title: How numpy.cov() function is implemented?
body: <p>I have my own implementation of the covariance function based on the equation:</p>
  
  <p><img src="https://i.stack.imgur.com/SwpYs.png" alt="enter image description here"></p>
  
  <pre><code>'''
  Calculate the covariance coefficient between two variables.
  '''
  
  import numpy as np
  
  X = np.array([171, 184, 210, 198, 166, 167])
  Y = np.array([78, 77, 98, 110, 80, 69])
  
  # Expected value function.
  def E(X, P):
      expectedValue = 0
      for i in np.arange(0, np.size(X)):
          expectedValue += X[i] * (P[i] / np.size(X))
      return expectedValue 
  
  # Covariance coefficient function.
  def covariance(X, Y):
      '''
      Calculate the product of the multiplication for each pair of variables
      values.
      '''
      XY = X * Y
  
      # Calculate the expected values for each variable and for the XY.
      EX = E(X, np.ones(np.size(X)))
      EY = E(Y, np.ones(np.size(Y)))
      EXY = E(XY, np.ones(np.size(XY)))
  
      # Calculate the covariance coefficient.
      return EXY - (EX * EY)
  
  # Display matrix of the covariance coefficient values.
  covMatrix = np.array([[covariance(X, X), covariance(X, Y)], 
  [covariance(Y, X), covariance(Y, Y)]])  
  print("My function:", covMatrix)
  
  # Display standard numpy.cov() covariance coefficient matrix.
  print("Numpy.cov() function:", np.cov([X, Y]))
  </code></pre>
  
  <p>But the problem is, that I'm getting different values from my function and from <code>numpy.cov()</code>, ie:</p>
  
  <pre><code>My function: [[ 273.88888889  190.61111111]
   [ 190.61111111  197.88888889]]
  Numpy.cov() function: [[ 328.66666667  228.73333333]
   [ 228.73333333  237.46666667]]
  </code></pre>
  
  <p>Why is that? How is <code>numpy.cov()</code> function implemented? If the function <code>numpy.cov()</code> is well-implemented, what am I doing wrong? I'll just say, that results from my function <code>covariance()</code> are consistent with the results from <code>paper</code> examples in the internet for calculating the covariance coefficient, eg <a href="http://www.naukowiec.org/wzory/statystyka/kowariancja_11.html" rel="noreferrer">http://www.naukowiec.org/wzory/statystyka/kowariancja_11.html</a>.</p>