More Univariate Data Quantitative Graphs

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More Univariate DataQuantitative Graphs
Describing Distributions with Numbers
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Quantitative Data

• Quantitative variables take numerical values for
  which it makes sense to do arithmetic operations
  like adding or averaging.
• Possible Graphs dotplots, stemplots,
• histograms, Cumulative frequency plots,
  boxplots

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Graphs

• Be sure to always
• Title your graphs
• Label your axis including units of
• measure
• number your axes in a consistent
• and reasonable manner

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Quantitative Graphs
Histograms A histograms vertical axis is counts
while a relative frequency histograms vertical
axis is percents.
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Quantitative Graphs
Stem Leaf This type of graph uses place values
as the stems units as the leaves. (Its very
hard to describe, we are going to make one for an
example.) We can also create whats called a
back-to-back stem plot with two data sets. It is
helpful for comparing to sets of univariate data.
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Quantitative Graphs
A histogram is preferred sometimes for larger
data sets. Its strongest asset is that it shows
shape well. Its weakness is that the individual
data values are lost. A stem leaf is
preferred sometimes because it retains all data
values but its very difficult to create for
large data sets.
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Quantitative Data

• The distribution of a variable tells us what
  values the variable typically takes and how often
  it takes them. It is a generalization about the
  variable values.

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• When describing any Quantitative distribution
• C Center
• U Unusual Features
• S Shape
• S Spread
• B Be
• S - Specific

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• Common Shapes of distributions/graphs
• Symmetric
• Skewed to the right
• Skewed to the left
• Bimodal
• Uniform

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• Once you have chosen a shape, you choose a
  measure of center and spread based on that shape.

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Center when the distribution is symmetric Mean
the average formula
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Measure Spread or Variability when the
distribution is Symmetric

• Standard deviation

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Measure of Center when the distribution is not
symmetric Median the middle value in an
ordered list. If there are two values in the
middle, then average them.
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Measure Spread or Variability when the
distribution is not Symmetric

• We can also examine spread by looking at the
  range of middle 50 of the data. This is called
  the
• Interquartile Range (IQR).
• IQR Q3 Q1

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We also need to talk about the 5-number summary.
The 5-number summary is made up of the minimum,
the first quartile, Q1 (where 25 of the data
lies below this value), the median, the third
quartile, Q3 (where 75 of the data lies below
this value), and the maximum.
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Another Measure of Spread or Variability

• Range the difference between the maximum and
  the minimum observations. This is the simplest
  measure of spread. We typically use this as
  preliminary information or if it is the only
  measure of spread we can calculate.

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Another measure of spread or variability

• Variance is the average of the squares of the
  deviations of the observations from their mean.
  It is the standard deviation squared.

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• An outlier is an individual observation in data
  that falls outside the overall pattern of the
  data.

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Using the IQR, we can perform a test for
outliers. Outlier Test Any value below Q1
1.5(IQR) or above Q3 1.5 (IQR) is
considered an outlier.
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Another Graph When we graph the five-number
summary along with outliers if present, it leads
to a modified boxplot.
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Measures that are not strongly affected by
extreme values are said to be resistant. The
median and IQR are more resistant than the mean
and standard deviation. The standard deviation,
is even less resistant than the mean.
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Measures of Spread or Variability Why? We
measure spread because its an important
description of what is happening with the data.
We need to know about the amount of variation we
can expect in a data set.