Prerequisite: Estimation of Variable | set 1

Terms related to Variability Metrics :

-> Deviation 
-> Variance
-> Standard Deviation
-> Mean Absolute Deviation
-> Meadian Absolute Deviation
-> Order Statistics
-> Range
-> Percentile 
-> Inter-quartile Range

• Median Absolute Deviation : Mean Absolute Devation, Variance and Standard deviation (discussed in previous section) are not robust to extreme values and outliers. We average the sum of deviations from the median.
  
  Example :

  Sequence : [2, 4, 6, 8] 
  Mean     = 5
  Deviation around mean = [-3, -1, 1, 3]
  
  Mean Absolute Deviation = (3 + 1 + 1 + 3)/ 4
  

  
  

  
  
  

  # Median Absolute Deviation    import numpy as np    def mad(data):     return np.median(np.absolute(             data - np.median(data)))        Sequence = [2, 4, 10, 6, 8, 11]     print ("Median Absolute Deviation : ", mad(Sequence))

  Output :

  Median Absolute Deviation :  3.0

• Order Statistics : This variability measurement approach is based on spread of ranked (sorted) data.
• Range : It is the most basic measurement belonging to Order Statistics. It is the difference between the largest and the smallest value of the dataset. It is good to know the spread of data but it is very sensitive to outliers. We can make it better by dropping the extreme values.
  Example :
  
  
  
  

  Sequence : [2, 30, 50, 46, 37, 91]
  Here, 2 and 91 are outliers
  
  Range = 91 - 2 = 89
  Range without outliers = 50 - 30 = 20
  

• Percentile: It is the a very good measure to measure the variablity in data, avoiding the outliers. P^th percentile in data is a value such that altleast P% or less values are lesser than it and atleast (100 – P)% values are more than P.
  Median is the 50th percentile of the data.
  Example :

  Sequence : [2, 30, 50, 46, 37, 91] 
  Sorted   : [2, 30, 37, 46, 50, 91]
  
  50th percentile = (37 + 46) / 2 = 41.5 
  

  Code –

  
  

  
  
  

  # Percentile    import numpy as np           Sequence = [2, 30, 50, 46, 37, 91]     print ("50th Percentile : ", np.percentile(Sequence, 50))        print ("60th Percentile : ", np.percentile(Sequence, 60))

  Output :

  50th Percentile :  41.5
  60th Percentile :  46.0

• Inter-Quartile Range(IQR) : It works for the ranked(sorted data). It has 3 quartiles dividing data – Q1(25^th percentile), Q2(50^th percentile) and Q3(75^th percentile). Inter-quartile Range is the difference between Q3 and Q1.

  Example :

  Sequence : [2, 30, 50, 46, 37, 91] 
  Q1 (25th percentile) : 31.75
  Q2 (50th percentile) : 41.5
  Q3 (75th percentile) : 49
  
  IQR = Q3 - Q1 = 17.25
  

  Code – 1

  
  

  
  
  

  # Inter-Quartile Range    import numpy as np from scipy.stats import iqr        Sequence = [2, 30, 50, 46, 37, 91]     print ("IQR : ", iqr(Sequence))

  Output :

  IQR :  17.25
  

  Code – 2

  
  

  
  
  

  import numpy as np    # Inter-Quartile Range iqr = np.subtract(*np.percentile(Sequence, [75, 25]))    print ("\nIQR : ", iqr)

  Output :

  IQR :  17.25
  

  

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