Normal Distribution 1

Normal Distribution

Author: Dr. Brajesh Kumar Jha

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Introduction

The Normal Distribution (also known as the Gaussian Distribution) is a continuous probability distribution that describes data clustering around a mean value in a symmetric, bell-shaped curve. It is widely used in statistics, science, and engineering to model real-world phenomena such as heights, weights, IQ, and measurement errors.

Definition and Formula

A random variable X is said to follow a Normal Distribution with mean μ and standard deviation σ if:

\f(x)=(1/(σ√(2π)))*e^(-½((x-μ)/σ)²),   -∞ < x < ∞

• μ = Mean (center of the curve)
• σ = Standard Deviation (spread)
• Total area under the curve = 1

Normal Distribution Curve

Properties of Normal Distribution

• Bell-shaped and symmetric about the mean.
• Mean = Median = Mode.
• Total area under the curve is 1.
• 68.26% of data lies within μ ± σ.
• 95.44% within μ ± 2σ.
• 99.73% within μ ± 3σ.
• Curve is asymptotic to the X-axis.

Standard Normal Distribution

When μ = 0 and σ = 1, the variable Z is defined as:

Z = (X - μ) / σ

Z is called the Standard Normal Variable. Probabilities are calculated using the Z-table.

Standard Normal Table (Partial)

Z	0.00	0.01	0.02	0.03	0.04
0.0	0.5000	0.5040	0.5080	0.5120	0.5160
0.1	0.5398	0.5438	0.5478	0.5517	0.5557
0.2	0.5793	0.5832	0.5871	0.5910	0.5948
0.3	0.6179	0.6217	0.6255	0.6293	0.6331
1.0	0.8413	0.8438	0.8461	0.8485	0.8508
1.5	0.9332	0.9345	0.9357	0.9370	0.9382
1.96	≈ 0.9750

Practical Applications

1. Education: Modeling students' test scores or IQ distribution.
2. Manufacturing: Analyzing product weights or diameters for quality control.
3. Biology: Studying variation in plant or animal sizes.
4. Finance: Modeling returns on investment portfolios.

Solved Examples

Example 1:

Find P(40 < X < 60) for X ~ N(50, 10).

Z₁ = (40 - 50)/10 = -1,   Z₂ = (60 - 50)/10 = 1
P(-1 < Z < 1) = 0.6826
Answer: 0.6826

Example 2:

Find P(X < 85) for X ~ N(100, 15).

Z = (85 - 100)/15 = -1
P(Z < -1) = 0.1587
Answer: 0.1587

Example 3:

Find P(X > 115) for X ~ N(100, 15).

Z = (115 - 100)/15 = 1
P(Z > 1) = 1 - 0.8413 = 0.1587
Answer: 0.1587

Example 4:

For X ~ N(70, 8), find P(62 < X < 78).

Z₁ = -1, Z₂ = 1 ⇒ P(-1 < Z < 1) = 0.6826
Answer: 0.6826

Summary

Range	Area Under Curve (%)
μ ± 1σ	68.26%
μ ± 2σ	95.44%
μ ± 3σ	99.73%

Application

The Normal Distribution is a cornerstone of probability and statistics. It is essential in data analysis, quality control, hypothesis testing, and predictive modeling across diverse scientific and engineering fields.