Understanding Data Distribution In Statistics And Machine Learning : A Complete Guide With Examples

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In machine learning, data distribution refers to how data points are distributed or spread out across a dataset. In the article, we’ll explore what Data Distribution in Statistics and Machine Learning is, and how to calculate it using Python.

Data Distribution in Statistics

Data distribution refers to how data points are spread out across a dataset. In other words, it shows us how the values are arranged—whether they are mostly clustered around a central point, spread out evenly, or skewed toward one side. Understanding the distribution of your data is very important because it can directly affect how well a machine learning algorithm performs.

To describe data distribution, we use statistical measures like mean, median, mode, standard deviation, and variance. These help us understand the central tendency (where the data is centered), the spread (how much the data varies), and the shape (such as symmetric, skewed, or bell-shaped) of the dataset.

Normal Distribution

Normal distribution, also called Gaussian distribution, is one of the most common and important probability distributions in statistics and machine learning. It has a smooth, bell-shaped curve that is symmetrical around the mean. This means most of the data points are close to the average, and fewer values appear as you move farther away from it.

The shape of a normal distribution is defined by two numbers: the mean (μ), which shows the center of the curve, and the standard deviation (σ), which tells us how spread out the data is.

In machine learning, the normal distribution is used in many ways. For example, it’s often used to model errors or noise in linear regression. It also plays a key role in hypothesis testing and building confidence intervals, which help us make decisions based on data.

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

# Generate a random sample of 1000 values from a normal distribution
mu = 0 # Mean
sigma = 1 # Standard deviation
sample = np.random.normal(mu, sigma, 1000)

# Calculate the PDF for the normal distribution
x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)

# Plot the histogram of the random sample and the PDF of the normal
distribution
pdf = norm.pdf(x, mu, sigma)
plt.figure(figsize=(7.5, 3.5))
plt.hist(sample, bins=30, density=True, alpha=0.5)
plt.plot(x, pdf)
plt.show()

import numpy as np

from scipy.stats import norm

import matplotlib.pyplot as plt

# Generate a random sample of 1000 values from a normal distribution

mu = 0 # Mean

sigma = 1 # Standard deviation

sample = np.random.normal(mu, sigma, 1000)

# Calculate the PDF for the normal distribution

x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)

# Plot the histogram of the random sample and the PDF of the normal

distribution

pdf = norm.pdf(x, mu, sigma)

plt.figure(figsize=(7.5, 3.5))

plt.hist(sample, bins=30, density=True, alpha=0.5)

plt.plot(x, pdf)

plt.show()

Skewed Distribution

In machine learning, a skewed distribution describes a dataset where the data is not spread evenly around the average (mean). Instead, most of the data points are clustered toward one side, while fewer points stretch out toward the other side.

There are two main types of skewed distributions: left-skewed and right-skewed. A left-skewed (or negative-skewed) distribution has a long tail on the left side, meaning there are some unusually low values, while most data points are grouped on the right. On the other hand, a right-skewed (or positive-skewed) distribution has a long tail on the right side, with most data points concentrated on the left.

Understanding skewness helps in choosing the right methods to analyze and prepare data for machine learning.

import numpy as np
import matplotlib.pyplot as plt

# Generate a skewed distribution using NumPy's random function
data = np.random.gamma(2, 1, 1000)

# Plot a histogram of the data to visualize the distribution
plt.figure(figsize=(7.5, 3.5))
plt.hist(data, bins=30)

# Add labels and title to the plot
plt.xlabel('Value')
plt.ylabel('Frequency')
plt.title('Skewed Distribution')

# Show the plot
plt.show()

import numpy as np

import matplotlib.pyplot as plt

# Generate a skewed distribution using NumPy's random function

data = np.random.gamma(2, 1, 1000)

# Plot a histogram of the data to visualize the distribution

plt.figure(figsize=(7.5, 3.5))

plt.hist(data, bins=30)

# Add labels and title to the plot

plt.xlabel('Value')

plt.ylabel('Frequency')

plt.title('Skewed Distribution')

# Show the plot

plt.show()

Uniform Distribution

In machine learning, a uniform distribution is a type of probability distribution where every value has an equal chance of occurring. This means the data is spread out evenly, with no particular value appearing more often than others. There’s no central peak or clustering around any point—everything is equally likely.

Uniform distribution is often used as a baseline to compare with other types of distributions. Since it represents completely random and fair sampling, it’s useful in situations like generating random numbers or choosing items from a set without bias. While real-world data often doesn’t follow a perfect uniform distribution, understanding it helps build a strong foundation in probability and data analysis.

import numpy as np
import matplotlib.pyplot as plt

# Generate 10,000 random numbers from a uniform distribution between 0 and 1
uniform_data = np.random.uniform(low=0, high=1, size=10000)

# Plot the histogram of the uniform data
plt.figure(figsize=(7.5, 3.5))
plt.hist(uniform_data, bins=50, density=True)

# Add labels and title to the plot
plt.xlabel('Value')
plt.ylabel('Frequency')
plt.title('Uniform Distribution')

# Show the plot
plt.show()

import numpy as np

import matplotlib.pyplot as plt

# Generate 10,000 random numbers from a uniform distribution between 0 and 1

uniform_data = np.random.uniform(low=0, high=1, size=10000)

# Plot the histogram of the uniform data

plt.figure(figsize=(7.5, 3.5))

plt.hist(uniform_data, bins=50, density=True)

# Add labels and title to the plot

plt.xlabel('Value')

plt.ylabel('Frequency')

plt.title('Uniform Distribution')

# Show the plot

plt.show()

Bimodal Distribution

A bimodal distribution is a type of probability distribution that has two distinct peaks or “modes”. This means the data has two areas where values occur most frequently, with a dip or gap in between where fewer values are found.

Bimodal distributions often appear when a dataset includes two different groups or patterns. For example, if you measured the heights of a group that includes both adults and children, the data might form two peaks—one for each group. Bimodal patterns can show up in many fields, such as biometric data, economic trends, or user behavior on social media.

Recognizing a bimodal distribution helps us understand that the data may contain multiple subpopulations or distinct behaviors, which can be important when building machine learning models.

import numpy as np
import matplotlib.pyplot as plt

# Generate 10,000 random numbers from a bimodal distribution
bimodal_data = np.concatenate((np.random.normal(loc=-2, scale=1, size=5000),
   np.random.normal(loc=2, scale=1, size=5000)))
 
# Plot the histogram of the bimodal data
plt.figure(figsize=(7.5, 3.5))
plt.hist(bimodal_data, bins=50, density=True)

# Add labels and title to the plot
plt.xlabel('Value')
plt.ylabel('Frequency')
plt.title('Bimodal Distribution')

# Show the plot
plt.show()