Data Science 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%)")

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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%)

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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: