By All Means

Introduction to Research Design and Statistics

By All Means

Recap on the Common Measures of Central Tendency

By now you are familiar with the three most common measures of central tendency, the mean, the median and the mode. The median is the middle-most score, the score above and below which 50% of the distribution falls. The mode is the value or class interval that occurs most frequently in the distribution. And the mean is the arithmetic average of the scores, and has the formula:

Mean Formula

These mean, median and mode are according to the different definitions the single values that are the 'most representative' of the distribution. There are, however, other ways of defining what is meant by 'most representative.' We will look at the harmonic mean, the geometric mean, the quadratic mean, trimmed mean, and the winsorized mean.

Harmonic Mean

The harmonic mean is obtained by dividing number by the sum of the reciprocal of each piece of data.

Harmonic Mean Formula

Geometric Mean

The geometric mean is obtained by taking the nth root of the product of the values.

Geometric Mean Formula

Quadratic Mean

The quadratic mean, also known as the root mean square (RMS), is obtained by obtained by taking the square root of the average sum of squared values.

Quadratic Mean Formula

Trimmed Mean

The trimmed mean is the arithmetic average after the x-proportion highest and lowest values are removed from the distribution.

Winsorized Mean

The winsorized mean is the arithmetic average after the x-proportion highest and lowest values are replaced by the next adjacent value in the distribution.

Example

Consider the sample: [1 3 4 6 7 8 9 10 10 15 20] 
     trimmed sample: [3 4 6 7 8 9 10 10 15]
  winsorized sample: [3 3 4 6 7 8 9 10 10 15 15]

                    mean =   8.455
                  median =   8.000
                    mode =  10.000
          harmonic mean  =   4.498
         geometric mean  =   6.630
         quadratic mean  =   9.913
      trimmed mean (.1)  =   8.000
   winsorized mean (.1)  =   8.182

Stata Example

input y
1 
3 
4 
6 
7 
8 
9 
10 
10 
15 
20
end

summarize y, detail

                              y
-------------------------------------------------------------
      Percentiles      Smallest
 1%            1              1
 5%            1              3
10%            3              4       Obs                  11
25%            4              6       Sum of Wgt.          11

50%            8                      Mean           8.454545
                        Largest       Std. Dev.      5.428879
75%           10             10
90%           15             10       Variance       29.47273
95%           20             15       Skewness       .7499235
99%           20             20       Kurtosis       3.042146

means y

Variable |    Type        Obs        Mean       [95% Conf. Interval]
---------+----------------------------------------------------------
       y | Arithmetic      11    8.454545        4.807375   12.10172 
         |  Geometric      11    6.630327        3.805889   11.55085 
         |   Harmonic      11     4.49781        2.473946   24.72288 
---------+----------------------------------------------------------

emeans y  /* Available from ATS via the Internet */

Variable |    Type        Obs        Mean       [95% Conf. Interval]
---------+----------------------------------------------------------
       y | Arithmetic      11    8.454545        4.807375   12.10172 
         |  Geometric      11    6.630327        3.805889   11.55085 
         |   Harmonic      11     4.49781        2.473946   24.72288 
         |  Quadratic      11     9.91326        4.363899   13.32298 
---------+----------------------------------------------------------

robmean y, trim(.1)  /* Available from ATS via the Internet */

.1 highest and lowest cases trimmed or winsorized

           y |     Obs     Estimate   Std. Dev.     Min       Max
-----------------------------------------------------------------
        mean |      11      8.4545      5.4289         1       20
      median |      11      8.0000
     trimmed |       9      8.0000      3.6056         3       15
  winsorized |      11      8.1818      4.2147         3       15
huber 1-step |       6      7.7008                     6       10
  multi-step |      11      7.1979                     1       20
-----------------------------------------------------------------

More on Robust Estimators of Location

Demonstration of meanpt, showing how resistant the median is to large outliers as compared to the mean.

The Crime Dataset

use http://www.philender.com/courses/data/crime, clear
  
graph7 murder, one box
  

  
kdensity murder, normal legend(off)
  

  
graph box murder, fxsize(25)
  

  
robmean murder
  
.2 highest and lowest cases trimmed or winsorized

      murder |     Obs     Estimate   Std. Dev.     Min       Max
-----------------------------------------------------------------
        mean |      51      8.7275     10.7176       1.6     78.5
      median |      51      6.8000
     trimmed |      31      7.1161      2.4439 3.5999999     11.3
  winsorized |      51      7.2471      3.0886 3.5999999     11.3
huber 1-step |      47      7.3470                   1.6     13.1
  multi-step |      50      7.1789                   1.6     20.3
-----------------------------------------------------------------

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Phil Ender, 30Jun98