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Chapter 2: Visualizations of Data

Created by: CK-12

Chapter Outline

Chapter Summary

Part One: Questions

  1. Which of the following can be inferred from this histogram?
    1. The mode is 1.
    2. mean < median
    3. median < mean
    4. The distribution is skewed left.
    5. None of the above can be inferred from this histogram.
  2. Sean was given the following relative frequency histogram to read. Unfortunately, the copier cut off the bin with the highest frequency. Which of the following could possibly be the relative frequency of the cut-off bin?
    1. 16
    2. 24
    3. 32
    4. 68
  3. Tianna was given a graph for a homework question in her statistics class, but she forgot to label the graph or the axes and couldn’t remember if it was a frequency polygon or an ogive plot. Here is her graph: Identify which of the two graphs she has and briefly explain why.

In questions 4-7, match the distribution with the choice of the correct real-world situation that best fits the graph.

    1. Endy collected and graphed the heights of all the 12^{\text{th}} grade students in his high school.
    2. Brittany asked each of the students in her statistics class to bring in 20 pennies selected at random from their pocket or piggy bank. She created a plot of the dates of the pennies.
    3. Thamar asked her friends what their favorite movie was this year and graphed the results.
    4. Jeno bought a large box of doughnut holes at the local pastry shop, weighed each of them, and then plotted their weights to the nearest tenth of a gram.
  1. Which of the following box plots matches the histogram?
  2. If a data set is roughly symmetric with no skewing or outliers, which of the following would be an appropriate sketch of the shape of the corresponding ogive plot?
  3. Which of the following scatterplots shows a strong, negative association?

Part Two: Open-Ended Questions

  1. The Burj Dubai will become the world’s tallest building when it is completed. It will be twice the height of the Empire State Building in New York.
Building City Height (ft)
Taipei 101 Tapei 1671
Shanghai World Financial Center Shanghai 1614
Petronas Tower Kuala Lumpur 1483
Sears Tower Chicago 1451
Jin Mao Tower Shanghai 1380
Two International Finance Center Hong Kong 1362
CITIC Plaza Guangzhou 1283
Shun Hing Square Shenzen 1260
Empire State Building New York 1250
Central Plaza Hong Kong 1227
Bank of China Tower Hong Kong 1205
Bank of America Tower New York 1200
Emirates Office Tower Dubai 1163
Tuntex Sky Tower Kaohsiung 1140

The chart lists the 15 tallest buildings in the world (as of 12/2007).

(a) Complete the table below, and draw an ogive plot of the resulting data.

Class Frequency Relative Frequency Cumulative Frequency Relative Cumulative Frequency

(b) Use your ogive plot to approximate the median height for this data.

(c) Use your ogive plot to approximate the upper and lower quartiles.

(d) Find the 90^{\text{th}} percentile for this data (i.e., the height that 90% of the data is less than).

  1. Recent reports have called attention to an inexplicable collapse of the Chinook Salmon population in western rivers (see http://www.nytimes.com/2008/03/17/science/earth/17salmon.html). The following data tracks the fall salmon population in the Sacramento River from 1971 to 2007.
Year ^* Adults Jacks
1971-1975 164,947 37,409
1976-1980 154,059 29,117
1981-1985 169,034 45,464
1986-1990 182,815 35,021
1991-1995 158,485 28,639
1996 299,590 40,078
1997 342,876 38,352
1998 238,059 31,701
1998 395,942 37,567
1999 416,789 21,994
2000 546,056 33,439
2001 775,499 46,526
2002 521,636 29,806
2003 283,554 67,660
2004 394,007 18,115
2005 267,908 8.048
2006 87,966 1,897

Figure: Total Fall Salmon Escapement in the Sacramento River. Source: http://www.pcouncil.org/newsreleases/Sacto_adult_and_jack_escapement_thru%202007.pdf

During the years from 1971 to 1995, only 5-year averages are available.

In case you are not up on your salmon facts, there are two terms in this chart that may be unfamiliar. Fish escapement refers to the number of fish who escape the hazards of the open ocean and return to their freshwater streams and rivers to spawn. A Jack salmon is a fish that returns to spawn before reaching full adulthood.

(a) Create one line graph that shows both the adult and jack populations for these years. The data from 1971 to 1995 represent the five-year averages. Devise an appropriate method for displaying this on your line plot while maintaining consistency.

(b) Write at least two complete sentences that explain what this graph tells you about the change in the salmon population over time.

