1. Lecture #3 Tuesday, August 30, 2016 Textbook: Sections 2.4 through 2.6
Statistics 200
Lecture #3 Tuesday, August 30, 2016
Textbook: Sections 2.4 through 2.6
Objectives (all relating to quantitative variables):
• Recognize and interpret two plots:
– Histogram
– Boxplot
• Calculate and interpret two measures of center
– Mean
– Median
• Calculate and interpret five-number summary
• Recognize and understand effects of outliers & skewness

2. Motivating example
A group of students was randomly assigned to one of two classes. One class was taught by teacher A and the other by teacher B. At the end of the semester, all students took the same exam.
Investigate whether there is any difference in exam scores between the two teachers.

3. Summarizing Quantitative Variables
The distribution of a quantitative variable is the overall pattern of how often the possible values occur.
Four key aspects of the distribution are:
Location: center, average
Spread: variability
Shape: symmetric, bell, skew
outliers
Let’s begin with the shape, which is best seen with a visual summary

4. Visual summaries for quantitative variables
Histogram
Boxplot
A chart of the data that shows how many observations are in each equally spaced interval.
Usually use 6-15 intervals
Can use frequency or relative frequency

5. Histograms
Teacher A Scores
Teacher B Scores

6. Outlier
An individual value that is unusual compared to the bulk of the other values.
Outlier!

7. Example
When considering study hours/week, what percent of the students spend:
at most 3 hours?
at least 11 hours?
between 5 and 9 hours?

8. Shapes of distributions
Symmetric the shape of the data is similar on both sides of the center.
Bell-shaped is a special case of symmetric
Skewed: Values are more spread out on one side than the other.
Left-skewed: lower values more spread out than higher values
Right-skewed: higher values more spread out than lower values.

9. Shape Examples: Symmetric
Question: What is the fastest you have ever driven a car?
Symmetric

10. Shape Examples: Right-skewed Left-skewed
Question: How many coins are you carrying?
Right-skewed
Left-skewed
Question: What is your grade point average?

11. Breakdown of Descriptive Statistical Methods: Quantitative Data
graphs
numbers: statistics
Measures
of center
did one: histogram
do now

12. Quantitative Data: Measures of Center
Mean:
___________ of all numbers
symbol for sample mean:
Value is sensitive to ______________
Median:
middle observation of ___________ data
value is resistant to ________________
Mode:
observation that occurs most frequently
don’t really use in this course
Average
Outliers
ordered
outliers

13. Example: Center and outliers
Sample 1 (n = 5)
Sample 2 (n = 6)
Sample
Mean
( )/5 =
32/5 =
Mean = ____
( )/5 =
55/5 =
Ordered
Data/
Median
Median = ____
Median = _____

14. Sensitive vs. Resistant statistics
Calculated using ALL observations
Affected by skewness and / or unusual observations.
Example: Mean
Sensitive Statistic
Resistant Statistic
Calculated using only some observations
Not affected much by outliers
Example: Median

15. Examples: mean = 94.8 mph median = 95 mph mean = 17.3 coins
median = 9 coins

16. Work together question:
Which is most likely true when considering salaries($) in a company that employs:
1. 20 factory workers and 2 very highly paid executives: one would find with the salaries that the:
mean > median
mean < median
mean ≈ median
2. 2 factory workers and 20 very highly paid executives: one would find with the salaries that the:

18. A percentile tells us how much of the data is below a specific value.
Percentiles
What is the value (in studyhrs/week) for the:
5th percentile?
90th percentile?

19. Percentiles of Interest
25th percentile:
___________Quartile (QL)
___________ Quartile (Q1)
Lower
First
50th percentile:
Second Quartile (Q2)
________
Median
75th percentile:
__________ Quartile (QU)
__________ Quartile (Q3)
Upper
Third

20. We use quartiles for the…
Five Number Summary
smallest
number
(min)
lower or
first
quartile
median
upper or
third
largest
(max)
Numerical method for summarizing quantitative data.

21. Example: 5-Number summary
Descriptive Statistics: Fastest_Speed
Variable N Minimum Q1 Median Q3 Maximum
Fastest_Speed
Fill-in the five number summary
25th
50th
75th
Min Q1
Median Q3
Max

22. Another look: 5-number summary
The 5-number summary divides your data into 4 quarters:

23. Approximately what percent of the fastest speeds:
Min Q1
Median Q3
Max 45 90 95
100
135
Approximately what percent of the fastest speeds:
are at least 100 mph?
are at most 90 mph?

24. Approximately what percent of the fastest speeds lie:
Min Q1
Median Q3
Max 45 90 95
100
135
Approximately what percent of the fastest speeds lie:
between 90 and 100 mph?
(at most 95) or (at least 100?) 45 90 95
100
135

25. Visual summaries for quantitative variables
Histogram
Boxplot
A chart of the data that shows how many observations are in each equally spaced interval.
Usually use 6-15 intervals
Can use frequency or relative frequency
Visualization of the 5-number summary
Shows Q1, Median, Q3
as lines around and through a middle box.
Identifies outliers.

26. Boxplots: Examples Max 135 mph Q3 100 mph Median 95 mph Q1 90 mph Min
80 coins Q3
25 coins
Median
9 coins Q1
5 coins
Min
0 coins

27. Boxplot shows same shape as histogram
Symmetric

28. Boxplot shows same shape as histogram
Right-skewed

29. Boxplot shows same shape as histogram
Left-skewed

30. Link measures of center to shape

31. Another example: Parties per month
Outliers!

32. Parties per month, without the outliers

33. Median: 50% of students surveyed partied less than 4.5 times per month.
Right-skewed  mean > median

34. Consider the variables Party and Year
Response
How many parties do you attend in a month?
What year are you in school?
Explanatory

35. Consider the variables Party and Year
How many parties do you attend in a month?
What year are you in school?
Quantitative
Categorical
(ordinal)

36. Explore relationship with boxplot

37. Which year has highest median?
Largest box?
Most outliers?
Do we observe a trend?

38. Review: If you understood today’s lecture, you should be able to solve
Objectives (all relating to quantitative variables):
• Recognize and interpret two plots:
– Histogram
– Boxplot
• Calculate and interpret two measures of center
– Mean
– Median
• Calculate and interpret five-number summary
• Recognize and understand effects of outliers & skewness