Normal Distribution Basics

For Elementary Students

What is Normal Distribution?

Normal distribution is a pattern that shows up EVERYWHERE in nature! It makes a shape like a bell — that's why it's called a "bell curve."

Think about it like this: Imagine you measure the heights of all the students in your school. Most kids will be about average height, some will be taller, and some will be shorter. When you graph it, it makes a bell shape!

The Bell Shape

        Number
        of
      People

        *
       ***
      *****
     *******
    *********
   ***********
  *************
───────┬───────────→ Height
     Average

Key ideas:

The middle (top of bell) = most common (average)

The sides = less common (taller or shorter than average)

The edges = very rare (extremely tall or extremely short)

Real-Life Examples

Heights:

• Most people are average height
• Few people are very tall or very short
• Makes a bell curve!

Test scores:

• Most students score in the middle (70s-80s)
• Few students score very low (30s) or very high (100)
• Bell curve!

Shoe sizes:

• Most adults wear sizes 7-10
• Few wear size 5 or size 15
• Bell curve!

The "Middle" is Most Common

When something follows a bell curve:

MOST things are in the middle (average)

SOME things are a little above or below average

VERY FEW things are way above or way below

Example: Basketball Free Throws

Let's say students practice free throws. Most make about 5 out of 10.

Students who make:
- 5 shots: LOTS of students (middle of bell)
- 4 or 6 shots: Some students
- 3 or 7 shots: Few students
- 1 or 9 shots: Very few students (edges of bell)

The 68-95-99.7 Rule (The Easy Way)

This is a special pattern for bell curves:

About 68% (most) of the data is close to average

About 95% (almost all) is pretty close to average

About 99.7% (basically everyone) is within a wider range

Think of it like this:

        68% here
      |-----------|

        *
       ***
      *****
     *******
    *********
   ***********
  *************
─────┴─────┴─────→
  Low  Avg  High

Example: Test Scores

Average score: 75

68% of students score between 67 and 83 (close to 75)

95% of students score between 59 and 91 (pretty close to 75)

99.7% of students score between 51 and 99 (almost everyone!)

This means scoring below 51 or above 99 is VERY rare!

What's "Normal"?

When we say something is normally distributed, we mean:

• Most things are average
• It makes a bell curve
• Very high or very low is rare

Not everything is normally distributed!

Examples of things that ARE:

• Heights of people
• Test scores (usually)
• Measurement errors

Examples of things that are NOT:

• Age (people are all different ages)
• Money people have (few people have LOTS)

Why Does This Matter?

Predictions: If we know something follows a bell curve, we can predict what's normal and what's unusual!

Example: If average height is 5 feet, and someone is 7 feet tall, that's unusual (on the edge of the bell curve)!

Understanding "Average"

In a normal distribution:

Average = Most common

If 100 students take a test and it's normally distributed:

• About 34 students score just below average
• About 34 students score just above average
• About 16 students score lower
• About 16 students score higher

For Junior High Students

Formal Definition

Normal distribution is a symmetric, bell-shaped probability distribution described by two parameters:

μ (mu): The mean (average) — center of the distribution

σ (sigma): The standard deviation — measures spread

Notation: N(μ, σ)

Example: N(100, 15) means mean = 100, standard deviation = 15

Properties of Normal Curve

Shape: Bell-shaped and symmetric

Center: Mean (μ) is at the peak

Symmetry: Left and right sides are mirror images

Mean = Median = Mode: All three measures of center are equal

Total area under curve: 100% (represents all data)

Tails: Extend infinitely but approach (never touch) the x-axis

Inflection points: Where curve changes from concave down to concave up, located at μ ± σ

Standard Deviation Controls Spread

Larger σ: Wider, flatter bell (more variation)

Smaller σ: Narrower, taller bell (less variation)

Example:

N`(50, 5)`:  Narrow bell
N`(50, 15)`: Wide bell

Both centered at 50, different spreads

The 68-95-99.7 Rule (Empirical Rule)

For ANY normal distribution:

68% of data falls within μ ± 1σ (one standard deviation from mean)

95% of data falls within μ ± 2σ (two standard deviations from mean)

99.7% of data falls within μ ± 3σ (three standard deviations from mean)

Also called: Empirical Rule or Three-Sigma Rule

Example 1: Heights

Adult male heights: N(170 cm, 10 cm)

μ = 170 cm, σ = 10 cm

68% of men between 160 cm and 180 cm (170 ± 10)

95% of men between 150 cm and 190 cm (170 ± 20)

99.7% of men between 140 cm and 200 cm (170 ± 30)

Example 2: SAT Scores

SAT: N(1000, 200)

μ = 1000, σ = 200

1σ interval: 800 to 1200 (68% of students)

2σ interval: 600 to 1400 (95% of students)

3σ interval: 400 to 1600 (99.7% of students)

Calculating Percentages Using Symmetry

By symmetry of normal curve:

50% below mean, 50% above mean

Example 1: What percent score above the mean?

Answer: 50%

Example 2: What percent between μ and μ + 1σ?

68% within ±1σ
By symmetry: 68% ÷ 2 = 34% on each side
Answer: 34%

Example 3: What percent above μ + 1σ?

68% within ±1σ
Outside ±1σ: 100% - 68% = 32%
Above μ + 1σ: 32% ÷ 2 = 16%
Answer: 16%

Example 4: What percent between μ + 1σ and μ + 2σ?

