Exploring Data

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Exploring Data

• The description of data is an important link
  between its collection and its interpretation.
• Exploratory data analysis combines numerical
  summary with graphical display in an attempt to
  discern patterns in data
• Data are often values of a numerical variable.
  The pattern of these values is called a
  distribution.

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Data Analysis

• Individuals Objects described by a set of data
• Variable any measured characteristics of an
  individual
• Explanatory variable Causes/Reasons
• Response variable Effects/Consequences
• The organizing principles
• Examine individual variables, then look for
  relationships between several variables
• Draw a graph and add to it numerical summaries
• Look first for an overall pattern and then for
  significant deviations from the pattern

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Numerical Summary

• Measure of center
• Mean Arithmetic average
• Median the midpoint of a set of observations
• Measures of spread or variability
• Range The difference between the highest and
  lowest observations
• Quartiles The first quartile is the midpoint of
  all data below the median. The third quartile is
  the midpoint of all data above the median.
• Standard deviation Square root of the variance
  it measures the spread of the data around the
  mean in the same units of measurement as the
  original data set
• Variance average of the squared differences
  between the individual observations and their mean

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Example

• For the following data set
• 85, 74, 64, 92, 76, 85, 32, 67,71 and 59.
• Find mean, median, quartiles, range, variance,
  standard deviation

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Standard Deviation

• The standard deviation gives the average distance
  of every observation from the mean. A small
  standard deviation means that the observations
  are clustered about the mean. A large standard
  deviation means that the data is scattered away
  from the mean.

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Standard Deviation

• The variance of a set of data is the average of
  the squares of their deviations from the mean.
• The standard deviation s is the square root of
  the variance

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Standard Deviation

• S measures spread about the mean, so we must use
  mean and not median.
• S0 only if there is no spread. Otherwise sgt0.
  The larger s, the more spread.
• S has the same units as the units of the
  observations.
• S is strongly influenced by a few extreme
  observations.

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Visual Summary of Data

• Histogram Bar graph
• Stemplot
• Boxplot
• Scatterplot points plotted on xy-plane

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Histograms

• A histogram is a common graph of the distribution
  of outcomes (often divided into classes) for a
  single variable. The number of each bar is the
  number of observations in the class of outcomes
  covered by the base of the bar
• How to make a histogram
• Divide the range of data into classes of equal
  width
• Count the number of individuals in each class
• Draw the histogram

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Interpreting Histograms

• Important features
• Overall Pattern Shape, center and spread
• Deviation Outlier
• Symmetric distribution A distribution with a
  histogram or stem plot in which the left and
  right sides are approximately mirror image of
  each other
• A distribution is skewed to the right if the
  right side extends much farther out than the left
  side
• A distribution is skewed to the left if the left
  side extends much farther out than the right side

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Stemplot

• A good way to represent data for small data sets
• Look like a histogram turned sideway.
• Easy to make
• Quicker to create than histogram and give more
  detailed information
• Preserve the actual values of data

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Stemplot

• To make a stemplot
• Separate each observation into a stem consisting
  of all but the final digit. Stems may have as
  many digits as needed, but each leaf contains
  only a single digit.
• Write the stems in a vertical column with the
  smallest at the top and draw a vertical line to
  the right of this column.
• Write each leaf in a row to the right of its
  stem, in increasing order from its stem

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Boxplot

• Five-number summary
• Min first quartile Median third
  quartile max
• ( Min Q1 M
  Q3 Max)
• Boxplot a visual display of the five-number
  summary
• A central box spans the quartiles
• A line in the box marks the median
• Lines extend from the box out to the smallest and
  largest observations

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Side-by-side boxplots comparing gas mileages of
minicompact and two-seater cars
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Mean and Standard Deviation vs Five-number
Summary

• The five-number summary is usually better than
  the mean and standard deviation for describing a
  skewed distribution or a distribution with
  outliers.
• Use the mean and standard deviation for
  reasonably symmetric distributions free of
  outliers.

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Scatterplot

• A useful device for examining relationship
  between 2 variables
• Consists of points in the plane These points
  represent pairs of values for the variables in
  question, one variable being plotted along the
  x-axis and the other along the y-axis

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Scatterplot

• A scatterplot shows the relationship between two
  numerical variables measured on the same
  individuals.
• The values of one variable are shown on the
  horizontal axis and the values of the other
  variable are shown on the vertical axis.
• Each point corresponds to coordinates on both
  axes.
• You can describe the overall pattern of a
  scatterplot by the form (what shape is it?) ,
  direction (up? Down?) , and strength (do the
  points lie close to the shape or are they
  scattered about it?) of the relationship

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Linear Relation

• Often, a scatterplot suggests a linear relation
  between 2 variables
• ?the points look as though they may be scattered
  about a line in the plan. Such a line is called
  regression line

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Homework assignment

• 3-9 odd, 13, 15,19, 21, 23, 37, 39, 41