The uniform distribution is one of the simplest and most intuitive probability distributions. If every outcome in a range is equally likely, you're dealing with a uniform distribution.

Using NumPy, Python makes it easy to generate, analyze, and visualize uniformly distributed data — a key skill for simulations, data analysis, and testing.

What is a Uniform Distribution?

A uniform distribution describes a situation where every value within a specified range is equally likely to occur.

There are two main types:

1. Discrete Uniform Distribution – finite, equally likely outcomes (e.g., rolling a die).

2. Continuous Uniform Distribution – all values in a real interval have equal probability (e.g., random float between 0 and 1).

This article focuses on the continuous uniform distribution as implemented in NumPy.

Uniform Distribution Formula

For a continuous uniform distribution between a and b, the probability density function (PDF) is:

f(x)=1b−afor a≤x≤bf(x) = \frac{1}{b - a} \quad \text{for } a \leq x \leq b

Real-Life Examples

NumPy’s Uniform Function

NumPy provides the uniform() method via its random number generator:

Syntax

numpy.random.Generator.uniform(low=0.0, high=1.0, size=None)

Parameters

✅ Returns

A float or NumPy array of uniformly distributed values.

✅ Example: Generate Uniform Random Numbers

import numpy as np

rng = np.random.default_rng(seed=42)
data = rng.uniform(low=0, high=10, size=1000)

This generates 1000 values uniformly distributed between 0 and 10.

Visualizing Uniform Distribution

import matplotlib.pyplot as plt
import seaborn as sns

sns.histplot(data, bins=20, kde=False, color='skyblue')
plt.title("Uniform Distribution (0 to 10)")
plt.xlabel("Value")
plt.ylabel("Frequency")
plt.grid(True)
plt.show()

You should see a fairly flat histogram — every value has roughly the same frequency.

Mean and Variance

For a continuous uniform distribution:

• Mean: (a+b)/2(a + b) / 2

• Variance: (b−a)2/12(b - a)^2 / 12

low, high = 0, 10
mean_theoretical = (low + high) / 2
var_theoretical = ((high - low) ** 2) / 12

mean_sample = np.mean(data)
var_sample = np.var(data)

print("Theoretical Mean:", mean_theoretical)
print("Sample Mean:", round(mean_sample, 2))
print("Theoretical Variance:", var_theoretical)
print("Sample Variance:", round(var_sample, 2))

Try Different Ranges

Uniform between -5 and 5

data = rng.uniform(low=-5, high=5, size=1000)

sns.histplot(data, bins=20, kde=False, color='lightgreen')
plt.title("Uniform Distribution (-5 to 5)")
plt.xlabel("Value")
plt.show()

Full Example: Simulating Random Delays

Let’s simulate 1000 random network delays that occur uniformly between 20ms and 120ms.

import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns

rng = np.random.default_rng(seed=123)

# Simulate delays
delays = rng.uniform(low=20, high=120, size=1000)

# Plot
sns.histplot(delays, bins=30, kde=False, color='orange')
plt.title("Simulated Network Delays (20ms to 120ms)")
plt.xlabel("Delay (ms)")
plt.ylabel("Frequency")
plt.grid(True)
plt.show()

# Stats
print("Average Delay:", round(np.mean(delays), 2), "ms")
print("Minimum Delay:", round(np.min(delays), 2), "ms")
print("Maximum Delay:", round(np.max(delays), 2), "ms")

Tips

⚠️ Common Pitfalls

Uniform vs Other Distributions

Conclusion

The uniform distribution is a foundational concept in statistics and simulation. With NumPy, it becomes effortless to:

• Generate random numbers across ranges

• Simulate real-world processes like delays or randomness

• Visualize and validate uniformity

• Perform statistical analysis

Key Takeaways

• Use uniform(low, high, size) for continuous uniform data.

• Use np.random.default_rng() for better random number generation.

• Validate your samples with mean/variance comparisons.

• Use histograms to visually confirm uniformity.