Normal Distributions

Introduction

The normal distribution (also called Gaussian distribution or bell curve) is the most important probability distribution in statistics. Many natural phenomena approximately follow a normal distribution, and it forms the foundation for much of statistical inference.

Characteristics of Normal Distributions

Shape Properties

1. Bell-shaped curve:

• Symmetric around the center
• Single peak at the mean
• Tails extend infinitely in both directions (approaching but never touching the x-axis)

2. Symmetric:

• Left side mirrors right side
• Mean = Median = Mode
• If folded at center, both halves match perfectly

3. Unimodal:

• Single peak (at the mean)
• Highest point at center
• Decreases smoothly on both sides

4. Asymptotic:

• Tails get closer and closer to x-axis
• Never actually reach zero
• Theoretically extends to $-\infty$ and $+\infty$

The Normal Curve Equation

Probability density function:

$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$

Don't memorize this! Just know:

• Defined by two parameters: $\mu$ (mean) and $\sigma$ (standard deviation)
• Shape determined entirely by $\mu$ and $\sigma$

Parameters: Mean and Standard Deviation

Mean ($\mu$)

Controls location:

• Center of distribution
• Peak of curve
• Balance point

Changing $\mu$:

• Shifts distribution left or right
• Doesn't change shape
• Doesn't change spread

Example:

• Distribution A: $\mu = 50$, centered at 50
• Distribution B: $\mu = 70$, centered at 70
• B is shifted 20 units right from A

Standard Deviation ($\sigma$)

Controls spread:

• How spread out distribution is
• Width of bell curve
• Distance from mean to inflection points

Changing $\sigma$:

• Larger $\sigma$ → wider, flatter curve
• Smaller $\sigma$ → narrower, taller curve
• Doesn't change center
• Total area under curve stays 1.0

Example:

• Distribution A: $\sigma = 5$, narrow and tall
• Distribution B: $\sigma = 15$, wide and flat
• Both centered at same $\mu$, but B more spread out

The Empirical Rule (68-95-99.7 Rule)

For normal distributions:

68% of data within 1 standard deviation of mean
$\mu - 1\sigma$ to $\mu + 1\sigma$

95% of data within 2 standard deviations of mean
$\mu - 2\sigma$ to $\mu + 2\sigma$

99.7% of data within 3 standard deviations of mean
$\mu - 3\sigma$ to $\mu + 3\sigma$

Example Application

IQ scores: $\mu = 100$, $\sigma = 15$

68% of people have IQ between: $100 - 15 = 85$ and $100 + 15 = 115$

95% of people have IQ between: $100 - 30 = 70$ and $100 + 30 = 130$

99.7% of people have IQ between: $100 - 45 = 55$ and $100 + 45 = 145$

Using the Empirical Rule

Quick mental calculations:

Example: Heights of adult males: $\mu = 70$ inches, $\sigma = 3$ inches

Q: What percentage between 67 and 73 inches?
A: 67 to 73 is $\mu \pm 1\sigma$ → 68%

Q: What percentage above 76 inches?
A: 76 is $\mu + 2\sigma$, so 95% are between 64 and 76
Above 76 = (100% - 95%) / 2 = 2.5%

Q: What percentage below 64 inches?
A: 64 is $\mu - 2\sigma$
Below 64 = (100% - 95%) / 2 = 2.5%

The Standard Normal Distribution (Z-distribution)

Definition

Standard Normal: Normal distribution with $\mu = 0$ and $\sigma = 1$

Denoted: $N(0, 1)$ or Z-distribution

Why it matters:

• Reference distribution
• All normal distributions can be standardized to this
• Tables and calculators use standard normal

Z-Scores

Z-score (standardized score): Number of standard deviations from the mean

Formula: $z = \frac{x - \mu}{\sigma}$

Where:

• $x$ = observed value
• $\mu$ = mean
• $\sigma$ = standard deviation

Interpretation:

• $z = 0$: At the mean
• $z = 1$: One SD above mean
• $z = -1$: One SD below mean
• $z = 2.5$: 2.5 SD above mean

Calculating Z-Scores

Example: Test scores with $\mu = 75$, $\sigma = 8$

Score of 83: $z = \frac{83 - 75}{8} = \frac{8}{8} = 1$

Score is 1 SD above mean

Score of 67: $z = \frac{67 - 75}{8} = \frac{-8}{8} = -1$

Score is 1 SD below mean

Score of 91: $z = \frac{91 - 75}{8} = \frac{16}{8} = 2$

Score is 2 SD above mean

Using Z-Scores

Purposes:

1. Standardize different distributions for comparison
2. Find probabilities using standard normal table
3. Identify unusual values (typically |z| > 2 or 3)
4. Compare across different scales

Example comparison:

Student A: Math score 85 (class $\mu = 75$, $\sigma = 8$)
$z = \frac{85-75}{8} = 1.25$

Student B: English score 88 (class $\mu = 80$, $\sigma = 5$)
$z = \frac{88-80}{5} = 1.6$

Student B performed better relative to their class (higher z-score)!

