Normal Data Distribution

Last modified on: October 19, 2024

<h3>Tutorial: Generating and Visualizing a Normal Distribution in Python</h3>
<h4>What is a Normal Distribution?</h4>
<p>A <strong>normal distribution</strong>, also known as a <strong>Gaussian distribution</strong>, is a bell-shaped curve where most data points cluster around the mean. It is symmetric about the mean, with the probability of data points decreasing as they move further from the mean.</p>
<p><strong>Key Characteristics</strong>:</p>
<ol>
<li><strong>Mean (&mu;)</strong>: The center of the distribution.</li>
<li><strong>Standard Deviation (&sigma;)</strong>: Controls the spread of the distribution. A larger standard deviation results in data being more spread out.</li>
</ol>
<h3>Steps to Generate a Normal Distribution in Python</h3>
<p>We will use the <code>numpy</code> library for generating the data and <code>matplotlib</code> for plotting the graph.</p>
<h4>1. <strong>Import Required Libraries</strong></h4>
<p>Begin by importing the necessary libraries:</p>
<pre class="language-python"><code>import numpy as np
import matplotlib.pyplot as plt</code></pre>
<ul>
<li><strong><code>numpy</code></strong>: Provides the function to generate random numbers.</li>
<li><strong><code>matplotlib</code></strong>: Used for creating the visual plot.</li>
</ul>
<h4>2. <strong>Generate Random Data</strong></h4>
<p>To generate data that follows a normal distribution, use the <code>np.random.normal()</code> function. You can specify the mean, standard deviation, and the number of data points.</p>
<ul>
<li><code>mean</code>: Data will cluster around this value (set to 0).</li>
<li><code>std_dev</code>: Most data will lie within 1 unit of the mean.</li>
<li><code>n_samples</code>: The number of random data points to generate.
<pre class="language-python"><code>mean = 0        # Mean of the distribution
std_dev = 1     # Standard deviation
n_samples = 1000  # Number of samples

data = np.random.normal(mean, std_dev, n_samples)</code></pre>
</li>
</ul>
<h4>3. <strong>Visualizing the Data Distribution</strong></h4>
<p>You can plot the generated data as a histogram using <code>matplotlib</code>:</p>
<ul>
<li><code>plt.hist()</code>: Creates the histogram.</li>
<li><code>bins=30</code>: Specifies the number of bins (intervals).</li>
<li><code>density=True</code>: Normalizes the histogram to show probability density.
<pre class="language-python"><code>plt.figure(figsize=(8, 6))   # Set the size of the plot

# Plot a histogram of the data
plt.hist(data, bins=30, edgecolor='black', alpha=0.7, density=True)

# Adding labels and title
plt.title('Normal Distribution with Mean=0, Standard Deviation=1')
plt.xlabel('Data Values')
plt.ylabel('Frequency')

# Display the plot
plt.show()</code></pre>
</li>
</ul>
<h4>4. <strong>Overlaying a Probability Density Function (PDF)</strong></h4>
<p>To make the plot more informative, overlay the <strong>Probability Density Function (PDF)</strong> of the normal distribution.</p>
<div class="dark bg-gray-950 contain-inline-size rounded-md border-[0.5px] border-token-border-medium relative">
<div class="flex items-center text-token-text-secondary bg-token-main-surface-secondary px-4 py-2 text-xs font-sans justify-between rounded-t-md h-9">
<pre class="language-python"><code># Generate values for the x-axis (from -4 to 4)
x = np.linspace(-4, 4, 1000)

# Compute the PDF of the normal distribution using the formula
pdf = (1 / (std_dev * np.sqrt(2 * np.pi))) * np.exp(-0.5 * ((x - mean) / std_dev)**2)

# Plot the PDF
plt.plot(x, pdf, color='red', label='PDF')

# Redraw the histogram for comparison
plt.hist(data, bins=30, edgecolor='black', alpha=0.6, density=True)

# Add labels and title
plt.title('Normal Distribution with PDF Overlay')
plt.xlabel('Data Values')
plt.ylabel('Density')
plt.legend()

# Display the plot
plt.show()</code></pre>
</div>
</div>
<p>This will overlay the normal distribution's PDF over the histogram, showing how well the data fits the theoretical curve.</p>
<h4>5. <strong>Customizing the Plot</strong></h4>
<p>You can further customize the plot by adjusting colors, adding grid lines, or modifying labels:</p>
<div class="dark bg-gray-950 contain-inline-size rounded-md border-[0.5px] border-token-border-medium relative">
<div class="flex items-center text-token-text-secondary bg-token-main-surface-secondary px-4 py-2 text-xs font-sans justify-between rounded-t-md h-9">
<pre class="language-python"><code>plt.figure(figsize=(10, 6))  # Set the figure size
plt.hist(data, bins=40, color='lightblue', edgecolor='black', alpha=0.7, density=True)

# Replot PDF
plt.plot(x, pdf, color='red', linewidth=2)

# Customizations
plt.title('Customized Normal Distribution Plot', fontsize=16)
plt.xlabel('Value', fontsize=14)
plt.ylabel('Probability Density', fontsize=14)
plt.grid(True)  # Add grid lines

# Show the plot
plt.show()</code></pre>
</div>
</div>
<h3>Conclusion</h3>
<p>In this tutorial, you learned how to:</p>
<ul>
<li>Generate random data following a normal distribution.</li>
<li>Plot the data using a histogram.</li>
<li>Overlay a probability density function (PDF) on the histogram.</li>
<li>Customize the plot for better visualization.</li>
</ul>
<p>This approach is fundamental in statistics and data science, and you can further experiment by changing the mean, standard deviation, and sample sizes to explore different normal distributions.</p>

Web Technology View More