Normal Distribution

The normal distribution, otherwise known as the Gaussian distribution, is one of the most signifcant probability distributions used for continuous data. It is defined by two parameters:

• Mean (μ): The central value of the distribution.
• Standard Deviation (σ): Computes the amount of variation or dispersion in the data.

Mathematically, the probability density function (PDF) used for the normal distribution is:

Where:

• x is the random variable
• μ is the mean
• σ is the standard deviation
• e is Euler’s number (approximately 2.71828)
• π is Pi (approximately 3.14159)

Key Properties

1. Bell-shaped and Symmetric: The distribution is perfectly symmetrical around its mean.
2. Mean, Median, and Mode are Equal: All three measures of central tendency have the same value.
3. Empirical Rule (68-95-99.7 Rule):
   • Approximately 68% of the given data falls within 1 standard deviation of the mean
   • Approximately 95% falls within 2 standard deviations
   • Approximately 99.7% falls within 3 standard deviations
4. Standardized Form: Any normal distribution can be converted to a standard normal distribution (μ=0, σ=1) using the formula z = (x-μ)/σ.

Applications

The normal distribution is broadly used in various fields:

• Finance: Modeling stock returns
• Natural Sciences: Measurement errors
• Social Sciences: IQ scores, heights, and other human characteristics
• Machine Learning: Assumptions in many algorithms
• Quality Control: Manufacturing processes

Example

The following code creates a sample of 1,000 normally distributed data points with a mean of 70 and a standard deviation of 10, and displays this data in a 2×2 grid of plots for analysis:

import numpy as np
import matplotlib.pyplot as plt
from scipy import stats
import seaborn as sns

# Set seed for reproducibility
np.random.seed(42)

# Generate random data from a normal distribution
# Parameters: mean=70, standard deviation=10, size=1000
data = np.random.normal(70, 10, 1000)

# Create visualizations
plt.figure(figsize=(12, 8))

# Histogram with density curve
plt.subplot(2, 2, 1)
sns.histplot(data, kde=True, stat="density")
plt.title('Histogram with Density Curve')
plt.xlabel('Value')
plt.ylabel('Density')

# Q-Q plot to check normality
plt.subplot(2, 2, 2)
stats.probplot(data, plot=plt)
plt.title('Q-Q Plot')

# Box plot
plt.subplot(2, 2, 3)
sns.boxplot(x=data)
plt.title('Box Plot')
plt.xlabel('Value')

# Verify the empirical rule
plt.subplot(2, 2, 4)
mean = np.mean(data)
std = np.std(data)
within_1_std = np.sum((mean - std <= data) & (data <= mean + std)) / len(data) * 100
within_2_std = np.sum((mean - 2*std <= data) & (data <= mean + 2*std)) / len(data) * 100
within_3_std = np.sum((mean - 3*std <= data) & (data <= mean + 3*std)) / len(data) * 100

bars = plt.bar(['1σ', '2σ', '3σ'], [within_1_std, within_2_std, within_3_std])
plt.axhline(y=68, color='r', linestyle='-', label='68% (theoretical)')
plt.axhline(y=95, color='g', linestyle='-', label='95% (theoretical)')
plt.axhline(y=99.7, color='b', linestyle='-', label='99.7% (theoretical)')
plt.title('Empirical Rule Verification')
plt.xlabel('Standard Deviation Range')
plt.ylabel('Percentage of Data (%)')
plt.legend()

plt.tight_layout()
plt.show()

# Statistical summary
print("Statistical Summary:")
print(f"Mean: {np.mean(data):.2f}")
print(f"Median: {np.median(data):.2f}")
print(f"Standard Deviation: {np.std(data):.2f}")
print(f"Skewness: {stats.skew(data):.4f}")
print(f"Kurtosis: {stats.kurtosis(data):.4f}")
print("\nEmpirical Rule Verification:")
print(f"Data within 1 standard deviation: {within_1_std:.2f}% (theoretical: 68%)")
print(f"Data within 2 standard deviations: {within_2_std:.2f}% (theoretical: 95%)")
print(f"Data within 3 standard deviations: {within_3_std:.2f}% (theoretical: 99.7%)")

The output of the above code will be:

Statistical Summary:
Mean: 70.19
Median: 70.25
Standard Deviation: 9.79
Skewness: 0.1168
Kurtosis: 0.0662

Empirical Rule Verification:
Data within 1 standard deviation: 68.60% (theoretical: 68%)
Data within 2 standard deviations: 95.60% (theoretical: 95%)
Data within 3 standard deviations: 99.70% (theoretical: 99.7%)

The histogram with density curve shows the bell-shaped curve characteristic of normal distributions:

Q-Q plot compares the data quantiles against theoretical normal distribution quantiles to check if the data follows a normal distribution (points following the diagonal line indicate normality):

Box plot visualizes the central tendency and spread of the data:

Bar chart tests whether the data follows the 68-95-99.7 rule by calculating the percentage of data points that fall within 1, 2, and 3 standard deviations: