Stor 155, Section 2, Last Time

• Course Organization & Websitehttp://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155-07Home.html

• What is Statistics?

• Data types and structure

• Get going in EXCEL

• Exploratory Data Analysis

• Bar Graphs

Reading In Textbook

Approximate Reading for Today’s Material:

Pages 14-23

Approximate Reading for Next Class:

Pages 40-55

Stat 31, Student Poll ResultsAs indicated on “Student Info” form:

Big changes from the past:

More Public …

More diversity

Stat 155, Section 2, Majors

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Busine

ss /

Man

.

Biolog

y

Public

Poli

cy /

Health

Pharm

/ Nur

sing

Jour

nalis

m /

Comm

.

Env. S

ci.

Other

Undec

ided

Fre

qu

ency

Stat 31, Student Poll Results

“Have you taken an AP Exam?”

Only ~10% had & grades generally low

So don’t worry if you haven’t…

Stat 31, Student Poll Results

Female: 48

Male: 53

Interesting Point:

Different from all of UNC: ~60 - 40

Lesson about which courses to take???

Major Concept: Distributions

“Distribution” = “Patterns of data”

= “Way data is spread out”

e.g. Bar Graph is visual display of categorical “distribution”

Exploratory Data Analysis 2

Visual Display of Quantitative Distributions:

1. Stem and Leaf Plots

(From last time:) Not Recommended

(Main motivation was pencil and paper statistical analysis, but now have better graphical methods readily accessible)

A limited special case of….

Visual Disp: Quantitative Dist’ns

2. Histograms

Idea: Apply bar graph idea,

By creating categories,

Called “class intervals” or “classes” or “bins”

Histograms

Idea: put numbers into “bins”,

bar heights are counts, or “frequencies”

1.3

3.6

1.9

3.1

1.5

Histograms

Idea: put numbers into “bins”,

bar heights Class Intervals:

1.3 (0,1], (1,2], (2,3], (3,4]

3.6

1.9

3.1

1.5 0 1 2 3 4

Histograms

Idea: put numbers into “bins”,

bar heights are counts, or “frequencies”

1.3

3.6

1.9

3.1

1.5 0 1 2 3 4

Histograms

Idea: put numbers into “bins”,

bar heights are counts, or “frequencies”

1.3

3.6

1.9

3.1

1.5 0 1 2 3 4

Buffalo Snowfall Data

Buffalo, N. Y. (Annual) Snowfall Data

Raw Data:http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg2Raw.xls

63 years, ranging from ~30 - ~120 (inches)

Histogram Analysis (pre-done):http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg2Done.xls

Buffalo Snowfall Data, I

A. EXCEL default (of bin edges)

• Unround numbers for bin edges– Harder to interpret

• Data “centered around 90”

• Most data between 50 and 130

• Assymetric Distribution

Buffalo Snowfall Data, II

B. Smaller bins

• Chosen by me

• Binwidth = 5, << ~13 from EXCEL default

• Nicer edge numbers

• Data centered around 84 (now more precise)

• Bar graph rougher (fewer points in each bin)

• Suggests 3 main groups

(called “modes” or “clusters”)

(can’t see this above: bin width is important)

Buffalo Snowfall Data, III

C. Larger bins

• Chosen by me

• Binwidth = 30, >> ~13 from EXCEL default

• Bar graph is “smooth”

(since many points in each bin)

• Only one mode (cluster)???

• Quite symmetric?

(different from above: bin width is important)

Buffalo Snowfall Data, IV

D. What’s under the hood (how to do this):

i. Tools Data Analysis Histogram (& Chart Out)

(may need Data Analysis “Add-in”)

ii. Massage pic (especially bar width)

iii. Sigma min, max

iv. Bin range: create first two & drag

v. Histogram, using input bin edges

Histogram HWHW: 1.33

• Use Excel and histograms

• Get data from CDrom

• Do both: – Excel Default bins

– Bins set to: 0,10,20,…,240

• Which gives answers closer to answers in back of book?

• Turn in only one page

And now for something completely different

Is this class too “monotone”?

• Easier to understand?

• Calm environment enhances learning?

• Or does it induce somnolence?

What is “somnolence”?

Google definition:

Sleepiness, a condition of

semiconsciousness approaching coma.

And now for something completely different

Recall last class’s Student Questionnaire…

I asked you for:

• Name

• Major

• Contact Info

• Background…

And now for something completely different

One response:

And now for something completely different

OK, will try to send your mind in a different

direction

Hopefully, a mental break …

(not on the Homework Assignment!)

And now for something completely different

An experiment:

• Pull out any coins you have with you

• How many of you have:

– >= 1 penny?

– >= 1 nickel?

– >= 1 dime?

– >= 1 quarter?

• Choose most frequent denomination

And now for something completely different

Collect data (into Spreadsheet):

• Years stamped on coins

(chosen denomination)

• Many as person has

• Enter into spreadsheet

• Look at “distribution” using histogram

And now for something completely different

• Predicted Answer

– From Text Book, Problem 1.32

• Distribution is Left Skewed

• Works out as predicted?

• Why?

• Note: most skewed dist’ns seem to be:

Right Skewed

Histogram Binwidths

Nice Example from the Webster West, U.S.C.:

http://www.stat.sc.edu/~west/applets/histogram.html

Control Binwidth with slider:

• Undersmoothing?

• About right?

• Oversmoothing?

(critical to visual impression)

Histogram Binwidth Example

Hidalgo Stamp Data

From Mexico in 1800s

How many sources of paper?

How many modes:

1, 2, 5, 7, 10?

Histogram Binwidth Example

How many modes (i.e. clusters)?

Caution: Answer depends on binwidth

(a serious and current

statistical research problem)

Have seen all of 2,3,5,7,10 in the literature!

Stamps Data Histogram

How many modes?

2nd Caution: Answer also depends on bin location

(i.e. “shift” of bins)

Histogram Bins

For this course:

Try several binwidths, to “get the idea”

Weakness of EXCEL (we will see several):

This process is inconvenient

Comparison of Histograms

Class Example: Study Habits Data

Idea: Compare Study Habits of Males vs. Females (measured by some “survey score”, perhaps of questionable value?)

http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg4Done.xls

Study Habits Data

EXCEL default histograms:

• Populations look similar???

• Careful: Binwidth very big…

• Careful: Different bin ranges…

• Need smaller binwidths, and common scales

Study Habits Data

Better Choice: Binwidths = 10, same bins for both

• Clear difference, easy to see

• Females higher “on average”

• Males are “more spread”

• 1 “exceptional value”, really true???

Things to look for (in histo’s)

1. Population Center Point (Study Habits Data)

2. Population Spread (Study Habits Data)

3. Shape - Symmetric vs. Skewed

Right Skewed:

Left Skewed:

1. Modes - Unexpected clusters

2. Outliers - “unusual data points”

Histogram Data ExamplesTextbook Applets: from Publisher’s Website

• One Variable Statistical Calculator

• Data Set: Service Times at a Call Center

• Histogram:

(hold mouse button, and slide left-right)

• Results:– Broad range of binwdiths (12 – 25 is “best”?)

– Single bin is useless

– Distribution is Right Skewed

– Clear Outlier

Comparison of Histograms HWHW: 1.35b, 1.34, 1.17• Work in this order• Get data from CDrom• Use EXCEL and histograms• Odd answers in back• You choose the bins

(if you miss something in answers, change this)• Turn in at most one page for each

1.31, 1.32

Exploratory Data Analysis 3

“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”

Class Time Series Example

Monthly Airline Passenger Numbershttp://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg5Done.xls

• Increasing Trend

(long term growth, over years)• Increasing Variation

(appears proportional to trend)• “Seasonal Effect” - 12 Month Cycle

(Peak in summer, less in winter)

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)

Airline Passengers Example

A look under the hoodhttp://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg5Raw.xls

• 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

Time Series HW

HW: 1.36, 1.37

• Use EXCEL