Slide 2 Lecture Unit 2 Graphical and Numerical Summaries of Data
UNIT OBJECTIVES At the conclusion of this unit you should be able
to: n 1)Construct graphs that appropriately describe data n
2)Calculate and interpret numerical summaries of a data set. n
3)Combine numerical methods with graphical methods to analyze a
data set. n 4)Apply graphical methods of summarizing data to choose
appropriate numerical summaries. n 5)Apply software and/or
calculators to automate graphical and numerical summary procedures.
Slide 3 Displaying Qualitative Data Section 2.1 Sometimes you can
see a lot just by looking. Yogi Berra Hall of Fame Catcher, NY
Yankees Slide 4 The three rules of data analysis wont be difficult
to remember n 1.Make a picture reveals aspects not obvious in the
raw data; enables you to think clearly about the patterns and
relationships that may be hiding in your data. n 2.Make a picture
to show important features of and patterns in the data. You may
also see things that you did not expect: the extraordinary
(possibly wrong) data values or unexpected patterns n 3.Make a
picture the best way to tell others about your data is with a
well-chosen picture. Slide 5 Bar Charts: show counts or relative
frequency for each category n Example: Titanic passenger/crew
distribution Slide 6 Pie Charts: shows proportions of the whole in
each category n Example: Titanic passenger/crew distribution Slide
7 Example: Top 10 causes of death in the United States 2001
RankCauses of deathCounts % of top 10s % of total deaths 1Heart
disease700,14237%28% 2Cancer553,76829%22%
3Cerebrovascular163,5389%6% 4Chronic respiratory123,0136%5%
5Accidents101,5375%4% 6Diabetes mellitus71,3724%3% 7Flu and
pneumonia62,0343%2% 8Alzheimers disease53,8523%2% 9Kidney
disorders39,4802% 10Septicemia32,2382%1% All other causes629,96725%
For each individual who died in the United States in 2001, we
record what was the cause of death. The table above is a summary of
that information. Slide 8 Top 10 causes of deaths in the United
States 2001 Top 10 causes of death: bar graph Each category is
represented by one bar. The bars height shows the count (or
sometimes the percentage) for that particular category. The number
of individuals who died of an accident in 2001 is approximately
100,000. Slide 9 Bar graph sorted by rank Easy to analyze Top 10
causes of deaths in the United States 2001 Sorted alphabetically
Much less useful Slide 10 1. United States $158 2. China $64.4 3.
Japan $54 4. Germany $24.4 5. Britain $23.5 6. France $19.3 7.
Brazil $14.2 8. Italy $13.1 9. Australia $12.8 10. India $11.9 1.
United States $137.9 2. Japan $23.4 3. Germany $20 4. Britain $16.8
5. France $12.6 6. Canada $7.3 7. Italy $6.3 8. China $5.4 9.
Netherlands $5.4 10. Australia $4.8 Software Sales 2009
($billions)Computer Hardware Sales 2009 ($billion) NY Times Slide
11 Percent of people dying from top 10 causes of death in the
United States in 2001 Top 10 causes of death: pie chart Each slice
represents a piece of one whole. The size of a slice depends on
what percent of the whole this category represents. Slide 12
Percent of deaths from top 10 causes Percent of deaths from all
causes Make sure your labels match the data. Make sure all percents
add up to 100. Slide 13 Slide 14 Student Debt North Carolina
Schools Slide 15 Child poverty before and after government
interventionUNICEF, 1996 What does this chart tell you? The United
States has the highest rate of child poverty among developed
nations (22% of under 18). Its government does almost the
leastthrough taxes and subsidiesto remedy the problem (size of
orange bars and percent difference between orange/blue bars). Could
you transform this bar graph to fit in 1 pie chart? In two pie
charts? Why? The poverty line is defined as 50% of national median
income. Slide 16 Slide 17 Unnecessary dimension in a pie chart
Slide 18 Contingency Tables: Categories for Two Variables n
Example: Survival and class on the Titanic Marginal distributions
marg. dist. of survival 710/2201 32.3% 1491/2201 67.7% marg. dist.
of class 885/2201 40.2% 325/2201 14.8% 285/2201 12.9% 706/2201
32.1% Slide 19 Marginal distribution of class. Bar chart. Slide 20
Marginal distribution of class: Pie chart Slide 21 Contingency
Tables: Categories for Two Variables (cont.) n Conditional
distributions. Given the class of a passenger, what is the chance
the passenger survived? Slide 22 Conditional distributions:
segmented bar chart Slide 23 Contingency Tables: Categories for Two
Variables (cont.) Questions: n What fraction of survivors were in
first class? n What fraction of passengers were in first class and
survivors ? n What fraction of the first class passengers survived?
