Normal Distribution¶

Standard Normal and QQ-Plots¶

In [1]:

import matplotlib.pyplot as plt
from scipy.stats import probplot, norm
import pandas as pd

fig, ax = plt.subplots(figsize=(4,4))
norm_sample = norm.rvs(size=100)
probplot(norm_sample, plot=ax)

import matplotlib.pyplot as plt from scipy.stats import probplot, norm import pandas as pd fig, ax = plt.subplots(figsize=(4,4)) norm_sample = norm.rvs(size=100) probplot(norm_sample, plot=ax)

Out[1]:

((array([-2.46203784, -2.12570747, -1.93122778, -1.79044653, -1.67819304,
         -1.58381122, -1.50174123, -1.42869743, -1.36256869, -1.30191411,
         -1.24570419, -1.19317644, -1.14374949, -1.09696931, -1.05247413,
         -1.00997067, -0.96921765, -0.93001393, -0.89218993, -0.85560121,
         -0.82012357, -0.78564937, -0.75208458, -0.71934648, -0.68736185,
         -0.65606548, -0.62539893, -0.59530962, -0.56574992, -0.53667655,
         -0.50804994, -0.47983378, -0.45199463, -0.42450149, -0.39732558,
         -0.37044003, -0.34381966, -0.31744076, -0.29128096, -0.26531902,
         -0.23953472, -0.21390872, -0.18842244, -0.16305799, -0.13779803,
         -0.1126257 , -0.08752455, -0.06247843, -0.03747145, -0.01248789,
          0.01248789,  0.03747145,  0.06247843,  0.08752455,  0.1126257 ,
          0.13779803,  0.16305799,  0.18842244,  0.21390872,  0.23953472,
          0.26531902,  0.29128096,  0.31744076,  0.34381966,  0.37044003,
          0.39732558,  0.42450149,  0.45199463,  0.47983378,  0.50804994,
          0.53667655,  0.56574992,  0.59530962,  0.62539893,  0.65606548,
          0.68736185,  0.71934648,  0.75208458,  0.78564937,  0.82012357,
          0.85560121,  0.89218993,  0.93001393,  0.96921765,  1.00997067,
          1.05247413,  1.09696931,  1.14374949,  1.19317644,  1.24570419,
          1.30191411,  1.36256869,  1.42869743,  1.50174123,  1.58381122,
          1.67819304,  1.79044653,  1.93122778,  2.12570747,  2.46203784]),
  array([-2.44830184, -2.02998617, -2.01832697, -1.88827648, -1.8829128 ,
         -1.8796135 , -1.6358162 , -1.55985054, -1.52759732, -1.51916628,
         -1.45915343, -1.45518746, -1.37631179, -1.34355166, -1.30785552,
         -1.25498437, -1.20370201, -1.12883074, -1.02799441, -1.02142495,
         -0.98536632, -0.92549378, -0.89739628, -0.84865924, -0.84004157,
         -0.80309913, -0.78787241, -0.72460806, -0.72362519, -0.65810047,
         -0.63201213, -0.60986112, -0.60155744, -0.60143905, -0.59571075,
         -0.56221674, -0.55689617, -0.54552105, -0.54364474, -0.51663865,
         -0.48852302, -0.47480099, -0.45639851, -0.44956551, -0.4402891 ,
         -0.41923124, -0.40817537, -0.40626393, -0.32432492, -0.31037607,
         -0.29542827, -0.26741507, -0.23194245, -0.18548701, -0.1694805 ,
         -0.14924765, -0.14730364, -0.13969416, -0.08049323, -0.06968245,
         -0.05378667,  0.0084616 ,  0.0273424 ,  0.1562903 ,  0.1642044 ,
          0.18881956,  0.19401566,  0.21096791,  0.23489804,  0.26099298,
          0.27488471,  0.33182229,  0.36487213,  0.40807126,  0.46939119,
          0.47178008,  0.48553555,  0.49873911,  0.50231611,  0.63153126,
          0.64858313,  0.65465624,  0.65651203,  0.70579422,  0.80309328,
          0.82924662,  0.87432743,  0.99881214,  1.05294332,  1.05721602,
          1.11938246,  1.28842505,  1.39150953,  1.44047513,  1.50378834,
          1.57025122,  1.62162806,  2.17055666,  2.39380068,  3.24126204])),
 (np.float64(1.0312192825303765),
  np.float64(-0.18989314332756427),
  np.float64(0.9909872901968868)))

In [2]:

loans = pd.read_csv("../data/loans_income.csv")
loan_values = loans["x"].dropna()
z_loan = (loan_values - loan_values.mean()) / loan_values.std()
z_loan

loans = pd.read_csv("../data/loans_income.csv") loan_values = loans["x"].dropna() z_loan = (loan_values - loan_values.mean()) / loan_values.std() z_loan

Out[2]:

0       -0.053557
1       -0.509872
2        0.950336
3        0.304255
4       -0.964939
           ...   
49995   -0.874924
49996   -0.449030
49997   -0.570714
49998    0.402758
49999    0.037706
Name: x, Length: 50000, dtype: float64

In [3]:

fig, ax = plt.subplots(figsize=(4,4))
probplot(z_loan, plot=ax)

fig, ax = plt.subplots(figsize=(4,4)) probplot(z_loan, plot=ax)

Out[3]:

((array([-4.19138481, -3.98563638, -3.8734641 , ...,  3.8734641 ,
          3.98563638,  4.19138481], shape=(50000,)),
  array([-1.97007936, -1.88185848, -1.87881638, ...,  3.94452245,
          3.96201452,  3.96201452], shape=(50000,))),
 (np.float64(0.9658789452608367),
  np.float64(-4.2414428944933633e-16),
  np.float64(0.9658085995476677)))

In [4]:

loan_dist = []
for _ in range(10000):
    sample = loans.sample(100)
    loan_dist.append(sample["x"].mean())

fig, ax = plt.subplots(figsize=(4,4))
probplot(x=loan_dist, plot=ax)

loan_dist = [] for _ in range(10000): sample = loans.sample(100) loan_dist.append(sample["x"].mean()) fig, ax = plt.subplots(figsize=(4,4)) probplot(x=loan_dist, plot=ax)

Out[4]:

((array([-3.81060943, -3.58545756, -3.46184782, ...,  3.46184782,
          3.58545756,  3.81060943], shape=(10000,)),
  array([57848.29, 57874.55, 58274.11, ..., 81005.91, 81481.89, 82403.12],
        shape=(10000,))),
 (np.float64(3271.061390190755),
  np.float64(68758.103299),
  np.float64(0.9996216628566901)))

We can see that sampling distribution of the sample mean is normally distributed.

In [5]:

import matplotlib.pyplot as plt
import seaborn as sns

plt.hist(loan_dist, bins=30, density=True, alpha=0.5, edgecolor='black')
sns.kdeplot(loan_dist, color='orange', linewidth=2)

import matplotlib.pyplot as plt import seaborn as sns plt.hist(loan_dist, bins=30, density=True, alpha=0.5, edgecolor='black') sns.kdeplot(loan_dist, color='orange', linewidth=2)

Out[5]:

<Axes: ylabel='Density'>