Descriptive Statistics-IV (Measures of Variation)

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Descriptive Statistics-IV(Measures of Variation)

• QSCI 381 Lecture 6
• (Larson and Farber, Sect 2.4)

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Deviation, Variance and Standard Deviation-I
Deviation

• The of a data entry xi in a
  population data set is the difference between xi
  and population mean ?, i.e.
• The sum of the deviations over all entries is
  zero.
• The is
  the sum of the squared deviations over all
  entries
• ? is the Greek letter sigma.

Population variance
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Deviation, Variance and Standard Deviation-II
Population standard deviation

• The
  is the square root of the
  population variance, i.e.
• Note these quantities relate to the population
  and not a sample from the population.
• Note sometimes the standard deviation is
  referred to as the standard error.

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The Sample variance and Standard Deviation

• The and
  the
  of a data set with n entries are given by

Sample variance
Sample standard deviation
Note the division by n -1 rather than N or n.
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Calculating Standard Deviations
Step Population Sample
Find the mean
Find the deviation for each entry
Square each deviation
Add to get the sum of squares (SSx)
Divide by N or (n -1) to get the variance
Take the square root to get the standard deviation
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Example

• Find the standard deviation of the following
  bowhead lengths (in m)
• (8.5, 8.4, 13.8, 9.3, 9.7)
• Key question (before doing anything) is this a
  sample or a population?

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Formulae in EXCEL

• Calculating Means Average(A1A10)
• Calculating Standard deviations Stdev(A1A10)
  this calculates the sample and not the
  population standard deviation!

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Standard Deviations-I
SD0
SD2.1
SD5.3
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Standard Deviations-II(Symmetric Bell-shaped
distributions)
k 2 proportion gt 75 k 3 proportion gt 88
Chebychevs Theorem The proportion of the data
lying within k standard deviations (k gt1) of
the mean is at least 1 - 1/k2
68
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95
13.5
99.7
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Standard Deviations-III(Grouped data)

• The standard deviation of a frequency
  distribution is
• Note where the frequency distribution consists
  of bins that are ranges, xi should be the
  midpoint of bin i (be careful of the first and
  last bins).

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Standard Deviations-IV(The shortcut formula)
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The Coefficient of Variation

• The is
  the standard deviation divided by the mean -
  often expressed as a percentage.
• The coefficient of variation is dimensionless and
  can be used to compare among data sets based on
  different units.

coefficient of variation
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Z-Scores
Standard (or Z) score

• The is calculated
  using the equation

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Outliers-I

• Outliers can lead to mis-interpretation of
  results. They can arise because of data errors
  (typing measurements in cm rather than in m) or
  because of unusual events.
• There are several rules for identifying outliers
• Outliers lt Q2-6(Q2-Q1) gt Q26(Q3-Q2)
• Strays lt Q2-3(Q2-Q1) gt Q23(Q3-Q2)

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Outliers-II

• Strays and outliers should be indicated on box
  and whisker plots
• Consider the data set of bowhead lengths, except
  that a length of 1 is added!

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10
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Length (m)
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Review of Symbols in this Lecture
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Summary

• We use descriptive statistics to get a feel for
  the data (also called exploratory data
  analysis). In general, we are using statistics
  from the sample to learn something about the
  population.