Module 4 Study Guide
The Normal Distribution
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1. Introduction to the Normal Distribution
Normal Distribution: A continuous probability distribution that is symmetric, bell-shaped, and completely described by two parameters: mean (μ) and standard deviation (σ). Also called the Gaussian distribution or bell curve.
Key Properties
1. Shape and Symmetry
- Bell-shaped: Single peak at the center, tapering off on both sides
- Symmetric: Left half is a mirror image of the right half
- Unimodal: Has exactly one peak (the mean)
- Mean = Median = Mode: All three measures of center are equal and located at the peak
2. Total Area
- The total area under the curve = 1 (or 100%)
- This represents the probability that X takes on some value
- Area under any portion of the curve = probability for that range
3. Parameters
- Mean (μ): Determines the center location of the distribution
- Standard deviation (σ): Determines the spread or width of the distribution
- Notation: X ~ N(μ, σ) means "X is normally distributed with mean μ and standard deviation σ"
4. Tail Behavior
- The curve extends infinitely in both directions (never touches the x-axis)
- The tails approach but never reach zero
- This means extreme values are possible but extremely unlikely
The Empirical Rule (68-95-99.7 Rule)
The Empirical Rule provides quick approximations for probabilities in a normal distribution:
| Range | Percentage of Data | Meaning |
|---|---|---|
| μ ± 1σ | ~68% | About 68% of data falls within 1 standard deviation of the mean |
| μ ± 2σ | ~95% | About 95% of data falls within 2 standard deviations of the mean |
| μ ± 3σ | ~99.7% | About 99.7% of data falls within 3 standard deviations of the mean |
Example: If SAT scores are N(1000, 100), then:
- 68% of scores fall between 900 and 1100 (μ ± 1σ)
- 95% of scores fall between 800 and 1200 (μ ± 2σ)
- 99.7% of scores fall between 700 and 1300 (μ ± 3σ)
Common Examples of Normal Distributions
- Heights of adults
- IQ scores
- Measurement errors
- Test scores (often approximately normal)
- Birth weights
- Blood pressure readings
2. Standard Normal Distribution & z-scores
The Standard Normal Distribution
Standard Normal Distribution: A special normal distribution with:
- Mean (μ) = 0
- Standard deviation (σ) = 1
Notation: Z ~ N(0, 1)
Why is this useful? By converting any normal distribution to the standard normal, we only need ONE probability table (the z-table) instead of separate tables for every possible combination of μ and σ!
The z-score Formula
z = (x − μ) / σ
Where: x = data value, μ = mean, σ = standard deviation
z-score (standard score): The number of standard deviations a data value is from the mean.
- Positive z-score: Value is above the mean
- Negative z-score: Value is below the mean
- z = 0: Value equals the mean
Interpreting z-scores
| z-score | Interpretation | Relative Position |
|---|---|---|
| z = 2.5 | 2.5 standard deviations above the mean | Unusually high (top 1%) |
| z = 1.0 | 1 standard deviation above the mean | Above average (top 16%) |
| z = 0 | Exactly at the mean | Average (50th percentile) |
| z = −1.0 | 1 standard deviation below the mean | Below average (bottom 16%) |
| z = −2.5 | 2.5 standard deviations below the mean | Unusually low (bottom 1%) |
Rule of Thumb: Values with |z| > 2 are considered unusual (less than 5% probability), and values with |z| > 3 are very unusual (less than 0.3% probability).
Example: On an exam with μ = 75 and σ = 10, a student scores x = 90.
- z = (90 − 75) / 10 = 15 / 10 = 1.5
- Interpretation: The student scored 1.5 standard deviations above the mean
- This is a good score, approximately in the top 7% of the class
Converting Back: Finding x from z
x = μ + zσ
Use this when: You know the z-score and need to find the original data value
Example: IQ scores are N(100, 15). What IQ corresponds to z = 2?
- x = μ + zσ = 100 + 2(15) = 100 + 30 = 130
- An IQ of 130 is 2 standard deviations above the mean
3. Finding Probabilities with the Normal Distribution
Area Under the Curve = Probability
For a continuous distribution:
- The total area under the curve = 1 (100%)
- The area to the left of a value = P(X < a)
- The area to the right of a value = P(X > a)
- The area between two values = P(a < X < b)
Important Note: For continuous distributions, P(X < a) = P(X ≤ a) because the probability of getting exactly one specific value is 0.
Using the Standard Normal (z) Table
Standard Normal Table (z-table): Provides the area to the LEFT of a given z-score (cumulative probability).
Steps to Find Probabilities
Type 1: P(X < a) – Probability Less Than
- Convert x to z-score: z = (x − μ) / σ
- Look up z in the standard normal table
- The table value IS the answer (area to the left)
Example: X ~ N(50, 10). Find P(X < 60).
- Step 1: z = (60 − 50) / 10 = 1.0
- Step 2: Look up z = 1.0 in table → 0.8413
- Answer: P(X < 60) = 0.8413 or 84.13%
Type 2: P(X > a) – Probability Greater Than
- Convert x to z-score: z = (x − μ) / σ
- Look up z in the table to get P(Z < z)
- Subtract from 1: P(Z > z) = 1 − P(Z < z)
Example: X ~ N(50, 10). Find P(X > 60).