202/710 202/2201 202/325 Slide 24 TV viewers during the Super Bowl
in 2007. What is the marginal distribution of those who watched the
commercials only? 1. 8.0% 2. 23.5% 3. 58.2% 4. 27.7% 10 Slide 25 TV
viewers during the Super Bowl in 2007. What percentage watched the
game and were female? 1. 41.8% 2. 38.8% 3. 51.2% 4. 19.8% 10 Slide
26 TV viewers during the Super Bowl in 2007. Given that a viewer
did not watch the Super Bowl, what percentage were male? 1. 45.2%
2. 48.8% 3. 26.8% 4. 27.7% 10 Slide 27 3-Way Tables n Example:
Georgia death-sentence data Slide 28 UC Berkeley Lawsuit Slide 29
LAWSUIT (cont.) Slide 30 Simpsons Paradox n The reversal of the
direction of a comparison or association when data from several
groups are combined to form a single group. Slide 31 Fly Alaska
Airlines, the on- time airline! Slide 32 American West Wins! Youre
a Hero! Slide 33 Section 2.2 Displaying Quantitative Data
Histograms Stem and Leaf Displays Slide 34 Relative Frequency
Histogram of Exam Grades 0.05.10.15.20.25.30 405060708090 Grade
Relative frequency 100 Slide 35 Frequency Histograms Slide 36 A
histogram shows three general types of information: n It provides
visual indication of where the approximate center of the data is. n
We can gain an understanding of the degree of spread, or variation,
in the data. n We can observe the shape of the distribution. Slide
37 All 200 m Races 20.2 secs or less Slide 38 Histograms Showing
Different Centers Slide 39 Histograms - Same Center, Different
Spread Slide 40 Frequency and Relative Frequency Histograms n
identify smallest and largest values in data set n divide interval
between largest and smallest values into between 5 and 20
subintervals called classes * each data value in one and only one
class * no data value is on a boundary Slide 41 How Many Classes?
Slide 42 Histogram Construction (cont.) * compute frequency or
relative frequency of observations in each class * x-axis: class
boundaries; y-axis: frequency or relative frequency scale * over
each class draw a rectangle with height corresponding to the
frequency or relative frequency in that class Slide 43 Example.
Number of daily employee absences from work n 106 obs; approx. no
of classes= {2(106)} 1/3 = {212} 1/3 = 5.69 1+ log(106)/log(2) = 1
+ 6.73 = 7.73 n There is no single correct answer for the number of
classes n For example, you can choose 6, 7, 8, or 9 classes; dont
choose 15 classes Slide 44 EXCEL Histogram Slide 45 Absences from
Work (cont.) n 6 classes n class width: (158-121)/6=37/6=6.17 7 n 6
classes, each of width 7; classes span 6(7)=42 units n data spans
158-121=37 units n classes overlap the span of the actual data
values by 42-37=5 n lower boundary of 1st class: (1/2)(5) units
below 121 = 121-2.5 = 118.5 Slide 46 EXCEL histogram Slide 47
Grades on a statistics exam Data: 75 66 77 66 64 73 91 65 59 86 61
86 61 58 70 77 80 58 94 78 62 79 83 54 52 45 82 48 67 55 Slide 48
Frequency Distribution of Grades Class Limits Frequency 40 up to 50
50 up to 60 60 up to 70 70 up to 80 80 up to 90 90 up to 100 Total
2 6 8 7 5 2 30 Slide 49 Relative Frequency Distribution of Grades
Class Limits Relative Frequency 40 up to 50 50 up to 60 60 up to 70
70 up to 80 80 up to 90 90 up to 100 2/30 =.067 6/30 =.200 8/30
=.267 7/30 =.233 5/30 =.167 2/30 =.067 Slide 50 Relative Frequency
Histogram of Grades 0.05.10.15.20.25.30 405060708090 Grade Relative
frequency 100 Slide 51 Based on the histo- gram, about what percent
of the values are between 47.5 and 52.5? 1. 50% 2. 5% 3. 17% 4. 30%
10 Slide 52 Stem and leaf displays n Have the following general
appearance stemleaf 18 9 21 2 8 9 9 32 3 8 9 40 1 56 7 64 Slide 53
Stem and Leaf Displays n Partition each no. in data into a stem and
leaf n Constructing stem and leaf display 1) deter. stem and leaf
partition (5-20 stems) 2) write stems in column with smallest stem
at top; include all stems in range of data 3) only 1 digit in
leaves; drop digits or round off 4) record leaf for each no. in
corresponding stem row; ordering the leaves in each row helps Slide
54 Example: employee ages at a small company 18 21 22 19 32 33 40
41 56 57 64 28 29 29 38 39; stem: 10s digit; leaf: 1s digit n 18:
stem=1; leaf=8; 18 = 1 | 8 stemleaf 18 9 21 2 8 9 9 32 3 8 9 40 1
56 7 64 Slide 55 Suppose a 95 yr. old is hired stemleaf 18 9 21 2 8
9 9 32 3 8 9 40 1 56 7 64 7 8 95 Slide 56 Number of TD passes by
NFL teams: 2012-2013 season ( stems are 10s digit) stemleaf 4343 03
247 26677789 201222233444 113467889 08 Slide 57 Pulse Rates n = 138
Slide 58 Advantages/Disadvantages of Stem-and-Leaf Displays n
Advantages 1) each measurement displayed 2) ascending order in each
stem row 3) relatively simple (data set not too large) n
Disadvantages display becomes unwieldy for large data sets Slide 59
Population of 185 US cities with between 100,000 and 500,000 n
Multiply stems by 100,000 Slide 60 Back-to-back stem-and-leaf
displays. TD passes by NFL teams: 1999-2000, 2012-13 multiply stems
by 10 1999-20002012-13 2403 637 2324 665526677789
43322221100201222233444 9998887666167889 4211134 08 Slide 61 Below
is a stem-and-leaf display for the pulse rates of 24 women at a
health clinic. How many pulses are between 67 and 77? Stems are 10s
digits 1. 4 2. 6 3. 8 4. 10 5. 12 10 Slide 62 Interpreting
Graphical Displays: Shape n A distribution is symmetric if the
right and left sides of the histogram are approximately mirror
images of each other. Symmetric distribution Complex, multimodal
distribution Not all distributions have a simple overall shape,
especially when there are few observations. Skewed distribution A
distribution is skewed to the right if the right side of the
histogram (side with larger values) extends much farther out than
the left side. It is skewed to the left if the left side of the
histogram extends much farther out than the right side. Slide 63
Shape (cont.)Female heart attack patients in New York state Age:
left-skewedCost: right-skewed Slide 64 AlaskaFlorida Shape (cont.):
Outliers An important kind of deviation is an outlier. Outliers are
observations that lie outside the overall pattern of a
distribution. Always look for outliers and try to explain them. The
overall pattern is fairly symmetrical except for 2 states clearly
not belonging to the main trend. Alaska and Florida have unusual
representation of the elderly in their population. A large gap in
the distribution is typically a sign of an outlier. Slide 65
Center: typical value of frozen personal pizza? ~$2.65 Slide 66
Spread: fuel efficiency 4, 8 cylinders 4 cylinders: more spread8
cylinders: less spread Slide 67 Other Graphical Methods for
Economic Data n Time plots plot observations in time order, with
time on the horizontal axis and the vari- able on the vertical axis
** Time series measurements are taken at regular intervals (monthly
unemployment, quarterly GDP, weather records, electricity demand,
etc.) Slide 68 Unemployment Rate, by Educational Attainment Slide
69 Water Use During Super Bowl Slide 70 Winning Times 100 M Dash
Slide 71 Annual Mean Temperature Slide 72 End of Section 2.2
