title: How do I estimate the right parameters for a cumulative gaussian fit?
body: <p>I'm trying to fit a cumulative Gaussian distribution to my data, however the fits are clearly wrong. Why am I getting wrong means and standard deviations? Below you find my code and output.</p>
  
  <pre><code>import numpy as np
  import matplotlib.pyplot as plt
  from scipy.stats import norm
  
  testrefratios=np.array([ 0.2,  0.4,  0.6,  0.8,  0.9,  1. ,  1.1,  1.2,  1.4,  1.6,  1.8])
  Pn_final=np.array([ 0. ,   0. ,   0.03 , 0.35 , 0.47,  0.57 , 0.68,  0.73,  0.76 , 0.85 , 0.91])
  Pd_final=np.array([ 0. ,   0.03,  0.36 , 0.85 , 0.97,  0.98 , 0.98 , 0.99 , 1.,    1.,    1.  ])
  
   # cumulative gaussian fit
  fg = plt.figure(1); fg.clf()
  ax = fg.add_subplot(1, 1, 1)
  t = np.linspace(0,2, 1000) 
  
  ax.grid(True)
  ax.set_ylabel("Cumulative Probability Density")
  ax.set_title("Fit to Normal Distribution")
  
  mu1,sigma1 = norm.fit(Pn_final) # classical fit
  ax.plot(t, norm.cdf(t, mu1, sigma1), alpha=.5)
  
  mu1,sigma1 = norm.fit(Pd_final) # classical fit
  ax.plot(t, norm.cdf(t, mu1, sigma1), alpha=.5)
  
  ax.plot(testrefratios, Pn_final, 'bo',label='numerosity comparison')
  ax.plot(testrefratios, Pd_final, 'ro', label='density comparison')
  
  plt.legend(loc='lower right')
  
  
  fg.canvas.draw()
  </code></pre>
  
  <p>Output:</p>
  
  <p><a href="https://i.stack.imgur.com/HFubX.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/HFubX.png" alt="Fit results with code shown"></a></p>