Ch4: Displaying Quantitative Data Dealing With a Lot of Numbers

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Ch4 Displaying Quantitative Data Dealing
With a Lot of Numbers

• Summarizing the data will help us when we look at
  large sets of quantitative data.
• Without summaries of the data, its hard to grasp
  what the data tell us. The best thing to do is to
  make a picture
• We dont use pie charts for quantitative data,
  since that is for categorical variables.
• Rounding Advice
• Dont round numbers in the middle of doing
  calculations.
• In writing the final answer, 3 decimal places
  should be sufficient
• Example 78.5 or .785 is fine. I dont need to
  see .784821.
• Also understand the relationship between
  percentages and decimals!
• Its much like http//www.youtube.com/watch?vGp0
  HyxQv97Qeurl

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Histograms Displaying the Distribution

• The chapter example discusses the changes in
  Enrons stock price from 1997 2001.
• First, slice up the entire span of values covered
  by the quantitative variable into equal-width
  piles called bins.
• The bins and the counts in each bin give the
  distribution of the quantitative variable.

• Monthly Price Changes in Enron Stock

• A histogram plots the bin counts as the heights
  of bars (It looks like a bar chart).

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Histograms Displaying the Distributionof Price
Changes (cont.)

• A relative frequency histogram displays the
  percentage of cases in each bin instead of the
  count.

• Relative Frequency Histogram
• Monthly Price Changes in Enron Stock

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Stem-and-Leaf Displays

• Stem-and-leaf displays show the distribution of a
  quantitative variable, like histograms do, while
  also preserving the individual values.
• Stem-and-leaf displays contain all the
  information found in a histogram and, when
  carefully drawn, satisfy the area principle and
  show the distribution.

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Constructing a Stem-and-Leaf Display

• First, cut each data value into leading digits
  (stems) and trailing digits (leaves).
• Use the stems to label the bins.
• Use only one digit for each leafeither round or
  truncate the data values to one decimal place
  after the stem.
• More detail can be obtained by breaking each stem
  into 2 lines as has been done here for low and
  high 60s, low and high 70s, etc. ( also see Book
  Pr 18)
• Its easy to learn by doing! (see example)

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Example

• Construct a Stem and Leaf for the following data
• Using stems of 10K
• Using stems of 5K
• 30-34, 35-39, etc

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Stem-and-Leaf vs. Histogram

• Compare the histogram and stem-and-leaf display
  for the pulse rates of 24 women at a health
  clinic. Which graphical display has more info?

Stem-and Leaf Plot Heart Rate for Women at
Health Clinic
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Dotplots
Kentucky Derby Winning Times, 1875-2004

• A dotplot is a simple display. It just places a
  dot along an axis for each case in the data.
• The dotplot to the right shows Kentucky Derby
  winning times, plotting each annual race with a
  dot.
• Dotplots can be displayed horizontally or
  vertically.
• What do you notice about the data in this dotplot?

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What is the Shape of the Distribution?
Shape, Center, and Spread

• When describing a distribution, make sure to
  always tell about three things shape, center,
  and spread

• Does the histogram have a single, central hump or
  several separated bumps?
• Is the histogram symmetric?
• Do any unusual features stick out?

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Humps and Bumps

• Does the histogram have a single, central hump or
  several separated bumps?
• Humps in a histogram are called modes.
• A histogram with one main peak is dubbed
  unimodal histograms with two peaks are bimodal
  histograms with three or more peaks are called
  multimodal.

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Humps and Bumps (cont.)

• A bimodal histogram has two apparent peaks

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Humps and Bumps (cont.)

• A histogram that doesnt appear to have any mode
  and in which all the bars are approximately the
  same height is called uniform

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Symmetry

• Is the histogram symmetric?
• If you can fold the histogram along a vertical
  line through the middle and have the edges match
  pretty closely, the histogram is symmetric.

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Symmetry (cont.)

• The (usually) thinner ends of a distribution are
  called the tails. If one tail stretches out
  farther than the other, the histogram is said to
  be skewed to the side of the longer tail.
• the blue histogram below is said to be skewed
  left, while the pink histogram is said to be
  skewed right.

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Anything Unusual?

