1. STAT 250 Dr. Kari Lock Morgan
Describing Data II
SECTIONS 2.3, 2.4, 2.5
One quantitative variable (2.3, 2.4)
One quantitative by one categorical (2.4)
Two quantitative (2.5)

2. The 95% Rule
The standard deviation for hours of sleep per night is closest to 1 2 4
I have no idea

3. z-score
The z-score for a data value, x, is 𝑧= 𝑥− 𝑥 𝑠 for sample data, and 𝑧= 𝑥−𝜇 𝜎 for population data.
z-score measures the number of standard deviations away from the mean

4. z-score A z-score puts values on a common scale
A z-score is the number of standard deviations a value falls from the mean
For symmetric, bell-shaped distributions, 95% of all z-scores fall between -2 and 2, so z-scores beyond these values can be considered extreme

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

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
> summary(study_hours)
Min. 1st Qu. Median 3rd Qu. Max.
The distribution of number of hours spent studying each week 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
Lines (“whiskers”) extend from each quartile to the most extreme value that is not an outlier Q3
Middle 50% of data
Median Q1
Minitab: Graph -> Boxplot -> One Y -> Simple

14. Boxplot
Outlier
*For boxplots, outliers are defined as any point more than 1.5 IQRs beyond the quartiles (although you don’t have to know that)

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

16. 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

17. Quantitative and Categorical Relationships
Interested in a quantitative variable broken down by categorical groups

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

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

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

21. 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

22. Difference in Means
Often, when comparing a quantitative variable across two categories, we compute the difference in means
𝑥 𝐹 − 𝑥 𝑀 = −24.466=1.12

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. Two Quantitative Variables
Summary Statistics: correlation
Visualization: scatterplot

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

26. 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

27. Exploring Associations
In the states data, explore the associations between obesity rate and the following variables:
PhysicalActivity: % doing physical activity in the past month
Smokers: % who smoke
Population: State population (in millions)
HouseholdIncome: Mean household income (in $)
McCainVote: % voting for McCain in 2008 election
IQ: Mean IQ score
Make your initial guesses…

28. Associations
Minitab: Graph -> Scatterplot -> Simple -> Multiple Graphs -> In separate panels of the same graph

29. Correlation
The correlation is a measure of the strength and direction of linear association between two quantitative variables
Sample correlation: r
Population correlation:  (“rho”)
Minitab: Stat -> Basic Statistics -> Correlation

30. What are the properties of correlation?
Correlations
What are the properties of correlation?

31. Correlation

32. Correlation Guessing Game
Enter PennState for the group ID.

33. Correlation
NFL Teams
r = 0.43

34. 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

35. TVs and Life Expectancy

36. 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!

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

38. Lots of Scatterplots
Minitab: Graph -> Matrix Plot

39. 3 Variables: Adding a Categorical Variable to a Scatterplot
Minitab: Graph -> Scatterplot -> With Groups

40. 3 Variables: Adding a Quantitative Variable to a Scatterplot
Minitab: Graph -> Bubble Plot -> Simple

41. Four Variables!: Adding a categorical and a quantitative variable to a scatterplot
Minitab: Graph -> Bubble Plot -> With Groups

42. Scatterplot with Histograms/Boxplots/Dotplots
Minitab: Graph -> Marginal Plot

43. To Do
Read Sections 2.4 and 2.5
Do Homework 2.2, 2.3, 2.4, 2.5 (due Friday, 2/6)