Numerical Summary Measures

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Numerical Summary Measures

• Lecture 03 Measures of Variation and
  Interpretation, and Measures of Relative Position

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Measures of Variation

• Consider the following three data sets
• Data 1 1, 2, 3, 4, 5
• Data 2 1, 1, 3, 5, 5
• Data 3 3, 3, 3, 3, 3
• For these data sets, the mean and the median are
  clearly identical.
• But, they are different data sets!
• The need to measure the variation in the data.

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On the Perils of an Average Value

• Situation Man has his head in a very hot
  compartment, and his feet feeling very cold.
• Question Mr., how are you feeling?
• Reply Oh, on the average, I am just fine!
• Crash! Dead!

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Sample Variance

• To measure degree of variation, one could look at
  the values of the deviations of the observations
  from its sample mean.
• The sample variance, denoted by S2, is defined to
  be the average of the squared deviations of the
  observations from its sample mean.

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Computational Formula

• Definitional formula not very efficient for
  purposes of computation of the sample variance.
• The computational formula is oftentimes used.

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Properties

• It has squared units which leads to defining
  the standard deviation.
• It is always nonnegative, and equals zero if and
  only if all the observations are identical.
• The larger the value, the more variation in the
  data.
• The divisor of (n-1) instead of n makes the
  sample variance unbiased for the population
  variance (s2) will be explained when we get
  into inference.

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Standard Deviation

• The sample standard deviation, denoted by S, is
  the positive square root of the sample variance.
• Purpose to have a measure with the same units of
  measurements as the original observations.

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Illustration of Computation

• Data set in the example for the mean and median.
• Data 122, 135, 110, 126, 100, 110, 110, 126,
  94, 124, 108, 110, 92, 98, 118, 110, 102, 108,
  126, 104, 110, 120, 110, 118, 100, 110, 120, 100,
  120, 92
• We illustrate computations using the definitional
  and computational formulas in a spreadsheet-type
  format.

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Example continued

• The spreadsheet-type table on the next slide is
  obtained from an Excel worksheet.
• The first three columns illustrates the
  computation using the definitional formula.
• The last column is used to illustrate the
  computation using the computational formula.
• Details will be provided in class!

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Explanations of Columns in the Sheet

• Column 1 contains the values of X, Sum of X,
  and Sample Mean.
• Column 2 contains the deviations, Dev
  X-SampleMean, and the Sum of Deviations.
• Column 3 contains the squared deviations, Sum of
  squared deviations, variance, and the standard
  deviation (via definitional formula).
• Column 4 contains the squared X sum of squared
  X, and the variance (via the computational
  formula).

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Population Parameters (Analogs)

• If the quantities are computed from the
  population values, then we obtain population
  parameters such as the mean, variance and
  standard deviations.
• The notation are as follows

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Information from Mean and Standard Deviation

• Empirical Rule For symmetric mound-shaped
  distributions
• Percentage of all observations within 1 standard
  deviation of the mean is approximately 68.
• Percentage of all observations within 2 standard
  deviations of the mean is approximately 95.
• Percentage of all observations within 3 standard
  deviations of the mean is approximately 100.
• Thus, usually no observations will be more than 3
  standard deviations of the mean!

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Information continued

• Chebyshevs Rule For any distribution (be it
  symmetric, skewed, bi-modal, etc.), we always
  have that
• Percentage of all observations within 1 standard
  deviation of the mean is at least 0.
• Percentage of all observations within 2 standard
  deviations of the mean is at least 75.
• Percentage of all observations within 3 standard
  deviations of the mean is at least 88.89.
• More generally, the percentage of observations
  within k standard deviations of the mean is at
  least (1 - 1/k2).

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Illustration of these Rules

• Consider the sample data with 30 observations
  considered earlier.
• Data 122, 135, 110, 126, 100, 110, 110, 126, 94,
  124, 108, 110, 92, 98, 118, 110, 102, 108, 126,
  104, 110, 120, 110, 118, 100, 110, 120, 100, 120,
  92
• Recall that
• Sample mean 111.1
• Sample standard deviation 11.11
• Percentages in the intervals of form
• Mean - kS, Mean kS

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Percentages in Certain Intervals
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Measure of Relative Standing Z-Score
Given a data set, the z-score, called the
standardized score, associated with an
observation whose value is x is given by

It measures the distance of x from the sample
mean in terms of the number of standard
deviations. A negative (positive) value indicates
the value x is smaller (larger) than the sample
mean.
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Percentiles

• Given a set of n observations, the 100pth
  percentile, where 0 lt p lt 1, is that value which
  is larger than 100p of all the observation, and
  less than 100(1-p) of the observations.
• For example, the 95th percentile is the value
  larger than 95 of all the observations and it is
  smaller than 5 of all the observations.

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Measures of Relative Standing Quartiles

• The first quartile, denoted by Q1, is the 25th
  percentile of the data set.
• The third quartile, denoted by Q3, is the 75th
  percentile of the data set.
• The second quartile, which is the 50th
  percentile, is simply the median of the data set,
  M.

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Computing the Quartiles

• Divide the arranged data set into two parts using
  the median as cut-off.
• If the sample size n is odd, then the median
  should be included in each group while if n is
  even then the median is not included in either
  group.
• First quartile (Q1) is the median of the lower
  group.
• Third quartile (Q3) is the median of the upper
  group.

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Example Quartile Computation

• Arranged Data
• 92, 92, 94, 98, 100, 100, 100, 102, 104, 108,
  108, 110, 110, 110, 110, 110, 110, 110, 110, 118,
  118, 120, 120, 120, 122, 124, 126, 126, 126, 135
• M 110 average of 15th and 16th values.
• Q1 in 8th position 102
• Q3 in 23rd position 120.

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Box Plots

• Another graphical summary of the data is provided
  by the boxplot. This provides information about
  the presence of outliers.
• Steps in constructing a boxplot are as follows
• Calculate M, Q1, Q3, and the minimum and maximum
  values.
• Form a box with left and right ends being at Q1
  and Q3, respectively.
• Draw a vertical line in the box at the location
  of the median.
• Connect the min and max values to the box by
  lines.

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The BoxPlot

• For the systolic blood pressure data set, the
  resulting boxplot, obtained using Minitab, is
  shown below.

HV
Q3
M
Q1
LV
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Comparative BoxPlots
The boxplot could also be used to make a
comparison of the distributions of different
groups. This could be achieved by presenting the
boxplots of the different groups in a
side-by-side manner. We demonstrate this idea
using the Beanie Babies Data on page 91. This
data set contains the following variable Name
name of beanie baby Age in months, since
9/98 Status Rretired, Ccurrent Value Value of
baby
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Comparative BoxPlots of Value by Status
Distributions for both groups very right-skewed!
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Comparative BoxPlots of Log(Value) by Status
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Relationship Between Age and Value
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Relationship Between Log(Age) and Value