AAEC 4302 ADVANCED STATISTICAL METHODS IN AGRICULTURAL RESEARCH

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AAEC 4302 ADVANCED STATISTICAL METHODS IN AGRICULTURAL RESEARCH Powered By Docstoc
					         AAEC 4302
       ADVANCED
 STATISTICAL METHODS IN
AGRICULTURAL RESEARCH


 Descriptive Statistics: Chapter 3
Univariate Statistics of Central Tendency

• They focus on a single variable which has n
  available observations; for example, deer
  weight in the biological data set

• Measures of central tendency attempt to
  measure the typical value taken by a given
  variable
Univariate Statistics of Central Tendency

• There are three alternative statistics (i.e.
  formulas) to measure the central tendency of
  a variable:
  * The Mean
  * The Median
  * The Mode
Univariate Statistics of Central Tendency

  • The mean (or average) is the most common
    and useful measure of central tendency
                    1 n
    * Mean of    X   Xi
                    n i 1

    * To calculate the mean, all of the observations
      (values) of X are added and the result is divided by
      the number of observations (mean deer weight =
      61.77 Kg)
Univariate Statistics of Central Tendency

  • Proof: The sum of the deviations of the
    observations from the mean is always equal to
    zero: d i  X i  X
     *
     *
         di       
                     Xi  X
             X X
                 i                  X  nX
                                         i

            nX   X               X  nX
            nX  nX  0
 Univariate Statistics of Central Tendency

• The median value of X (Xmed) is simply the value
  taken by the middle observation on X after the
  observations have been ordered.
   * If there is an odd number of observations the
     median is unambiguous
   * If there is an even number of observations, there is
     no single middle observations
   * In the later case, by convention, the median is
     calculated by averaging out the values of the two
     middle observations on X:
      (median deer weight = (64+64)/2 = 64 Kg)
 Univariate Statistics of Central Tendency

• The mode is the most frequently occurring
  value of X, which may not be unique
• Mode of X is 66
Univariate Statistics of Central Tendency

• In statistics, the mean is the most common
  measure of the central tendency or typical value
  taken by a given variable, while the median and
  the mode are mostly neglected.

• However, the median can sometimes be more
  useful to describe the typical value of X, since the
  mean is very sensitive to extreme values of X.
Univariate Statistics of Central Tendency

• For example, if the 15 smallest deer weights
  are ignored; the mean increases markedly
  from 61.77 Kg to 64.0 Kg while the median
  only goes from 64 Kg to 65Kg

• The mode may be a useful statistic in the
  case of a discrete variable, but not for
  continuous variables because each
  observation value is likely to be unique
   Univariate Statistics of Dispersion
p 45



• A measure of dispersion is a statistic (formula) that
  indicates how spread (i.e. disperse) the values of a
  given variable are

• The range is a measure of dispersion given by the
  difference between the greatest and the smallest
  value of X in the n observations available

       For example, in the Deer Data Set, the range is
        61, the difference between the maximum weight of
        93Kg and the minimum weight of 32Kg.
 Univariate Statistics: Dispersion

• As demonstrated before, the mean or
  average deviation of X from its mean
        di     (X  X)
                     i 
       n          n     
                        

 is always zero (the positive and
 negative deviations cancel out in the
 summation), which makes it a useless
 measure of dispersion.
    Univariate Statistics: Dispersion

• The mean absolute deviation (MAD),
  calculated by:
           
            d i     (X  X) 
                             i  
            n
                        n      
                                
                               

 solves the “canceling out” problem.
   Univariate Statistics: Dispersion


MAD in deer weight = 9.00 Kg;
 max absolute deer weight deviation is
 93 Kg - 61.77 Kg = 31.23 Kg
 min absolute deer weight deviation is
 32 Kg – 61.77 Kg = -29.77 Kg

It has an intuitive appeal since it represents the
 “typical deviation without regard to sign”
   Univariate Statistics: Dispersion


• An alternative way to address the
  canceling out problem is by squaring the
  deviations from the mean to obtain the
  mean squared deviation (MSD):
           di
                2
                           X  X   2

                             i

            n                     n
  MSD=143.54
   Univariate Statistics: Dispersion


• Problem of squaring can be solved by taking
  the square root of the MSD to obtain the root
  mean squared deviation (RMSD):
                         X  X   2

  RMSD  MSD               i           = 11.98
                                n
• When calculating the RMSD, the squaring of
  the deviations gives a greater importance to
  the deviations that are larger in absolute value,
  which may or may not be desirable
    Univariate Statistics: Dispersion

• For statistical reasons, it turns out that a
  slight variation of the RMSD, known as the
  standard deviation (S or SX), is more
  desirable as a measure of dispersion.

                  X  X  2

      sX            i

                    n  1     = 12.01   (3.6)
          Univariate Statistics of Dispersion
p 46




       • n-1 is known as the degrees of freedom
         in calculating SX: Intuitively, once X is
         known, only n-1 observation values are
         free to vary, one is predetermined by X

       • When a sample of data is taken to learn
         about the population from which it is
         drawn, SX is often the best estimate of
         the degree of dispersion of the data in
         the population

				
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posted:4/12/2013
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