Normal Distribution Calculator

The normal distribution is the foundation of statistical analysis.

The Bell Curve

The probability density function (PDF) is:

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

Where:

• μ (mu) = mean
• σ (sigma) = standard deviation
• e ≈ 2.71828 (Euler's number)
• π ≈ 3.14159

Key Parameters

Mean (μ): The center of the distribution

• Shifts the curve left or right
• Equals the median and mode

Standard Deviation (σ): The spread of the distribution

• Larger σ = wider, flatter curve
• Smaller σ = narrower, taller curve

Standard Normal Distribution

When μ = 0 and σ = 1, we have the standard normal distribution:

• Z ~ N(0, 1)
• Used as reference for all normal calculations
• Z-tables are based on this

Converting to Standard Normal

Any normal variable X ~ N(μ, σ²) can be standardized:

Z = (X - μ) / σ

This allows using standard normal tables for any normal distribution.

A quick way to understand normal distribution spread.

The Rule

For any normal distribution:

Visual Interpretation

         |
       __|__
      /  |  \
     /   |   \
    /    |    \
   /     |     \
__|______|______|__
-3σ -2σ -1σ μ +1σ +2σ +3σ

   |---- 68% ----|
|------- 95% -------|
|-------- 99.7% --------|

Practical Example

SAT scores: μ = 1000, σ = 200

• 68% score between 800 and 1200
• 95% score between 600 and 1400
• 99.7% score between 400 and 1600

Beyond 3σ

Six Sigma quality control aims for defect rates at ± 6σ.

Z-scores measure how many standard deviations a value is from the mean.

Z-Score Formula

Z = (X - μ) / σ

Where:

• X = raw score
• μ = population mean
• σ = population standard deviation

Interpretation

Example: Test Scores

Test with μ = 75, σ = 10

Your score: 90 Z = (90 - 75) / 10 = 1.5

You scored 1.5 standard deviations above average.

Percentile: P(Z < 1.5) ≈ 93.3%

You scored better than about 93% of test-takers.

Why Z-Scores Are Useful

1. Compare across distributions: A z = 1.5 in English vs. z = 1.2 in Math
2. Identify outliers: |Z| > 3 is usually considered unusual
3. Probability calculation: Use standard normal tables
4. Data analysis: Standardization centers and scales data

Reverse Calculation

Given Z, find X: X = μ + Z × σ

If Z = 1.5, μ = 75, σ = 10: X = 75 + 1.5 × 10 = 90

Finding the probability that a normal random variable falls in a given range.

Types of Probability Questions

1. Left-tail: P(X ≤ x) Area under curve from -∞ to x

2. Right-tail: P(X ≥ x) Area under curve from x to +∞ = 1 - P(X ≤ x)

3. Between: P(a ≤ X ≤ b) Area between two values = P(X ≤ b) - P(X ≤ a)

4. Outside: P(X < a or X > b) = 1 - P(a ≤ X ≤ b)

Step-by-Step Process

1. Convert X to Z: Z = (X - μ) / σ
2. Use standard normal table or calculator
3. Apply appropriate formula for tail direction

Example

Heights: μ = 170 cm, σ = 10 cm

P(Height < 180 cm)?

1. Z = (180 - 170) / 10 = 1.0
2. P(Z < 1.0) = 0.8413
3. 84.13% of people are shorter than 180 cm

P(Height > 185 cm)?

1. Z = (185 - 170) / 10 = 1.5
2. P(Z > 1.5) = 1 - 0.9332 = 0.0668
3. 6.68% of people are taller than 185 cm

P(160 < Height < 180)?

1. Z₁ = (160 - 170) / 10 = -1.0
2. Z₂ = (180 - 170) / 10 = 1.0
3. P(-1 < Z < 1) = 0.8413 - 0.1587 = 0.6826
4. 68.26% of people are between 160-180 cm

Finding the value that corresponds to a given probability.

Definition

The p-th percentile is the value x such that P(X ≤ x) = p/100

• Median = 50th percentile
• Quartiles = 25th, 50th, 75th percentiles
• Deciles = 10th, 20th, ..., 90th percentiles

Inverse Normal Calculation

Given probability p, find x:

1. Find z such that P(Z ≤ z) = p
2. Convert back: x = μ + z × σ

Common Z-Values for Percentiles

Example: Finding Percentiles

SAT scores: μ = 1000, σ = 200

What score is the 90th percentile?

1. Z for 90th percentile = 1.282
2. X = 1000 + 1.282 × 200
3. X = 1256

To be in the top 10%, you need to score at least 1256.

What percentile is a score of 1150?

1. Z = (1150 - 1000) / 200 = 0.75
2. P(Z < 0.75) = 0.7734
3. A score of 1150 is the 77th percentile

Applications

• Setting cutoff scores for programs
• Determining "normal" ranges in medicine
• Grading on a curve
• Quality control limits

The normal distribution appears throughout statistics and real-world applications.

Natural Phenomena

Many measurements follow approximately normal distributions:

• Human heights and weights
• Blood pressure and cholesterol
• IQ scores (designed to be normal)
• Measurement errors
• Temperature variations

Statistical Inference

Confidence Intervals: For sample mean with large n: CI = x̄ ± z × (σ / √n)

Common z-values:

• 90% CI: z = 1.645
• 95% CI: z = 1.960
• 99% CI: z = 2.576

Hypothesis Testing: Many tests assume normality or use normal approximations.

Quality Control

Control Charts: Upper Control Limit = μ + 3σ Lower Control Limit = μ - 3σ

Points outside these limits signal a problem.

Six Sigma: Aims for 3.4 defects per million (beyond ± 6σ)

Finance

Value at Risk (VaR): VaR at 95% = μ - 1.645σ

Estimates potential loss in worst 5% of cases.

Option Pricing: Black-Scholes model assumes log-normal stock returns.

Central Limit Theorem

Sample means approach normal distribution as n increases, regardless of population distribution. This makes the normal distribution fundamental to statistical inference.

Sample mean: x̄ ~ N(μ, σ²/n)

Even non-normal data can be analyzed using normal-based methods for large samples.