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

Last Updated : 24 Jan, 2021

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') 