1. Describing Data: Two Variables
STAT 250
Dr. Kari Lock Morgan
Describing Data: Two Variables
SECTIONS 2.4, 2.5
One quantitative variable (2.4)
One quantitative and one categorical (2.4)
Two quantitative (2.5)

2. z-score
Which is better, an ACT score of 28 or a combined SAT score of 2100?
ACT: μ = 21, σ = 5
SAT: μ = 1500, σ = 325
Assume ACT and SAT scores have approximately bell-shaped distributions
ACT score of 28
SAT score of 2100
I don’t know

3. Honeybee Waggle Dances

4. Honeybee Waggle Dance
Honeybee scouts investigate new home or food source options; the scouts communicate the information to the hive with a “waggle dance”
Scientists took bees to an island with only two possible options for nesting: one of very high quality and one of low quality. They recorded
Quality of nesting site
Distance to nesting site
Number of waggle dance circuits performed
Duration of waggle dance B
Seeley, T., Honeybee Democracy, Princeton University Press, Princeton, NJ, 2010, p. 128

5. Questions of the Day
How many circuits of the waggle dance do honey bees do?
How is this related to quality of a nesting site?
How is duration of the dance related to distance to a nesting site?

6. Other Measures of Location
Maximum = largest data value
Minimum = smallest data value
Quartiles:
Q1 = median of the values below m.
Q3 = median of the values above m.

7. Five Number Summary Five Number Summary: Min Max Q1 Q3 m 25%
Minitab: Stat -> Basic Statistics -> Display Descriptive Statistics

8. Five Number Summary The distribution of number of circuits is
Symmetric
Right-skewed
Left-skewed
Impossible to tell

9. The Pth percentile is the value which is greater than P% of the data
We already used z-scores to determine whether an SAT score of 2100 or an ACT score of 28 is better
We could also have used percentiles:
ACT score of 28: 91st percentile
SAT score of 2100: 97th percentile

10. Five Number Summary Five Number Summary: Min Max Q1 Q3 m 25%
0th percentile
25th percentile
50th percentile
75th percentile
100th percentile

11. Measures of Spread Range = Max – Min
Interquartile Range (IQR) = Q3 – Q1
Is the range resistant to outliers?
Yes No
Is the IQR resistant to outliers?

12. Comparing Statistics Measures of Center: Measures of Spread:
Mean (not resistant)
Median (resistant)
Measures of Spread:
Standard deviation (not resistant)
IQR (resistant)
Range (not resistant)
Most often, we use the mean and the standard deviation, because they are calculated based on all the data values, so use all the available information

13. Boxplot Middle 50% of data Median
*For boxplots, outliers are defined as any point more than 1.5 IQRs beyond the quartiles (although you don’t have to know that)
Outlier
Outlier
Lines (“whiskers”) extend from each quartile to the most extreme value that is not an outlier
Middle 50% of data Q3
Median Q1
Minitab: Graph -> Boxplot -> One Y -> Simple

14. Boxplot This boxplot shows a distribution that is Symmetric
Left-skewed
Right-skewed

15. One Quantitative and One Categorical
How is number of waggle circuits related to the quality of the nesting site?
Two variables
One quantitative (number of circuits)
One categorical (quality – low or high)
Can do anything for one quantitative variable, broken down by categorical groups

16. Side-by-Side Boxplots
Minitab: Graph -> Boxplot -> One Y -> With Groups

17. Stacked Dotplots
Minitab: Graph -> Dotplot -> One Y -> With Groups

18. Overlaid Histograms
Minitab: Graph -> Histogram -> With Groups

19. Quantitative Statistics by a Categorical Variable
Any of the statistics we use for a quantitative variable can be looked at separately for each level of a categorical variable
Minitab: Stat -> Basic Statistics -> Display Descriptive Statistics -> By variables

20. Difference in Means
Often, when comparing a quantitative variable across two categories, we compute the difference in means
Honeybees perform 60.5 circuits more, on average, for the high quality site as opposed to the low quality site.

21. Association?
Does there appear to be an association between number of waggle circuits and quality of potential nesting site?
Yes No

22. Summary: One Quantitative and One Categorical
Summary Statistics
Any summary statistics for quantitative variables, broken down by groups
Difference in means
Visualization
Side-by-side graphs

23. Two Quantitative Variables
How is duration of the dance related to distance to a nesting site?
Two quantitative variables
Summary Statistics: correlation
Visualization: scatterplot

24. Scatterplot
A scatterplot is the graph of the relationship between two quantitative variables.
Minitab: Graph -> Scatterplot -> Simple

25. Direction of Association
A positive association means that values of one variable tend to be higher when values of the other variable are higher
A negative association means that values of one variable tend to be lower when values of the other variable are higher
Two variables are not associated if knowing the value of one variable does not give you any information about the value of the other variable

26. Correlation
The correlation is a measure of the strength and direction of linear association between two quantitative variables
Sample correlation: r
Population correlation: ρ (“rho”)
r = for duration of dance and distance to site
Minitab: Stat -> Basic Statistics -> Correlation

27. Correlation -1 ≤ r ≤ 1 The sign indicates the direction of association
positive association: r > 0
negative association: r < 0
no linear association: r ≠ 0
The closer r is to ±1, the stronger the linear association
r has no units and does not depend on the units of measurement
The correlation between X and Y is the same as the correlation between Y and X

28. Correlation Guessing Game
Enter PennState for the group ID.
Highest scorer in the class by the first exam gets one extra credit point on Exam 1!

29. Correlation
NFL Teams
r = 0.43

30. Correlation Cautions
Correlation can be heavily affected by outliers. Always plot your data!

31. Testosterone Levels and Time
What is the correlation between testosterone levels and hour of the day?
Positive
Negative
About 0
Are testosterone level and hour of the day associated?
Yes No

32. Correlation Cautions
Correlation can be heavily affected by outliers. Always plot your data!
r = 0 means no linear association. The variables could still be otherwise associated. Always plot your data!

33. TVs and Life Expectancy

34. Correlation Cautions
Correlation can be heavily affected by outliers. Always plot your data!
r = 0 means no linear association. The variables could still be otherwise associated. Always plot your data!
Correlation does not imply causation!

35. Summary: Two Quantitative Variables
Summary Statistics: correlation
Visualization: scatterplot

36. To Do
Read Sections 2.4 and 2.5
Do HW 2.2, 2.3, 2.4, 2.5 (due Friday, 9/18)