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Lecture Unit 2 Graphical and Numerical Summaries of Data

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### Displaying Qualitative Data

Lecture Unit 2Graphical and Numerical Summaries of Data

UNIT OBJECTIVES

At the conclusion of this unit you should be able to:

- 1) Construct graphs that appropriately describe data
- 2) Calculate and interpret numerical summaries of a data set.
- 3) Combine numerical methods with graphical methods to analyze a data set.
- 4) Apply graphical methods of summarizing data to choose appropriate numerical summaries.
- 5) Apply software and/or calculators to automate graphical and numerical summary procedures.

Section 2.1

“Sometimes you can see a lot just by looking.”

Yogi Berra

Hall of Fame Catcher, NY Yankees

The three rules of data analysis won’t be difficult to remember

- 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.
- 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
- 3. Make a picture —the best way to tellothers about your data is with a well-chosen picture.

Bar Charts: show counts or relative frequency for each category

- Example: Titanic passenger/crew distribution

Pie Charts: shows proportions of the whole in each category

- Example: Titanic passenger/crew distribution

Example: Top 10 causes of death in the United States 2001

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.

The number of individuals who died of an accident in 2001 is approximately 100,000.

Top 10 causes of death: bar graph

Each category is represented by one bar. The bar’s height shows the count (or sometimes the percentage) for that particular category.

Top 10 causes of deaths in the United States 2001

Top 10 causes of deaths in the United States 2001

Bar graph sorted by rank

Easy to analyze

Sorted alphabetically

Much less useful

Computer Hardware Sales 2009 ($billion)

Software Sales 2009 ($billions)

1. United States $1582. China $64.43. Japan $544. Germany $24.45. Britain $23.56. France $19.37. Brazil $14.28. Italy $13.19. Australia $12.810. India $11.9

1. United States $137.92. Japan $23.43. Germany $204. Britain $16.85. France $12.66. Canada $7.37. Italy $6.38. China $5.4 9. Netherlands $5.410. Australia $4.8

NY Times

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.

Percent of people dying from

top 10 causes of death in the United States in 2001

labels match

the data.

Make sure

all percents

add up to 100.

Percent of deaths from top 10 causes

Percent of deaths from all causes

Child poverty before and after government intervention—UNICEF, 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 least—through taxes and subsidies—to 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.

710/2201 32.3%

1491/2201 67.7%

885/2201 40.2%

325/2201 14.8%

285/2201 12.9%

706/2201 32.1%

marg. dist. of class

Contingency Tables: Categories for Two Variables- Example: Survival and class on the Titanic

Marginal distributions

Contingency Tables: Categories for Two Variables (cont.)

- Conditional distributions.

Given the class of a passenger, what is the chance the passenger survived?

Contingency Tables: Categories for Two Variables (cont.)

Questions:

- What fraction of survivors were in first class?
- What fraction of passengers were in first class and survivors ?
- What fraction of the first class passengers survived?

202/710

202/2201

202/325

TV viewers during the Super Bowl in 2007. What is the marginal distribution of those who watched the commercials only?

- 8.0%
- 23.5%
- 58.2%
- 27.7%

10

TV viewers during the Super Bowl in 2007. What percentage watched the game and were female?

- 41.8%
- 38.8%
- 51.2%
- 19.8%

10

TV viewers during the Super Bowl in 2007. Given that a viewer did not watch the Super Bowl, what percentage were male?

- 45.2%
- 48.8%
- 26.8%
- 27.7%

10

3-Way Tables

- Example: Georgia death-sentence data

Simpson’s Paradox

- The reversal of the direction of a comparison or association when data from several groups are combined to form a single group.

Relative Frequency Histogram of Exam Grades

.30

.25

.20

Relative frequency

.15

.10

.05

0

40

50

60

70

80

90

100

Grade

Frequency Histograms

A histogram shows three general types of information:

- It provides visual indication of where the approximate center of the data is.
- We can gain an understanding of the degree of spread, or variation, in the data.
- We can observe the shape of the distribution.

Frequency and Relative Frequency Histograms

- identify smallest and largest values in data set
- 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

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

Example. Number of daily employee absences from work

- 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

- There is no single “correct” answer for the number of classes
- For example, you can choose 6, 7, 8, or 9 classes; don’t choose 15 classes

Absences from Work (cont.)

- 6 classes
- class width: (158-121)/6=37/6=6.17 7
- 6 classes, each of width 7; classes span 6(7)=42 units
- data spans 158-121=37 units
- classes overlap the span of the actual data values by 42-37=5
- lower boundary of 1st class: (1/2)(5) units below 121 = 121-2.5 = 118.5

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

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

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

Stem and leaf displays

- Have the following general appearance

stem leaf

1 8 9

2 1 2 8 9 9

3 2 3 8 9

4 0 1

5 6 7

6 4

Stem and Leaf Displays

- Partition each no. in data into a “stem” and “leaf”
- 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

Example: employee ages at a small company

18 21 22 19 32 33 40 41 56 57 64 28 29 29 38 39; stem: 10’s digit; leaf: 1’s digit

- 18: stem=1; leaf=8; 18 = 1 | 8

stem leaf

1 8 9

2 1 2 8 9 9

3 2 3 8 9

4 0 1

5 6 7

6 4

Advantages/Disadvantages of Stem-and-Leaf Displays

- Advantages

1) each measurement displayed

2) ascending order in each stem row

3) relatively simple (data set not too large)

- Disadvantages

display becomes unwieldy for large data sets

Population of 185 US cities with between 100,000 and 500,000

- Multiply stems by 100,000

Back-to-back stem-and-leaf displays. TD passes by NFL teams: 1999-2000, 2012-13multiply stems by 10

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 10’s digits

- 4
- 6
- 8
- 10
- 12

10

- A distribution is skewed to the rightif the right side of the histogram (side with larger values) extends much farther out than the left side. It is skewed to the leftif the left side of the histogram extends much farther out than the right side.

Skewed distribution

Complex, multimodal distribution

- Not all distributions have a simple overall shape, especially when there are few observations.

- A distribution is symmetricif the right and left sides of the histogram are approximately mirror images of each other.

Shape (cont.): Outliers

An important kind of deviation is an outlier. Outliersare 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.

Alaska

Florida

Other Graphical Methods for Economic Data

- 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.)

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