Normal Distribution

🎯 The Normal Distribution: Nature’s Favorite Pattern

The Story of the Bell Curve

Imagine you’re at a school fair, and everyone throws a ball at a target. Some throw perfectly. Some miss left. Some miss right. If you drew where all the balls landed, you’d see something magical: a beautiful bell shape.

This is the Normal Distribution — nature’s favorite way to organize things!

🔔 What is the Normal Distribution?

Think of measuring the height of everyone in your class.

• Most kids are around the middle height
• A few are really tall
• A few are really short

When you draw this on a graph, it looks like a bell — tall in the middle, sloping down on both sides.

graph TD
    A["📊 Heights of Kids"] --> B["Most kids: Medium height"]
    A --> C["Some kids: Taller"]
    A --> D["Some kids: Shorter"]
    B --> E["🔔 Bell Shape!"]
    C --> E
    D --> E

Key Features:

Real Example: Test scores in a class of 100 students

• Mean score: 75 points
• Most students score 70-80
• Few score very high (95+) or very low (50-)

📏 Standard Normal Distribution

What if we had a special ruler that works for ALL bell curves?

That’s the Standard Normal Distribution!

• Mean = 0 (the center)
• Standard Deviation = 1 (our measuring unit)

It’s like converting feet to meters — we’re converting ANY normal curve into ONE universal curve.

Why is this helpful?

One table works for EVERYTHING! 🎉

🔢 Z-Score: Your Magic Translator

A Z-score tells you: How far is this value from average?

The Formula:

Z = (Your Value - Mean) ÷ Standard Deviation

Or simply:

Z = (X - μ) ÷ σ

Example: Test Score Translation

Sarah scored 85 on a test.

• Class average (μ): 70
• Standard deviation (σ): 10

Z = (85 - 70) ÷ 10 = 15 ÷ 10 = 1.5

Sarah is 1.5 standard deviations ABOVE average! ⭐

What Z-scores tell you:

📊 Using Z-Tables: Your Treasure Map

A Z-table is like a treasure map. Give it a Z-score, and it tells you what percentage of people are below that score!

How to Read It:

1. Find your Z-score’s row (like 1.5)
2. Find the column for extra decimals (like .00)
3. Read the probability!

Example: Z = 1.50

Looking at the table → 0.9332

This means: 93.32% of people scored below Sarah!

Quick Reference:

🎯 Normal Percentiles: Finding Your Rank

A percentile answers: What percent of people are below me?

Example: Height Percentile

Tommy is in the 75th percentile for height.

This means: 75% of kids his age are shorter than him!

From Z-score to Percentile:

graph TD
    A["Your Value"] --> B["Calculate Z-score"]
    B --> C["Look up in Z-table"]
    C --> D["Get Percentile!"]

Finding a Percentile Value (Reverse!)

Question: What score is at the 90th percentile?

1. Look up 0.90 in the Z-table body
2. Find Z ≈ 1.28
3. Convert back: X = μ + Z × σ

Example: Test with mean 70, SD 10

X = 70 + (1.28 × 10) = 70 + 12.8 = 82.8

The 90th percentile score is about 83!

🌟 The Empirical Rule: 68-95-99.7

This is the most powerful shortcut in statistics!

For ANY normal distribution:

Visual Story:

        ←——— 99.7% ———→
      ←—— 95% ——→
    ←— 68% —→

|.....|.....|.....|.....|.....|
-3σ   -2σ   -1σ   μ    +1σ   +2σ   +3σ

Real Example: IQ Scores

IQ has mean = 100, SD = 15

• 68% of people: IQ between 85-115
• 95% of people: IQ between 70-130
• 99.7% of people: IQ between 55-145

If someone has IQ 145, they’re in the top 0.15%! 🧠

🔄 Normal to Binomial: When Coin Flips Meet Bells

Sometimes, counting problems (binomial) can be solved with the normal curve!

When Can We Switch?

The magic rule: np ≥ 10 AND n(1-p) ≥ 10

Where:

• n = number of trials
• p = probability of success

Example: Coin Flips

Flip a coin 100 times. How likely to get 45-55 heads?

Check: 100 × 0.5 = 50 ✓ (≥10)

Convert to Normal:

• Mean: μ = np = 100 × 0.5 = 50
• SD: σ = √(np(1-p)) = √(100×0.5×0.5) = 5

Continuity Correction (The Secret Sauce!)

Since we’re turning counting numbers into a smooth curve, we add 0.5:

For P(45 ≤ X ≤ 55):

• Use 44.5 to 55.5

Calculate:

Z₁ = (44.5 - 50) ÷ 5 = -1.1
Z₂ = (55.5 - 50) ÷ 5 = 1.1

Look up: P(-1.1 < Z < 1.1) ≈ 72.9%

graph TD
    A["Binomial Problem"] --> B{Check: np ≥ 10?}
    B -->|Yes| C["Calculate μ = np"]
    B -->|No| D["Use Binomial Formula"]
    C --> E["Calculate σ = √np·1-p·"]
    E --> F["Apply Continuity Correction"]
    F --> G["Use Normal Z-table!"]

🎮 Quick Practice

Scenario: A factory makes bolts with mean length 10cm and SD 0.5cm.

1. Z-score: A bolt is 11cm. Z = (11-10)/0.5 = 2
2. Percentile: Z=2 → 97.72% are shorter
3. 68-95-99.7: 95% of bolts are between 9cm and 11cm

🚀 Key Takeaways

💡 Remember

The normal distribution is everywhere — heights, test scores, measurement errors, even the number of chips in your snack bag!

Once you master the Z-score, you unlock the power to analyze any normal distribution using just one simple table.

You’ve got this! 🌟