Measures of Variability

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Measures of Variability

• MATH 224

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Last Class

• Mean and Median
• Both are statistics
• Called Measures of Central Tendency
• report something about the middle of the data

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This class

• Measures of Variability (or Spread)
• e.g.

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Dataset

• Model dataset is one of the assignment datasets,
  FinalGrades2007.xls

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Mean The Formula

• Formula for the mean is

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Variance The Formula

• Spread can be measured as either
• distance between data points
• distance from data to center

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Variance The Formula
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Variance The Formula
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Variance and Standard Deviation

• Variance is in units of
• can be hard to interpret
• If we take square root of variance, we get the
  datas standard deviation
• Units of standard deviation are same as data (x)
• Guarantee 75 of the data will lie within 2
  standard deviations of the mean
• Usually most (up to 95) will lie in that range

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Example - Marks

• In MATLAB,
• Variance
• var( ltdata columngt)
• Standard deviation
• sqrt(var( ltdatagt )), or
• std( ltdatagt )

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Computing mean 2 std. devs

• m mean( ltdatagt )
• stdev std( ltdatagt )
• upper m 2 stdev
• lower m 2 stdev

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On Histogram
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In reports

• Mean 2 standard deviations often reported
• e.g. Marks were distributed as 69 37
• Breaking strength of a beam was 1598 N 350 N
• It is understood that a small fraction values
  will lie outside the given range

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Determining Percentage Inside 2 standard
deviations

• Consider a vector of values, v, and single
  threshold values
• Count number of values
• length(v)
• Match of values above a threshold
• v gt threshold
• Count of values above a threshold
• sum( v gt threshold)
• logical AND
• (v gt threshold1) (v lt threshold2)
• Count matching both conditions
• sum( (v gt threshold1) (v lt threshold2))
• Percent matching both conditions
• N length(v)
• perc (sum( (v gt threshold1) (v lt
  threshold2)))/N

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Example Marks

• Percentage within 2 standard deviations of mean
  is

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Problems with variance, standard deviation

• As with mean, both variance and standard
  deviation are sensitive to exceptional values
• variance even worse than mean, because of squared
  difference in formula

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Interquartile Range (IQR)

• Quartiles are points which divide sorted data
  into quarters
• Also known as 25th, 50th, 75th percentiles
• prctile(v, 25, 50, 75)
• 50th percentile median
• 75th 25th percentile interquartile range
• IQR is in same units as data
• We expect roughly half the data to lie within the
  IQR

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Interquartile Example
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Interquartile Range

• This range is called a robust statistic because
  small sets of extreme values have little effect
  on it
• Median is a robust statistic for measuring the
  center of data
• Mean, variance, standard deviation are not robust
• Why trimmed mean is sometimes used instead of
  mean

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Boxplots

• Measures of spread are often not interesting on
  their own
• Typically used with mean to compare datasets
• lower variance usually better in manufactured
  goods
• high variability makes predictions difficult in
  environmental data
• Graphical approach to this uses boxplots

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Boxplots

• More challenging to format in MATLAB than
  histograms
• Will cover how to generate them in tutorials next
  week

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Administration

• Assignment Format
• Labs will be for QA about assignments, help with
  MATLAB
• No lab Thursday both labs on Friday
• Tutorials will cover more advanced MATLAB
• summaries will be posted
• MATLAB purchasing
• Have put late order in with bookstore, let you
  know what I hear back
• Can order directly from MathWorks see course web
  site