CHAPTER 4 Displaying and Summarizing Quantitative Data

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CHAPTER 4 Displaying and Summarizing Quantitative
Data

• Slice up the entire span of values in piles
  called bins (or classes)
• Then count the number of values that fall in each
  bin
• The bins and the counts in each bin give the
  distribution of the quantitative variable

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Histogram

• Display the counts in each bin in a histogram.
• Like a bar chart, a histogram plots the bin
  counts as the heights of bars.
• No spaces between bins. (different from a bar
  chart)
• Relative frequency histogram displays percentage
  of cases in each bin instead of the count.

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Stem and Leaf Display

• Shows the distribution as well as the individual
  values.
• Very Convenient easy to make by hand.
• Make a Steam and Leaf Display of the data set of
  exercise 40 (page 82)

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Shape, Center, and Spread

• How many Modes (humps)?
• Histograms with
• One peak Unimodal
• Two peaks Bimodal
• Three or more Multimodal
• A histogram that doesnt appear to have any mode
  and in which all the bars are approximately the
  same height is called Uniform
• Exercise 7 Page 78

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Symmetry

• A distribution is symmetric if the two halves on
  either side of the center look approximately like
  mirror images of each other.

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Skewed Distributions

• Tails The thinner ends of a distribution are
  called tails. If one tail stretches out farther
  than the other the histogram is said to be skewed
  to the side of the longer tail
• Skew to the left Skew to the right

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Outliers

• Outliers are values that stand off away from the
  body of the distribution
• Gaps in the distribution warn us that the data
  may not be homogeneous. They may come from
  different sources or contain more than one group.
• (Example on page 52)

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Center of the Distribution

• For unimodal and symmetric distributions
• In the middle
• For skewed and more than one mode is harder to
  find
• (split in groups)

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How Spread is the Distribution?

• Just Checking page 56
• Comparing Distributions
• Do men and women tend to get heart attacks at
  different ages?

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Summarizing Distributions

• Center
• Midrange
• Median The middle value that divides the
  histogram into two equal areas
• Order the values first
• If n is odd the median is the middle value.
  Position (n1)/2
• If n is even then take the average of the two
  middle values, that is the average of positions
  n/2 and n/21

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Summarizing Distributions (cont.)

• Spread
• Range Max Min
• Quartiles
• Find the median, then find the median of each
  half. (Note If n is odd include the median of
  the complete set to calculate the median of each
  half)
• These are called the Lower quartile and Upper
  quartile and are denoted by Q1 and Q3
  respectively.

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The Interquartile Range

• IQR Q3 Q1
• The lower and upper quartiles are also called the
  25th and 75th percentiles
• Q1 25th percentile
• Median 50th percentile
• Q3 75th Percentile

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Summarizing Distributions (cont.)

• Summarizing Symmetric Distributions
• If the shape of the distribution is symmetric,
  the mean (average) is a good alternative to
  summarize the distribution
• Remember Symmetric and no outliers
• Mean

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Mean or Median

• The mean is the point at which the histogram
  would balance.
• Outliers will pull the mean in that direction.
• For skewed data its better to report the median
  than the mean as a measure of center

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What About Spread?The Standard Deviation

• Standard Deviation
• It takes into account how far each value is from
  the mean
• Appropriate only for symmetric data
• Deviation Distance from each data value to the
  mean
• Variance
• Standard Deviation

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Shape, Center and Spread

• Report always center and spread
• Which measure for center and which measure for
  spread?
• Skewed Median and IQR
• Symmetric Mean and Standard Deviation
• If there are outliers report the mean and
  standard deviations with and without the
  outliers. Median and IQR are not likely to be
  affected.

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Chapter 5 Understanding and Comparing
Distributions

• After you have the five number summary you can
  create a display called a BoxPlot

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Box Plots

• Place the Median and quartiles over a line
  spanning the range of the data. (as shown in the
  board)
• Locate the Upper and lower fences
• Upper Fence Q3 1.5 IQR
• Lower Fence Q1 1.5 IQR
• Then draw the Whiskers (Most Extreme data value
  Found within the fences)
• Display Outliers

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Exercise

• Comparing Groups (Page 93)

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Time Plot

• Displays data that changes over time
• (What is wrong with the time plot on page 104?)