Module 4 Quick Reference Card

       KEY PROPERTIES: Bell-shaped • Symmetric • Mean = Median = Mode • Total area = 1
    

Essential Formulas

z-score: z = (x − μ) / σ

Converts to standard normal

Inverse: x = μ + zσ

Converts z-score to x-value

Notation

X ~ N(μ, σ) – normal distribution
Z ~ N(0, 1) – standard normal
z > 0 → above mean
z < 0 → below mean

Empirical Rule (68-95-99.7)

Range	% of Data
μ ± 1σ	~68%
μ ± 2σ	~95%
μ ± 3σ	~99.7%

Use for QUICK estimates without z-table!

Common z-values

Percentile	z-value	Area Left
50th	0.00	0.5000
90th	1.645	0.9000
95th	1.96	0.9500
97.5th	1.96	0.9750
99th	2.576	0.9900

Memorize these for faster work!

Checking Normality

Visual Methods

Histogram: Bell-shaped? Symmetric?
Q-Q Plot: Points on straight line?

Range Check

If normal, expect:

~68% in [x̄ − s, x̄ + s]
~95% in [x̄ − 2s, x̄ + 2s]

Finding Probabilities

Type 1: P(X < a)

Calculate z = (a − μ) / σ
Look up z in table → answer

Type 2: P(X > a)

Calculate z = (a − μ) / σ
Look up z → get P(Z < z)
Answer = 1 − P(Z < z)

Type 3: P(a < X < b)

z₁ = (a − μ) / σ, z₂ = (b − μ) / σ
Look up both z-values
Answer = P(Z < z₂) − P(Z < z₁)

Type 4: Finding Percentiles

Find probability inside table → z
x = μ + zσ

z-table Usage Guide

Standard z-table gives:

Area to the LEFT of z
This equals P(Z < z)
For z = 1.5 → 0.9332 means 93.32% below

Reading the table:

Row = first two digits (e.g., 1.2)
Column = second decimal (e.g., 0.05)
Example: z = 1.25 → row 1.2, col 0.05

Decision Tree

Problem Type?

Given x, find P? → x to z, use table
Given P, find x? → table to z, z to x
"Less than"? → Direct table lookup
"Greater than"? → 1 − table value
"Between"? → Subtract two table values
Percentile? → Inverse lookup

Common Mistakes

Forgetting to convert x to z first
Wrong formula: (μ − x)/σ instead of (x − μ)/σ
Not subtracting from 1 for P(X > a)
Assuming all data is normal
Sign errors: negative z = below mean
Using σ² instead of σ in formula

Quick Tips for Success

Always draw a picture and shade the area
Check: probabilities must be 0 to 1
Memorize Empirical Rule (68-95-99.7)
|z| > 2 is unusual (< 5%)
|z| > 3 is very unusual (< 0.3%)
For continuous: P(X < a) = P(X ≤ a)
Check normality before using methods
Practice reading z-table both ways

Example Walkthrough

Problem: IQ ~ N(100, 15). Find P(IQ > 120).
Step 1: z = (120 − 100) / 15 = 20/15 = 1.33
Step 2: Table: P(Z < 1.33) = 0.9082
Step 3: P(Z > 1.33) = 1 − 0.9082 = 0.0918
Answer: 9.18% of people have IQ > 120

Normal Distribution Quick Reference Card