Incomes and Other Quantities

1
Section 5.1

• Incomes and Other Quantities

2
Examples of Categorical Variables

• What is your gender?
• Did you see Toni Morrison last night?
• How confident are you that youll be able to find
  a job in your major upon graduation?
• not confident at all somewhat confident
• confident very confident

3
Numerical Summaries of Categorical Data

• What is your gender?
• There were 10 females and 20 males in the sample.
• How can we numerically summarize this data
  besides reporting raw counts?

4
Summarizing Categorical Data

• So the proportion of females is 33.33 and the
  proportion of males is 67.77.
• Are there other appropriate numeric measurements?
  Does it makes sense to describe gender by an
  average? NO.

5
Job Confidence Results

• 10 say not confident
• 15 say somewhat confident
• 30 say confident
• 20 say very confident
• How can we numerically summarize this data?
  Using proportions.

6
Graphical Displays of Categorical Data

• Bar Graphs
• Pie Charts

7
Numeric Variables

• Numeric data consists of numbers representing
  measurements.
• The text calls numeric data, number line
  data.
• Examples
• Weights of football players
• Prices of college textbooks
• Age of US Presidents at inauguration

8
Looking Ahead

• Chapters 5-7 examine many of the same ideas that
  we studied in Chapters 1-4, except from the
  point of view of numeric variables.
• Similar to before, well look at numerical and
  graphical summaries of data, sampling
  distributions of statistics, confidence
  intervals, and hypothesis tests.

9
Overview of Chapter 5 (in part)

• Numerical Summaries of Numeric Variables
• Measures of center What is the center value?
• Measures of spread Is the data set close to the
  center or spread out?

10
Numerical Summaries for Numeric Data

• Prices of college textbooks
• 82.50, 75.50, 27.50, 88.25, 79.00, 120.50,
  90.25, 68.50, 85.50, 90.25
• How should we summarize this numeric data?
• Does computing proportions make sense?

11
Measurements of Center for Numeric Data

• Three common measures of center are
• Mean arithmetic average
• Median middle value
• Mode most frequent

12
CO2 Pollution of the 8 Largest Nations

• The Pew Center on Global Climate Change reports
  that possible global warming is due in large part
  to human activity that produces carbon dioxide
  emissions and other greenhouse gases. The CO2
  emissions from fossil fuel combustion are the
  result of the generation of electricity, heating,
  and gas consumption in cars.

13
Which countries are most populated?

• http//www.aneki.com/populated.html

14
Per capita CO2 emissions for the 8 largest
countries in population size (metric tons/person)

• China 2.3
• India 1.1
• USA 19.7
• Indonesia 1.2
• Russia 9.8
• Brazil 1.8
• Pakistan 0.7
• Bangladesh 0.2

15
Dotplot of the CO2 emissions data
16
Mean

• Defn The sum of the data values divided by the
  number of data values.
• Ex Find the mean the 8 countries
• Ans

17
Median

• Defn Center value of ordered data.
• Ex Find the median of the data set.
• Begin by ordering the data.
• 0.2, 0.7, 1.1, 1.2, 1.8, 2.3, 9.8, 19.7
• Since there are an even number of data
  points, the median is the mean of the middle two
  values, 1.2 and 1.8. So the median is 1.5.

18
Why two measures of center?

• The mean and median are usually different so
  journalists have an opportunity to mislead you by
  which one is reported.
• Ex In 2004 the median household income was 44,
  389 and the mean household income was 60,528.

19
Mean vs. Median

• Median is below about half of its observations.
• Its possible for the mean to be below most of
  the observations.
• Ex http//bcs.whfreeman.com/ips5e/default.asp?s
  nivons0uid0rau0

20
Describing the Shape of a Histogram

• Mean is the balance point.
• If a histogram is symmetrical, its balance point
  is the middle observation. In this case,
  meanmedian.
• Distributions that are not symmetrical are skewed
  either to the right (tail extends out further
  to the right than the left) or to the left (tail
  extends out further to the left than to the
  right.)

21
Skewed RightHow much cash do you have on
you?Median 15 Mean 35.82
22
Skewed left
23
Number of States VisitedMedian 15 Mean
16.43
24
Mean follows skewness

• If a distribution of data is skewed, the mean
  will be farther towards the tail than the median.

25
Exercises

• The workers and management of a company are
  having a labor dispute. Explain why workers
  might use the median income of all employees to
  justify a raise but management might use the mean
  to argue that a raise is not needed.
• The mean age of four people in a room is 30
  years. A new person whose age is 55 years enters
  the room. What is the mean age of the five
  people in the room?