 50
81-100 1 51
121+ 52

17. Cumulative Frequencies
The histogram for a cumulative frequency distribution is called an “ogive”

18. Cumulative Frequencies
Data table: n = 8 A B A B A C B B Cum Freq Histogram: distribution: A: 3 A or B: 3+4=7 A,B or C: 8

19. Cumulative Frequencies
Note that the last category in a cumulative frequency ALWAYS has the value n

20. Cumulative Frequencies
Note also a cumulative frequency cannot get smaller as you move up the categories

21. Cumulative Frequencies
Note also a cumulative frequency cannot get smaller as you move up the categories It can stay the same (if the category count is 0)

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

23. Questions?

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

25. 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

26. Fractiles
quartile - four pieces percentile pieces

27. Step 1: Find n! FRACTILES IN-CLASS PROBLEM 6
Step 1: Find n!

28. n = 12 What’s next? FRACTILES IN-CLASS PROBLEM 6
n = 12
What’s next?

29. What if you split it into equal halves?
FRACTILES
IN-CLASS PROBLEM 7
Order the data!
What if you split it into equal halves?
How many observations would be in each half?

30. 6 observations in each half! This is the 50th percentile
FRACTILES
IN-CLASS PROBLEM 8
Poof!
6 observations in each half!
This is the 50th percentile
or the “median”

31. The 50th percentile or the “median” 33+41 2 = = 37 FRACTILES
IN-CLASS PROBLEM 9,12
The 50th percentile
or the “median”
33+41 2
= = 37

32. What if you wanted quartiles?
FRACTILES
IN-CLASS PROBLEM 10
What if you wanted quartiles?
How many observations would be in each quartile?
Where would the splits be?

33. 3 observations in each quartile!
FRACTILES
IN-CLASS PROBLEM 11,13
Poof!
3 observations in each quartile!

34. 1st quartile = = 23.5 30+17 2 3rd quartile = = 58 62+54 FRACTILES
IN-CLASS PROBLEM 11,13
1st quartile = = 23.5
3rd quartile = = 58
30+17 2
62+54

35. 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”

36. Questions?

37. Variability
Another measure of variability:

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

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

40. 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

41. 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

42. 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

43. Questions?

44. Continuous Distributions
You can think of smooth quantitative data graphs as a series of skinnier and skinnier bars

45. Continuous Distributions
When the width of the bars reach “zero” the graph is perfectly smooth

46. Continuous Distributions
SO, a smooth quantitative (continuous) graph can be thought of as a bar chart where the bars have width zero

47. Normal Distribution
The most popular continuous graph in statistics is the NORMAL DISTRIBUTION

48. Normal Distribution
Two descriptive statistics completely define the shape of a normal distribution: Mean µ Standard deviation σ

51. Suppose we have a normal distribution, µ = 12 σ = 2

52. Normal Distribution
If µ = 12 12

53. Normal Distribution
If µ = 12 σ = 2

54. ? Suppose we have a normal distribution, µ = 10 Normal Distribution
PROJECT QUESTION
Suppose we have a normal distribution, µ = 10 ?

55. ? ? ? 10 ? ? ? Suppose we have a normal distribution, µ = 10 σ = 5
PROJECT QUESTION
Suppose we have a normal distribution, µ = 10 σ = 5
? ? ? ? ? ?

56. -5 0 5 10 15 20 25 Suppose we have a normal distribution, µ = 10 σ = 5
PROJECT QUESTION
Suppose we have a normal distribution, µ = 10 σ = 5

57. Normal Distribution
We can change any normally-distributed variable into a standard normal One with: mean = 0 standard deviation = 1

58. ? ? ? ? ? ? ? For the standard normal distribution, µ = 0 σ = 1
PROJECT QUESTION
For the standard normal distribution, µ = 0 σ = 1
? ? ? ? ? ? ?

59. Normal Distribution -3 -2 -1 0 1 2 3
For the standard normal distribution, µ = 0 σ = 1

60. The standard normal is also called “z”
Normal Distribution
The standard normal is also called “z”

61. Normal Distribution
To calculate a “z-score”: Take your value x Subtract the mean µ Divide by the standard deviation σ

62. Normal Distribution
z = (x - µ)/σ

63. Normal Distribution
IN-CLASS PROBLEMS
Suppose we have a normal distribution, µ = 10 σ = 2 z = (x - µ)/σ = (x-10)/2 Calculate the z values for x = 9, 10, 15

64. z = (x - µ)/σ = (x-10)/2 x . 9 z = (9-10)/2 = -1/2
Normal Distribution
IN-CLASS PROBLEMS
z = (x - µ)/σ = (x-10)/2 x . 9 z = (9-10)/2 = -1/2

65. Normal Distribution
IN-CLASS PROBLEMS
z = (x - µ)/σ = (x-10)/2 x . 9 z = (9-10)/2 = -1/2 10 z = (10-10)/2 = 0

66. Normal Distribution
IN-CLASS PROBLEMS
z = (x - µ)/σ = (x-10)/2 x . 9 z = (9-10)/2 = -1/2 10 z = (10-10)/2 = 0 15 z = (15-10)/2 = 5/2

67. On the graph:
| |
-1/ /2

68. Empirical Rule

69. Questions?

70. You survived! Turn in your classwork! Don’t forget your homework
due next week!
Have a great
rest of the week!