Chi-Square Distribution in NumPy

The Chi-Square Distribution appears when you add up the squares of independent standard normal random variables. It is widely used in hypothesis testing, goodness-of-fit tests, variance testing, and statistical modeling. In NumPy, we generate Chi-Square values using numpy.random.chisquare().

Example: Here, we generate one Chi-Square random value using df = 2 (degrees of freedom).

import numpy as np
x = np.random.chisquare(df=2)
print(x)

0.3396810372458067

Explanation: np.random.chisquare(df=2) generates one value formed by summing the squares of 2 standard normal variables.

Syntax

numpy.random.chisquare(df, size=None)

Parameters:

• df: Degrees of freedom (controls the shape of the curve)
• size: Shape of output array

Examples

Example 1: In this example, we generate 5 Chi-Square random values with df = 2.

import numpy as np
arr = np.random.chisquare(df=2, size=5)
print(arr)

[0.9664276  0.9718178  0.05315296 5.76413224 0.17793754]

Explanation: np.random.chisquare(..., size=5) returns an array of 5 simulated Chi-Square outcomes.

Example 2: Here, we generate Chi-Square values with a higher degree of freedom (df = 5).

import numpy as np
x = np.random.chisquare(5, size=4)
print(x)

[5.5554776  8.39987081 1.96062558 1.15995049]

Explanation: np.random.chisquare(5) produces values based on 5 summed squares, giving a wider and more symmetric shape.

Example 3: In this example, we generate a 2×3 matrix of Chi-Square random values.

import numpy as np
m = np.random.chisquare(3, size=(2, 3))
print(m)

[[0.04478657 4.99709829 0.51742473]
 [0.87978466 1.96904351 3.51846182]]

Explanation: size=(2,3) creates a matrix where each element is a Chi-Square-distributed value.

Visualizing the Chi-Square Distribution

Visualizing the generated numbers helps in understanding how the Chi-Square curve behaves for different degrees of freedom.

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import chi2

df = 2
size = 1000

data = np.random.chisquare(df, size)
plt.hist(data, bins=30, density=True, edgecolor='black', alpha=0.7, label='Histogram')

x = np.linspace(0, max(data), 200)
pdf = chi2.pdf(x, df)
plt.plot(x, pdf, color='red', label='Theoretical PDF')

plt.title(f"Chi-Square Distribution (df={df})")
plt.xlabel("Value")
plt.ylabel("Density")
plt.legend()
plt.grid(True)
plt.show()

Output

Explanation:

• np.random.chisquare(df, size) simulates 1000 Chi-Square values.
• plt.hist(..., density=True) displays their frequency.
• chi2.pdf(x, df) computes the true theoretical curve.
• The red line shows how the actual Chi-Square distribution should look for df = 2.
• Chi-Square distributions are right-skewed, especially at lower degrees of freedom.