1. Stat 31, Section 1, Last Time
Distributions (how are data “spread out”?)
Visual Display: Histograms
Binwidth is critical
Bivariate display: scatterplot
Course Organization & Website

2. Exploratory Data Analysis 4
“Time Plots”, i.e. “Time Series:
Idea: when time structure is important,
plot variable as a function of time:
variable
time
Often useful to “connect the dots”

3. Class Time Series Example
Monthly Airline Passenger Numbers
Increasing Trend
(long term growth, over years)
Increasing Variation
(appears proportional to trend)
“Seasonal Effect” Month Cycle
(Peak in summer, less in winter)

4. Airline Passengers Example
Interesting variation: log transformation
Stabilizes variation
Since log of product is sum
Shows changing variation prop’l to trend
Log10 is “most interpretable”
(log10(1000) = 3, …)
Generally useful trick (there are others)

5. Airline Passengers Example
A look under the hood
Use Chart Wizard
Chart Type: Line (or could do XY)
Use subtype for points & lines
Use menu for first log10
Although could just type it in
Drag down to repeat for whole column

6. Time Series HW
HW: 1.36, 1.37
Use EXCEL

7. Exploratory Data Analysis 5
Numerical Summaries of Quant. Variables:
Idea: Summarize distributional information
(“center”, “spread”, “skewed”)
In Text, Sec. 1.2
for data
(subscripts allow “indexing numbers” in list)

8. Numerical Summaries “Centers” (note there are several)
“Mean” = Average =
Greek letter “Sigma”, for “sum”
In EXCEL, use “AVERAGE” function

9. Numerical Summaries of Center
“Median” = Value in middle (of sorted list)
Unsorted E.g: Sorted E.g:
27 “in middle”? (no) 2 better “middle”!
EXCEL: use function “MEDIAN”

10. Difference Betw’n Mean & Median
Symmetric Distribution: Essentially no difference
Right Skewed:
50% area % area M
bigger since “feels tails more strongly”

11. Difference Betw’n Mean & Median
Outliers (unusual values):
Nice Web Example:
Mean feels outliers much more strongly
Leaves “range of most of data”
Good notion of “center”? (perhaps not)
Median affected very minimally
Robustness Terminology:
Median is “resistant to the effect of outliers”

12. Difference Betw’n Mean & Median
A more flexible web example:
Get various dist’ns, by manipulating bar heights
See Mean, Median and more
Similar for symmetric distributions
Very different when skewed
“Big Gap”, can make median jump a lot
But mean is less sensitive (more “continuous”)

13. Numerical Centerpoint HW
HW: a (but make histograms), b
Use EXCEL

14. Numerical Summaries (cont.)
“Spreads” (again there are several)
1. Range = biggest smallest
range
Problems:
Feels only “outliers”
Not “bulk of data”
Very non-resistant to outliers

15. Numerical Summaries of Spread
Variance =
= “average squared distance to “
EXCEL: VAR
Drawback: units are wrong
e. g. For in feet  is in square feet

16. Numerical Summaries of Spread
Standard Deviation
EXCEL: STDEV
Scale is right
But not resistant to outliers
Will use quite a lot later
(for reasons described later)

17. Interactive View of S. D. Revisit flexible web example:
Note SD range centered at mean
Can put SD “right near middle” (densely packed data)
Can put SD at “edges of data” (U shaped data)
Can put SD “outside of data” (big spike + outlier)
But generally “sensible measure of spread”

18. Variance – S. D. HW HW: for both data sets in 1.49, find the:
Standard Deviation (26.4, )
Use EXCEL

19. Numerical Summaries of Spread
Interquartile Range = IQR
Based on “quartiles”, Q1 and Q3
(idea: shows where are 25% & 75% “through the data”)
25% % % %
Q Q2 = median Q3
IQR = Q3 – Q1

20. Quartiles Example Revisit flexible web example: Right skewness gives:
Right skewness gives:
Median < Mean
(mean “feels farther points more strongly”)
Q1 near median
Q3 quite far
(makes sense from histogram)

21. Tools  Data Analysis  Descriptive Stats
Quartiles Example
A look under the hood:
Can compute as separate functions for each
Or use:
Tools  Data Analysis  Descriptive Stats
Which gives many other measures as well
Use “k-th largest & smallest” to get quartiles

22. 5 Number Summary Summarize Information About: Minimum
Q st Quartile
Median
Q rd Quartile
Maximum
Summarize Information About:
Center - from 3
Spread - from 2 & 4 (maybe 1 & 6)
Skewness - from 2, 3 & 4
Outliers - from 1 & 5

23. 5 Number Summary How to Compute? EXCEL function QUARTILE
EXCEL function QUARTILE
“One stop shopping”
IQR seems to need explicit calculation

24. Rule for Defining “Outliers”
Caution: There are many of these
Textbook version:
Above Q * IQR
Below Q1 – 1.5 * IQR
For stamps data:
No outliers at “low end”
Some that “high end”

25. Box Plot Additional Visual Display Device
Again legacy from pencil & paper days
Not supported in EXCEL
We will skip

26. 5 Number Sum. & Outliers HW
1.49 c, d
and add:
(d) How much does the mean change if you omit Montana and Wyoming?