Finding Areas Under the Normal Curve

Methods

1. Empirical Rule (for z = ±1, ±2, ±3)

2. Standard Normal Table (z-table)

• Gives area to LEFT of z-score
• Also called cumulative probability

3. Calculator

• normalcdf function
• More accurate, easier

Using the Table

Area to the left of z:

• Look up z in table directly
• Example: z = 1.23 → area = 0.8907
• Meaning: 89.07% of data below z = 1.23

Area to the right of z:

• Area to right = 1 - area to left
• Example: z = 1.23 → area to left = 0.8907
• Area to right = 1 - 0.8907 = 0.1093 (10.93%)

Area between two z-scores:

• Find area to left of each
• Subtract smaller from larger
• Example: Between z = -1 and z = 1
  • Area left of 1: 0.8413
  • Area left of -1: 0.1587
  • Between: 0.8413 - 0.1587 = 0.6826 (68.26%)

Calculator Method

TI-83/84:

normalcdf(lower, upper, mean, SD)

Examples:

Area between 65 and 75 ($\mu = 70$, $\sigma = 5$):
normalcdf(65, 75, 70, 5) → 0.6827

Area above 80:
normalcdf(80, 999999, 70, 5) → 0.0228

Area below 60:
normalcdf(-999999, 60, 70, 5) → 0.0228

Finding Values from Areas (Inverse Normal)

The Inverse Problem

Given: Probability (area)
Find: Corresponding x-value or z-score

Example: Find the score such that 90% of students score below it

Calculator Method

TI-83/84:

invNorm(area to left, mean, SD)

Examples:

90th percentile ($\mu = 70$, $\sigma = 5$):
invNorm(0.90, 70, 5) → 76.4

Meaning: 90% score below 76.4

25th percentile (Q1):
invNorm(0.25, 70, 5) → 66.6

75th percentile (Q3):
invNorm(0.75, 70, 5) → 73.4

Assessing Normality

Why It Matters

Many statistical methods assume normality. We need to check if data is approximately normal before applying these methods.

Methods to Assess Normality

1. Histogram/Dotplot:

• Look for bell shape
• Check for symmetry
• Quick visual check

2. Normal Probability Plot (Q-Q Plot):

• Plot observed values vs. expected normal values
• If roughly linear → approximately normal
• If curved or non-linear → not normal

3. Numerical Checks:

• Mean ≈ Median (symmetry)
• Few outliers by 1.5 × IQR rule
• Most data within $\mu \pm 2\sigma$

What to Look For

Approximately normal: ✓ Bell-shaped histogram
✓ Linear normal probability plot
✓ Mean ≈ Median
✓ About 68% within 1 SD, 95% within 2 SD

Not normal: ❌ Skewed histogram
❌ Curved normal probability plot
❌ Mean ≠ Median
❌ Many outliers
❌ Gaps or multiple peaks

Common Mistakes

❌ Assuming all data is normal
Many distributions are NOT normal!

❌ Confusing z-scores with original values
z-scores are standardized, no units

❌ Using empirical rule for non-normal data
Only valid for normal distributions

❌ Forgetting to standardize before using table
Must convert to z-scores first

❌ Reading wrong side of table
Most tables give area to LEFT

❌ Not checking normality assumption
Methods based on normality won't work if data isn't normal

Quick Reference

Normal Distribution:

• Parameters: $\mu$ (mean), $\sigma$ (SD)
• Notation: $N(\mu, \sigma)$
• Properties: Symmetric, bell-shaped, unimodal

Empirical Rule (68-95-99.7):

• 68% within $\mu \pm 1\sigma$
• 95% within $\mu \pm 2\sigma$
• 99.7% within $\mu \pm 3\sigma$

Z-Score:

• Formula: $z = \frac{x - \mu}{\sigma}$
• Interpretation: # of SDs from mean
• Standard normal: $\mu = 0$, $\sigma = 1$

Calculator Commands:

• normalcdf(lower, upper, μ, σ) for area/probability
• invNorm(area, μ, σ) for x-value

Assessing Normality:

• Histogram: bell-shaped?
• Normal plot: linear?
• Mean ≈ Median?

Remember: The normal distribution is powerful but not universal. Always check if the normality assumption is reasonable before using methods that require it!