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. Honeybee Waggle Dances

3. 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

4. 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?

5. 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.

6. Five Number Summary
Five Number Summary:
Min
Max Q1 Q3 m
25%

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

8. 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

9. Small for Gestational Age (SGA)
Children are diagnosed as Small for Gestational Age (SGA) if, when compared to other children of the same gestational age at birth, their birth weight…
Has a z-score less than -2
Is less than the 3rd percentile
When do we expect these two criteria to be roughly the same?
For any distribution
For any symmetric distribution
Only for bell-shaped distributions
Never

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. Summary: One Quantitative Variable
Summary Statistics
Center: mean, median
Spread: standard deviation, range, IQR
5 number summary
Percentiles
Visualization
Dotplot
Histogram
Boxplot
Other concepts
Shape: symmetric, skewed, bell-shaped
Outliers, resistance
z-scores

16. 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?

17. 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

18. Side-by-Side Boxplots

19. Stacked Dotplots

20. 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

21. 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.

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

23. 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

24. 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?

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

26. Scatterplot
A scatterplot is the graph of the relationship between two quantitative variables.

27. 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

28. Correlation
The correlation is a measure of the strength and direction of linear association between two quantitative variables
Sample correlation: r
Population correlation:  (“rho”)
r = for duration of dance and distance to site

29. 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

30. 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!

31. Correlation
NFL Teams
r = 0.43

32. 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

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. Variable(s) Visualization Summary Statistics Categorical bar chart,
pie chart
frequency table,
relative frequency table, proportion
Quantitative
dotplot,
histogram,
boxplot
mean, median, max, min, standard deviation, range, IQR,
five number summary
Categorical vs Categorical
side-by-side bar chart, segmented bar chart
two-way table,
difference in proportions
Quantitative vs Categorical
side-by-side boxplots,
Stacked dotplots
statistics by group,
difference in means
Quantitative vs Quantitative
scatterplot
correlation

37. Descriptive Statistics
Think of a topic or question you would like to use data to help you answer.
What would the cases be?
What would the variables be? (Limit to one or two variables)
How would you visualize or summarize the variable or association between variables?

38. Descriptive Statistics
How would you visualize and summarize the variable or relationship between variables?
bar chart/pie chart, proportions, frequency table/relative frequency table
dotplot/histogram/boxplot, mean/median, sd/range/IQR, five number summary
side-by-side bar charts, difference in proportions, two-way table
side-by-side boxplot, difference in means
scatterplot, correlation

39. To Do
HW 2.2, 2.3, 2.4, 2.5 (due Monday, 2/6)