  1. The following data set about Galapagos land area was used in the first chapter.
Island Approximate Area (sq. km)
Baltra 8
Darwin 1.1
Española 60
Fernandina 642
Floreana 173
Genovesa 14
Isabela 4640
Marchena 130
North Seymour 1.9
Pinta 60
Pinzón 18
Rabida 4.9
San Cristóbal 558
Santa Cruz 986
Santa Fe 24
Santiago 585
South Plaza 0.13
Wolf 1.3

Figure: Land Area of Major Islands in the Galapagos Archipelago. Source: http://en.wikipedia.org/wiki/Gal%C3%A1pagos_Islands

(a) Choose two methods for representing this data, one categorical, and one numerical, and draw the plot using your chosen method.

(b) Write a few sentences commenting on the shape, spread, and center of the distribution in the context of the original data. You may use summary statistics to back up your statements.

  1. Investigation: The National Weather Service maintains a vast array of data on a variety of topics. Go to: http://lwf.ncdc.noaa.gov/oa/climate/online/ccd/snowfall.html. You will find records for the mean snowfall for various cities across the US.
    1. Create a back-to-back stem-and-leaf plot for all the cities located in each of two geographic regions. (Use the simplistic breakdown found at http://library.thinkquest.org/4552/ to classify the states by region.)
    2. Write a few sentences that compare the two distributions, commenting on the shape, spread, and center in the context of the original data. You may use summary statistics to back up your statements.

Keywords

Back-to-back stem plots
Stem plots can also be a useful tool for comparing two distributions when placed next to each other. These are commonly called back-to-back stem plots.
Bar graph
the bars in a bar graph usually are separated slightly. The graph is just a series of disjoint categories.
Bias
The systematic error in sampling is called bias.
Bivariate data
The goal of examining bivariate data is usually to show some sort of relationship or association between the two variables.
Box-and-whisker plot
A box-and-whisker plot is a very convenient and informative way to represent single-variable data.
Cumulative frequency histogram
A relative cumulative frequency histogram would be the same, except that the vertical bars would represent the relative cumulative frequencies of the data
Density curves
The most important feature of a density curve is symmetry. The first density curve above is symmetric and mound-shaped.
Dot plot
A dot plot is one of the simplest ways to represent numerical data.
Explanatory variable
the time in years is considered the explanatory variable, or independent variable.
Five-number summary
The five-number summary is a numerical description of a data set comprised of the following measures (in order): minimum value, lower quartile, median, upper quartile, maximum value.
Frequency polygon
A frequency polygon is similar to a histogram, but instead of using bins, a polygon is created by plotting the frequencies and connecting those points with a series of line segments.
Frequency tables
to create meaningful and useful categories for a frequency table.
Histogram
A histogram is a graphical representation of a frequency table (either actual or relative frequency).
Modified box plot
This box plot will show an outlier as a single, disconnected point and will stop the whisker at the previous point.
Mound-shaped
it has a single large concentration of data that appears like a mountain. A data set that is shaped in this way is typically referred to as mound-shaped.
Negative linear association
If the ellipse cloud were trending down in this manner, we would say the data had a negative linear association.
Ogive plot
This plot is commonly referred to as an ogive plot.
Pie graph
data that can be represented in a bar graph can also be shown using a pie graph (also commonly called a circle graph or pie chart).
Positive linear association
Data that are oriented in this manner are said to have a positive linear association. That is, as one variable increases, the other variable also increases.
Relative cumulative frequency histogram
it is helpful to know how the data accumulate over the range of the distribution.
Relative cumulative frequency plot
In a relative cumulative frequency plot, we use the point on the right side of each bin.
Relative frequency histogram
A relative frequency histogram is just like a regular histogram, but instead of labeling the frequencies on the vertical axis, we use the percentage of the total data that is present in that bin.
Response variable
can be identified as having an impact on the value of the other variable, the response (dependent) variable.
Scatterplot
Bivariate data can be represented using a scatterplot to show what, if any, association there is between the two variables.
Skewed left
the left tail of the distribution is stretched out, so this distribution is skewed left.
Skewed right
The right side of the data is spread out across a wider area. This type of distribution is referred to as skewed right.
Stem-and-leaf plot
A stem-and-leaf plot is a similar plot in which it is much easier to read the actual data values.
Symmetric
A data set that is mound shaped can be classified as either symmetric or skewed.
Tail
It is the direction of the long, spread out section of data, called the tail.

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