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How to calculate and plot a Cumulative Distribution function with Matplotlib in Python ?

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  • Last Updated : 24 Jan, 2021
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Prerequisites: Matplotlib 

Matplotlib is a library in Python and it is a numerical — mathematical extension for the NumPy library.  The cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

Properties of CDF:

  • Every cumulative distribution function F(X) is non-decreasing
  • If maximum value of the cdf function is at x, F(x) = 1.
  • The CDF ranges from 0 to 1.

Method 1: Using the histogram

CDF can be calculated using PDF (Probability Distribution Function). Each point of random variable will contribute cumulatively to form CDF.

Example : 

A combination set containing 2 balls which can be either red or blue can be in the following set.

{RR, RB, BR, BB}

t -> No of red balls.

P(x = t) -> t = 0 : 1 / 4 [BB] 

            t = 1 : 2 / 4 [RB, BR]

            t = 2 : 1 / 4 [RR]

            

CDF :

F(x) = P(x<=t)

x = 0 : P(0)               -> 1 / 4

x = 1 : P(1) + P(0)        -> 3 / 4

x = 2 : P(2) + P(1) + P(0) -> 1

Approach

  • Import modules
  • Declare number of data points
  • Initialize random values
  • Plot histogram using above data
  • Get histogram data
  • Finding PDF using histogram data
  • Calculate CDF
  • Plot CDF

Example:

Python3




# defining the libraries
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
%matplotlib inline
  
# No of Data points
N = 500
  
# initializing random values
data = np.random.randn(N)
  
# getting data of the histogram
count, bins_count = np.histogram(data, bins=10)
  
# finding the PDF of the histogram using count values
pdf = count / sum(count)
  
# using numpy np.cumsum to calculate the CDF
# We can also find using the PDF values by looping and adding
cdf = np.cumsum(pdf)
  
# plotting PDF and CDF
plt.plot(bins_count[1:], pdf, color="red", label="PDF")
plt.plot(bins_count[1:], cdf, label="CDF")
plt.legend()

Output:

Histogram plot of the PDF and CDF :

Plotted CDF:

CDF plotting

Method 2: Data sort 

This method depicts how CDF can be calculated and plotted using sorted data. For this, we first sort the data and then handle further calculations.

Approach

  • Import module
  • Declare number of data points
  • Create data
  • Sort data in ascending order
  • Get CDF
  • Plot CDF
  • Display plot

Example:

Python3




# defining the libraries
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
%matplotlib inline
  
# No of data points used
N = 500
  
# normal distribution
data = np.random.randn(N)
  
# sort the data in ascending order
x = np.sort(data)
  
# get the cdf values of y
y = np.arange(N) / float(N)
  
# plotting
plt.xlabel('x-axis')
plt.ylabel('y-axis')
  
plt.title('CDF using sorting the data')
  
plt.plot(x, y, marker='o')

Output:


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