Within ±2σ: 95%
Within ±1σ: 68%
Between 1σ and 2σ on one side: (95% - 68%) ÷ 2 = 13.5%
Answer: 13.5%

Z-Scores (Standard Scores)

Z-score: Measures how many standard deviations a value is from the mean

Formula: z = (x - μ)/σ

Where:

• x = data value
• μ = mean
• σ = standard deviation

Interpretation:

z = 0: Value equals the mean

z > 0: Value is above the mean

z < 0: Value is below the mean

|z|: Distance from mean in standard deviation units

Example 1: Calculate Z-Score

Heights: μ = 170 cm, σ = 10 cm

Person is 185 cm tall. Find z-score.

z = (185 - 170)/10
z = 15/10
z = 1.5

Interpretation: This person is 1.5 standard deviations above average height.

Example 2: Negative Z-Score

Test scores: μ = 75, σ = 8

Student scores 63. Find z-score.

z = (63 - 75)/8
z = -12/8
z = -1.5

Interpretation: This score is 1.5 standard deviations below the mean.

Example 3: Converting from Z to X

Given: μ = 70 kg, σ = 12 kg, z = -2

Find the actual weight (x):

Formula: x = μ + z·σ

x = 70 + (-2)(12)
x = 70 - 24
x = 46 kg

Using Z-Scores to Compare

Z-scores allow comparison across different distributions

Example: Which Test Performance is Better?

Math test: Score = 85, μ = 75, σ = 5

z_math = (85 - 75)/5 = 10/5 = 2

English test: Score = 90, μ = 82, σ = 6

z_english = (90 - 82)/6 = 8/6 ≈ 1.33

Conclusion: Math score is better relative to class (higher z-score means better relative performance)

Standard Normal Distribution

Standard normal distribution: N(0, 1)

• Mean = 0
• Standard deviation = 1

Any normal distribution can be standardized using z-scores

Purpose: Simplifies calculations and allows use of standard normal tables

Example: Standardizing

Original: N(100, 15)

Convert x = 115 to standard normal:

z = (115 - 100)/15 = 1

Now we use standard normal N(0, 1) with z = 1

Identifying Unusual Values

Common rule: A value is unusual if |z| > 2

Meaning: More than 2 standard deviations from mean

Reasoning: About 95% of data within ±2σ, so only ~5% outside

Example: Is a Value Unusual?

Heights: μ = 170 cm, σ = 10 cm

Is 195 cm unusual?

z = (195 - 170)/10 = 2.5
|2.5| > 2, so YES, unusual

Is 175 cm unusual?

z = (175 - 170)/10 = 0.5
|0.5| < 2, so NO, not unusual

Percentiles and Normal Distribution

Percentile: Percentage of data below a value

Using 68-95-99.7 rule:

• 50th percentile: μ (median)
• 84th percentile: μ + 1σ (50% + 34%)
• 16th percentile: μ - 1σ (50% - 34%)
• 97.5th percentile: μ + 2σ (50% + 47.5%)
• 2.5th percentile: μ - 2σ (50% - 47.5%)

Example: Find Percentile Rank

Test score is 1σ above mean

Below mean: 50%
Between mean and 1σ: 34%
Total: 50% + 34% = 84%

Answer: 84th percentile

When is Data Normally Distributed?

Data tends to be normal when:

• Natural biological measurements (heights, weights)
• Test scores from large populations
• Measurement errors
• Many small independent factors affect outcome

Data is NOT normal when:

• Highly skewed (income, house prices)
• Discrete counts with small values
• Data with hard boundaries (percentages bounded by 0-100)
• Bimodal (two peaks)

Real-World Applications

Education: SAT, ACT, IQ tests designed to be normally distributed

Medicine: Blood pressure, cholesterol levels, bone density

Manufacturing: Quality control — product dimensions, weights

Finance: Stock returns (approximately normal)

Science: Measurement errors, experimental data

Example: Quality Control

Bottle filling machine: μ = 500 mL, σ = 5 mL

Question: What percent of bottles contain less than 490 mL?

Analysis:

490 = μ - 2σ
Below μ - 2σ: Half of the 5% outside ±2σ
Answer: 2.5% underfilled

Example: College Admissions

University accepts top 16% of applicants

SAT: μ = 1000, σ = 200

What score needed?

Top 16% = above 84th percentile ≈ μ + 1σ

Cutoff = 1000 + 200 = 1200

Need approximately 1200 SAT score

Common Errors

Error 1: Assuming all data is normal

❌ Income is normally distributed
✓ Income is right-skewed (few very high incomes)

Error 2: Miscalculating z-scores

❌ z = x - μ (forgot to divide by σ)
✓ z = (x - μ)/σ

Error 3: Confusing percentages

❌ 68% above mean
✓ 68% within ±1σ (34% on each side of mean)

Error 4: Using rule for non-normal data

❌ Apply 68-95-99.7 to skewed data
✓ Only applies to normal distributions

Tips for Success

Tip 1: Always check if data is approximately normal before using normal distribution methods

Tip 2: Remember z-score formula: z = (x - μ)/σ

Tip 3: Use symmetry: 50% below mean, 50% above

Tip 4: 68-95-99.7 rule is your friend for quick estimates

Tip 5: |z| > 2 indicates unusual values

Tip 6: Sketch the bell curve to visualize problems

Tip 7: Standard normal has μ = 0, σ = 1

Summary

Normal distribution: Bell-shaped, symmetric distribution

Notation: N(μ, σ)

Properties:

• Symmetric around mean
• Mean = median = mode
• Defined by μ (center) and σ (spread)

68-95-99.7 Rule:

• 68% within μ ± 1σ
• 95% within μ ± 2σ
• 99.7% within μ ± 3σ

Z-score: z = (x - μ)/σ

• Measures distance from mean in standard deviations
• Allows comparison across distributions

Standard normal: N(0, 1), obtained by standardizing

Applications: Test scores, biological measurements, quality control, many natural phenomena

Practice