• Do any unusual features stick out?
• Sometimes its the unusual features that tell us
  something interesting or exciting about the data.
• You should always mention any stragglers, or
  outliers, that stand off away from the body of
  the distribution.
• Are there any gaps in the distribution? If so, we
  might have data from more than one group.

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Anything Unusual? (cont.)

• The following histogram has outliersthere are
  three cities in the leftmost bar
• What do you think is happening?

Histogram People per Housing Unit in Selected
Cities
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Where is the Center of the Distribution?

• If you had to pick a single number to describe
  all the data what would you pick?
• Its easy to find the center when a histogram is
  unimodal and symmetricits right in the middle.
• On the other hand, its not so easy to find the
  center of a skewed histogram or a histogram with
  more than one mode.
• For now, we will eyeball the center of the
  distribution. In the next chapter we will find
  the center numerically.

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How Spread Out is the Distribution?

• Variation matters, and Statistics is about
  variation.
• Are the values of the distribution tightly
  clustered around the center or more spread out?
• In the next two chapters, we will talk about
  spread

Stay Tuned
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Comparing Distributions

• Often we would like to compare two or more
  distributions instead of looking at one
  distribution by itself.
• When looking at two or more distributions, it is
  very important that the histograms have been put
  on the same scale. Otherwise, we cannot really
  compare the two distributions.
• When we compare distributions, we talk about the
  shape, center, and spread of each distribution.

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Comparing Distributions (cont.)

• Compare the following 2 Charts
• What do they have in common
• How do they differ?
• Title Distributions of Ages for Female and
  Male Heart Attack Patients

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Time-plots Order, Please!

• For some data sets, we are interested in how the
  data behave over time. In these cases, we
  construct time-plots of the data.
• What do we notice about Enrons stock over time?

Changes in Price of Enron Stock, 1997-2002
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Think Before You Draw, Again

• Remember the Make a picture rule?
• Now that we have options for data displays, you
  need to Think carefully about which type of
  display to make.
• Before making a stem-and-leaf display, a
  histogram, or a dotplot, check the
• Quantitative Data Condition The data are values
  of a quantitative variable whose units are known.
• Does it make sense to make a histogram of
  students in this class as broken down by the last
  4 digits of their cel phone numbers? (Bin 1
  (0000-0999), Bin2 (1000-1999)) What information
  would that tell us?

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What Can Go Wrong?

• Dont make a histogram of a categorical
  variablebar charts or pie charts should be used
  for categorical data.
• Dont look for shape,
  center, and spread
  of
  a bar chart.
• Although they look
• alike, dont confuse
• Histograms
• with Bar Charts

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What Can Go Wrong? (cont.)

• Dont use bars in every displaysave them for
  histograms and bar charts.
• Below is a badly drawn timeplot and the proper
  histogram entitled Number of Eagles Sighted in a
  Collection of Weeks
• What does it look like the first graphic tells
  us?
• What would be the correct way to convey this
  information?

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What Can Go Wrong? (cont.)

• Avoid inconsistent scales, either within the
  display or when comparing two displays.
• Label clearly so a reader knows what the plot
  displays.
• Good intentions, bad plot
• At least it has a title

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What Can Go Wrong? (cont.)

• Y-axis values need to be shown, or at the very
  least, all items should be drawn to scale, as
  this wonderfully bad USA Today Stats-shot fails
  to do.

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What have we learned?

• Weve learned how to make a picture for
  quantitative data to help us see the story the
  data have to Tell.
• We can display the distribution of quantitative
  data with a histogram, stem-and-leaf display, or
  dot-plot.
• Tell about a distribution (of quantitative data)
  by talking about shape, center, spread, and any
  unusual features.
• We can compare two quantitative distributions by
  looking at side-by-side displays (plotted on the
  same scale).
• Trends in a quantitative variable can be
  displayed in a time-plot.

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Examples

• Its a good idea to think about what a
  distribution might look like before we collect
  the data. What is your estimate of the
  following?
• Number of miles run by Saturday morning joggers
  in Golden Gate Park?
• Hours spent by all US adults watching football on
  Thanksgiving Day?
• Amount of winnings of all people who bought Lotto
  tickets last week?
• Ages of SFSU professors?
• Last digit of all SFSU campus extension phone
  numbers