Normal Distribution

The Normal Distribution

The normal (Gaussian) distribution is the most important probability distribution in statistics. It's characterized by its bell-shaped curve and is defined by two parameters:

• Mean (μ): Center of the distribution
• Standard Deviation (σ): Spread of the distribution

Probability Density Function (PDF)

The height of the normal curve at any point x:

f(x) = (1 / (σ√(2π))) × e^(-(x-μ)²/(2σ²))

• Maximum density occurs at x = μ (the mean)
• Inflection points occur at μ ± σ
• Total area under curve = 1 (100%)

Cumulative Distribution Function (CDF)

The area under the curve from -∞ to x gives the probability P(X ≤ x):

• CDF at mean (μ) = 0.5 (50th percentile)
• CDF values range from 0 to 1
• Cannot be expressed in closed form (requires numerical approximation)

Standard Normal Distribution

Special case with μ = 0 and σ = 1. Any normal distribution can be converted to standard normal using:

Z = (X - μ) / σ

68-95-99.7 Rule (Empirical Rule)

For any normal distribution:

• 68% of data falls within μ ± 1σ
• 95% of data falls within μ ± 2σ (more precisely, ±1.96σ)
• 99.7% of data falls within μ ± 3σ

Properties of Normal Distribution

• Symmetric around the mean
• Mean = Median = Mode
• Bell-shaped curve
• Asymptotic (approaches but never touches x-axis)
• Total area = 1 (represents 100% probability)

Calculating Probabilities

P(X < a): Area to the left of a

P(X > a): 1 - P(X < a) = Area to the right of a

P(a < X < b): P(X < b) - P(X < a) = Area between a and b

Real-World Examples

Variable	Mean (μ)	Std Dev (σ)
IQ Scores	100	15
SAT Scores (old)	500	100
Adult Male Height (US, cm)	175	7
Adult Female Height (US, cm)	162	7
Birth Weight (kg)	3.4	0.5

Applications

• Quality Control: Manufacturing tolerances, defect rates
• Finance: Stock returns, risk models
• Science: Measurement errors, experimental data
• Psychology: Test scores, personality traits
• Medicine: Blood pressure, cholesterol levels
• Education: Standardized test scoring, grade curves

Central Limit Theorem

Why the normal distribution is so important: For large sample sizes, the distribution of sample means approaches a normal distribution, regardless of the shape of the population distribution.

When NOT to Use Normal Distribution

• Data is heavily skewed (use log-normal or other distributions)
• Data has outliers or heavy tails (use t-distribution)
• Count data (use Poisson distribution)
• Binary outcomes (use binomial distribution)
• Small sample sizes (n < 30) with unknown distribution

Example Calculation

Question: IQ scores are normally distributed with μ = 100, σ = 15. What percentage of people have IQ > 130?

Solution:

1. Calculate z-score: z = (130 - 100) / 15 = 2.0
2. Find P(Z < 2.0) = 0.9772 from z-table
3. P(Z > 2.0) = 1 - 0.9772 = 0.0228 = 2.28%

Answer: About 2.28% of people have IQ scores above 130.

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