Displaying Distributions with Graphs

1

• Displaying Distributions with Graphs

2
Interesting Problems

• Poker games, lottery,
• Sports statistics,
• Political voting, poll, survey,
• Business, stock market,
• Census,
• Marketing,
• Biological, medical, psychological,
• Practical for decision making

3
Recall

• Statistics is the science of data.
• Collecting
• Analyzing
• Decision making
• Data
• Individuals
• Variables
• Categorical variables
• Quantitative variables

4
NBA Draft 2003 Top 5 Picks
5
Students in STAT 31

• Class Roll
• Variables PID, College, Class, Degree, Major -
  Categorical
• How many categories? How many students in each
  category?
• Equivalently, what is the distribution for each
  variable?

6
Distributions of Variables

• The distribution of a variable indicates what
  values a variable takes and how often it takes
  these values.
• For a categorical variable, distribution
• For a quantitative variable, distribution

7
Variable Class
8
Exploratory Data Analysis (EDA)

• Use statistical tools and ideas to help us
  examine data
• Goal to describe the main features of the data
• NEVER skip this
• EDA
• Displaying distributions with
• Displaying distributions with

9
Basic Strategies for EDA

• Graphical visualizations
• Numerical summaries
• One variable at a time
• Relationships among the variables

10
Graphic Techniques for Categorical Variables

• Bar Graph uses bars to represent the frequencies
  (or relative frequencies) such that the height of
  each bar equals the frequency or relative
  frequency of each category.
• Frequencies counts
• Relative frequencies percent
• Bar Graph height indicates count or percent

11
Graphic Techniques for Categorical Variables
12
Graphic Techniques for Quantitative Variables

• Stemplot (Stem-and-Leaf Plot)
• Histogram
• Time plot

13
Example Midterm Scores

• The following data set contains the midterm
• exam scores
• 74 76 78 88 87 87 53 95 82 79 79 78
• 62 80 77 70 60 60 84 95 85 93 79 84
• 71 85 100 77 72 95 79 83 97 87 73 84
• 74 83 85 95 62 50 86 83 86 36
• Type of variable?

14
Stemplot

• Separate each observation into a stem consisting
  of all but the final (rightmost) digit and a
  leaf, the final digit. Stems may have as many
  digits as needed, but each leaf contains only a
  single digit.
• Write the stems in a vertical column with the
  smallest at the top, and draw a vertical line at
  the right of this column.
• Write each leaf in the row to the right of its
  stem, in increasing order out from the stem.

15
Example Midterm Scores of STAT 101

• The following data set contains the midterm
• exam scores
• 74 76 78 88 87 87 53 95 82 79 79 78
• 62 80 77 70 60 60 84 95 85 93 79 84
• 71 85 100 77 72 95 79 83 97 87 73 84
• 74 83 85 95 62 50 86 83 86 36

16
Example Midterm Scores of STAT 101

• A stem-and-leaf display is follows
• 3 6
• 4
• 5 03 Leaf last digit
• 6 0022 Stem remaining digit(s)
• 7 012344677889999
• 8 02333444555667778
• 9 355557
• 10 0

17
Back-to-back Stemplot

• Babe Ruth (New York Yankees) 1920-1934
• 54 59 35 41 46 25 47 60 54 46 49 46 41 34 22
• Mark McGwire (St. Louis Cardinals) 19862001
• 3 49 32 33 39 22 42 9 9 39 52 58 70 65 32 29

18
Back-to-back Stemplot

• Ruth McGwire
• 0 3 9 9
• 1
• 5 2 2 2 9
• 5 4 3 2 2 3 9 9
• 9 7 6 6 6 1 1 4 2 9
• 9 4 4 5 2 8
• 0 6 5
• 7 0

19
Splitting stems rounding

• For a moderate number of obs,
• Split each stem into two one with leaves 0-4 and
  the other with leaves 5-9
• Increase of stems, reduce of leaves
• Rounding
• If many stems have no leaves or only one leaf,
  rounding may help.

20
Spending at a supermarket
21
Example A study on litter size

• Data (170 observations)
• 4 6 5 6 7 3 6 4 4 6 4 4 9 5 10 6
  6 5 6 8 2 7 7 7 9 3 7 5 7 7 4 5
  5 6 7 6 7 8 6 6 7 6 6 7 5 4 5 6 6
  1 3 4 7 5 4 7 5 8 8 5 6 8 5 5 4
  9 6 7 3 7 7 5 4 6 9 6 7 7 5 7 3 7
  6 5 3 7 10 5 6 8 7 5 5 7 5 5 8 9
  7 5 7 5 5 5 6 3 7 8 7 7 6 3 4 4 4
  7 2 7 8 5 8 6 6 5 6 4 7 5 5 6 9
  3 5 4 8 3 9 8 3 6 5 4 7 8 4 8 6 8
  5 6 4 3 8 8 6 9 5 5 6 6 7 6 8 6
  11 6 5 6 6 3

22
Stem-and-leaf plot for pups

• 0122333333333333344 (35)
• 0555555555555555555555555... (132)
• 1 001

23
Limitations of Stemplot

• Awkward for large data sets
• Splitting stem/rounding is not very helpful.

24
Histogram

• breaks the range of the values of a quantitative
  variable into intervals and displays only the
  count or percent of the observations that fall
  into each interval.
• You can choose any convenient number of
  intervals.
• Intervals must be of equal width.

25
Example A study on litter size
26
Example Call Center Data

• Financial firm call center
• Calls handled by AVI within 60 seconds
• October 666
• December 523
• Avi Service Time Data

27
October
28
December
29
Notes for Making Histogram

• Choose the number of classes sensibly
• Too few classes skyscraper graph
• Too many pancake graph
• Sturges rule
• Choose number of classes k such that
• where n is the sample size
• Intervals must be of equal width.
• Areas of the bars are proportional to the
  frequency.

30
Examining Distributions

• Overall Pattern
• Shape
• Center (numerical, Lecture 3)
• midpoint
• Spread (numerical, Lecture 3)
• range
• Deviations
• Outliers some values that fall outside the
  overall pattern.

31
Shapes of Distributions

• Graphs can help to determine shapes.
• Modes peaks of a distribution.
• Unimodal one peak
• Bimodal two peaks
• Symmetric or skewed?

32
Shakespeares Words
33
A unimodal histogram
34
Tuition and fees
35
A bimodal histogram
36
Shakespeares Words
37
Skewness
Left skewed
Right skewed
38
Iowa Test of Basic Skills vocabulary scores
39
A study on litter size
40
Bell-shaped Histograms
41
Summary Shapes of Distributions

• Symmetric
• histogram in which the right half is a mirror
  image of the left half.
• Skewed to the right
• histogram in which the right tail is more
  stretched out than the left.(long tail to the
  right)
• Skewed to the left
• histogram the left tail is more stretched out
  than the right.(long tail to the left)
• Number of modal classes
• the number of distinct peaks in a histogram
• Bell-shaped
• A histogram looks like a bell.

42
Time plots

• A time plot of a variable plots each obs against
  the time at which it was measured.
• Time x-axis
• Variable y-axis
• Examples stock price, unemployment rate, daily
  temperature
• Great for identifying changing patterns related
  to time.
• What to look for
• .
• .
• .

43
Example Number of Suicides in USA (1900-1970)
44
Call Center Daily Call Volume in Sep. 2002
45
Call Center Monthly Call Volume in 2002
46
Outliers

• Observations that lie outside the overall pattern
  of a distribution.
• Possible reasons
• error in data entry (most likely reason)
• extraordinary individuals (Jordans salary)

47
Handling Outliers

• Detect it using graphical and numerical methods.
• Check the data to make sure correct entry.
• Reducing influence of outlier
• delete the observation (BE CAREFUL!)
• Use transformations, robust methods.

48
Speed of Light (Histogram)
49
Speed of Light (Time plot)
50
Remember

• Distribution of variables
• Examine distributions
• Overall pattern
• Shape
• Symmetric or skewed
• How many modes?
• Bell-shaped
• Outliers
• Graphical tools for categorical data
• Bar graph
• Pie chart
• Graphical tools for quantitative data
• Stemplot
• Histograms
• Time plots