DATA DESCRIPTION

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DATA DESCRIPTION
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Units

• Unit entity we are studying, subject if human
  being
• Each unit/subject has certain parameters, e.g., a
  student (subject) has his age, weight, height,
  home address, number of units taken, and so on.

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Variables

• These parameters are called variables.
• In statistics variables are stored in columns,
  each variable occupying a column.

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Cross-sectional and time-series analyses

• In a cross-sectional analysis a unit/subject will
  be the entity you are studying. For example, if
  you study the housing market in San Diego, a unit
  will be a house, and variables will be price,
  size, age, etc., of a house.
• In a time-series analysis the unit is a time
  unit, say, hour, day, month, etc.

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Data Types

• Nominal data male/female, colors,
• Ordinal data excellent/good/bad,
• Interval data temperature, GMAT scores,
• Ratio data distance to school, price,

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Two forms

• GRAPHICAL form
• NUMERICAL SUMMARY form

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Graphical forms

• Sequence plots
• Histograms (frequency distributions)
• Scatter plots

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Sequence plots

• To describe a time series
• The horizontal axis is always related to the
  sequence in which data were collected
• The vertical axis is the value of the variable

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Example sequence plot
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Histograms I

• A histogram (frequency distribution) shows how
  many values are in a certain range.
• It is used for cross-sectional analysis.
• the potential observation values are divided into
  groups (called classes).
• The number of observations falling into each
  class is called frequency.
• When we say an observation falls into a class, we
  mean its value is greater than or equal to the
  lower bound but less than the upper bound of the
  class.

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Example histogram

• A commercial bank is studying the time a customer
  spends in line. They recorded waiting times (in
  minutes) of 28 customers
• 5.9 7.6 5.3 9.7 1.6 3.5 7.4
• 4.0 1.6 7.3 8.2 8.4 6.5 8.9
• 1.1 8.6 4.3 1.2 3.3 2.1 8.4
• 1.1 6.7 5.0 4.5 9.4 6.3 6.4

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Example histogram
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Histogram II

• The relative frequency distribution depicts the
  ratio of the frequency and the total number of
  observations.
• The cumulative distribution depicts the
  percentage of observations that are less than a
  specific value.

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Example relative frequency distribution

• A relative frequency distribution plots the
  fraction (or percentage) of observations in each
  class instead of the actual number. For this
  problem, the relative frequency of the first
  class is 6/280.214. The remaining relative
  frequencies are 0.179, 0.250, 0.286 and 0.071. A
  graph similar to the above one can then be
  plotted.

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Example cumulative distribution

• In the previous example, the percentage of
  observations that are less than 3 minutes is
  0.214, the percentage of observations that are
  less than 5 is 0.2140.1790.393, less than 7 is
  0.2140.1790.250.643, less than 9 is
  0.2140.1790.250.2860.929, and that less than
  11 is 1.0.

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Example cumulative distribution
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Histogram III

• The summation of all the relative frequencies is
  always 1.
• The cumulative distribution is non-decreasing.
• The last value of the cumulative distribution is
  always 1.
• A cumulative distribution can be derived from the
  corresponding relative distribution, and vice
  versa.

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Probability

• A random variable is a variable whose values
  cannot predetermined but governed by some random
  mechanism.
• Although we cannot predict precisely the value of
  a random variable, we might be able to tell the
  possibility of a random variable being in a
  certain interval.
• The relative frequency is also the probability of
  a random variable falling in the corresponding
  class.
• The relative frequency distribution is also the
  probability distribution.

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Scatter plots

• A scatter plot shows the relationship between two
  variables.

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Example scatter plot

• . The following are the height and foot size
  measurements of 8 men arbitrarily selected from
  students in the cafeteria. Heights and foot sizes
  are in centimeters.
• man 1 2 3 4 5 6 7 8
• Height 155 160 149 175 182 145 177 164
• foot 23.3 21.8 22.1 26.3 28.0 20.7 25.3 24.9

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Example scatter plot
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Numerical Summary Forms

• Central locations mean, median, and mode.
• Dispersion standard deviation and variance.
• Correlation.

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Mean

• Mean/average is the summation of the observations
  divided by the number of observations
• 27 22 26 24 27 20 23 24 18 32
• Sum (27 22 26 24 27 20 23 24
  18 32) 243
• Mean 243/10 24.3

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Median

• Median is the value of the central observation
  (the one in the middle), when the observations
  are listed in ascending or descending order.
• When there is an even number of values, the
  median is given by the average of the middle two
  values.
• When there is an odd number of values, the median
  is given by the middle number.

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Example median

• 18 20 22 23 24 24 26 27 27 32

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Compare mean and median

• The median is less sensitive to outliers than the
  mean. Check the mean and median for the
  following two data sets
• 18 20 22 23 24 24 26 27 27
  32
• 18 20 22 23 24 24 26 27 27
  320

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Mode

• Mode is the most frequently occurring value(s).

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Symmetry and skew

• A frequency distribution in which the area to the
  left of the mean is a mirror image of the area to
  the right is called a symmetrical distribution.
• A distribution that has a longer tail on the
  right hand side than on the left is called
  positively skewed or skewed to the right. A
  distribution that has a longer tail on the left
  is called negatively skewed.
• If a distribution is positively skewed, the mean
  exceeds the median. For a negatively skewed
  distribution, the mean is less than the median.

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Range

• The range is the difference in the maximum and
  minimum values of the observations.

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Standard deviation and variance

• The standard deviation is used to describe the
  dispersion of the data.
• The variance is the squared standard deviation.

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Calculation of S.D.

• Calculate the mean
• calculate the deviations
• calculate the squares of the deviations and sum
  them up
• Divide the sum by n-1 and take the square root.

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Example S.D.

• Sample 27 22 26 24 27 20
  23 24 18 32
• Deviation 2.7 -2.3 1.7 -0.3 2.7
  -4.3 -1.3 -.3 -6.3 7.7
• Sq of Dev 7.29 5.29 2.89 .09 7.29 19.5
  1.69 .09 39.7 59.3
• Sum of 7.29 5.29 ..... 59.3 142.1
• Std. Dev.

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Empirical rules

• If the distribution is symmetrical and
  bell-shaped,
• Approximately 68 of the observations will be
  within plus and minus one standard deviation from
  he mean.
• Approximately 95 observations will be within two
  standard deviation of the mean.
• Approximately 99.7 observations will be within
  three standard deviations of the mean.

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Percentiles

• The 75th percentile is the value such that 75 of
  the numbers are less than or equal to this value
  and the remaining 25 are larger than this value.
• The k-th percentile is the value such that k of
  the numbers are less than or equal to this value
  and the remaining 1-k are larger than this value.

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Correlation coefficient

• The Correlation coefficient measures how closely
  two variables are (linearly) related to each
  other. It has a value between -1 to 1.
• Positive and negative linear relationships.
• If two variables are not linearly related, the
  correlation coefficient will be zero if they are
  closely related, the correlation coefficient will
  be close to 1 or -1.