- Step 1: z = (60 − 50) / 10 = 1.0
- Step 2: Table gives P(Z < 1.0) = 0.8413
- Step 3: P(Z > 1.0) = 1 − 0.8413 = 0.1587
- Answer: P(X > 60) = 0.1587 or 15.87%
Type 3: P(a < X < b) – Probability Between Two Values
- Convert both x-values to z-scores
- Find P(Z < z₂) and P(Z < z₁) from the table
- Subtract: P(z₁ < Z < z₂) = P(Z < z₂) − P(Z < z₁)
Example: X ~ N(50, 10). Find P(45 < X < 60).
- Step 1: z₁ = (45 − 50) / 10 = −0.5, z₂ = (60 − 50) / 10 = 1.0
- Step 2: P(Z < 1.0) = 0.8413, P(Z < −0.5) = 0.3085
- Step 3: P(−0.5 < Z < 1.0) = 0.8413 − 0.3085 = 0.5328
- Answer: P(45 < X < 60) = 0.5328 or 53.28%
Finding Percentiles (Inverse Normal)
Percentile: The value below which a given percentage of observations fall.
Steps to Find a Percentile
- Identify the desired cumulative probability (area to the left)
- Look up this probability INSIDE the z-table to find the corresponding z-score
- Convert z back to x: x = μ + zσ
Example: X ~ N(100, 15). Find the 90th percentile.
- Step 1: We want P(X < ?) = 0.90
- Step 2: Look for 0.90 inside the table → z ≈ 1.28
- Step 3: x = 100 + 1.28(15) = 100 + 19.2 = 119.2
- Answer: The 90th percentile is approximately 119.2
Common z-values to Memorize
| Confidence Level | Area in Middle | z-score | Usage |
|---|---|---|---|
| 90% | 0.90 | z = 1.645 | 90th percentile = μ + 1.645σ |
| 95% | 0.95 | z = 1.96 | 95th percentile = μ + 1.96σ |
| 99% | 0.99 | z = 2.576 | 99th percentile = μ + 2.576σ |
4. Applications of the Normal Distribution
Checking for Normality
Why check? Many statistical methods assume data is normally distributed. Before using these methods, we should verify this assumption.
Visual Methods
1. Histogram
- Good signs: Bell-shaped, symmetric, single peak near center
- Warning signs: Skewed, multiple peaks, gaps, extreme outliers
2. Normal Probability Plot (Q-Q Plot)
- Plots observed data against expected normal values
- Good sign: Points fall close to a straight diagonal line
- Warning signs: Systematic curvature, points far from the line
Numerical Method: Range Check
If data is normal, we expect:
- About 68% of values in [x̄ − s, x̄ + s]
- About 95% of values in [x̄ − 2s, x̄ + 2s]
- About 99.7% of values in [x̄ − 3s, x̄ + 3s]
Where x̄ = sample mean, s = sample standard deviation
When NOT to Use the Normal Distribution
Do NOT use normal distribution methods if:
- Data is strongly skewed (not symmetric)
- Data has multiple peaks (bimodal or multimodal)
- Data has extreme outliers
- Sample size is very small (n < 30) and distribution is clearly non-normal
- Data is discrete with very few possible values
Normal Approximation to Other Distributions
Central Limit Theorem (Preview): Under certain conditions, even if the original data is NOT normal, the distribution of sample means becomes approximately normal as sample size increases.
Practical Applications
1. Quality Control
- Manufacturing tolerances (products within specifications)
- Defect rates
- Process control charts
2. Standardized Testing
- SAT, ACT, GRE scores often designed to be normal
- Percentile rankings
- Comparing scores across different tests
3. Scientific Research
- Measurement error
- Natural variation in biological measurements
- Hypothesis testing foundations
4. Finance
- Stock returns (approximately normal in short term)
- Risk assessment
- Option pricing models
5. Quick Reference: All Formulas
z-score Formula
z = (x − μ) / σ
Converts any normal distribution to standard normal
Inverse z-score Formula
x = μ + zσ
Converts z-score back to original value
Empirical Rule (68-95-99.7 Rule)
- 68% of data: μ ± 1σ
- 95% of data: μ ± 2σ
- 99.7% of data: μ ± 3σ
Probability Calculations
P(X < a): Convert to z, look up in table
P(X > a): 1 − P(X < a)
P(a < X < b): P(X < b) − P(X < a)
Key Properties
- Total area under curve = 1
- Mean = Median = Mode (all at center)
- Symmetric around mean
- P(X < a) = P(X ≤ a) for continuous distributions
Tips for Success
Common Mistakes to Avoid
- Forgetting to convert to z-score first before using the table
- Using the wrong formula: z = (x − μ) / σ, NOT (μ − x) / σ
- Reading the z-table incorrectly – make sure you understand whether your table gives left-tail or right-tail probabilities
- Forgetting to subtract from 1 when finding P(X > a)
- Assuming all data is normal – always check first!
- Confusing percentile with percentage – the 90th percentile means 90% of data is below it
- Sign errors with z-scores – negative z means below the mean, positive z means above
Study Strategies
- Master the z-score formula – it's the foundation for everything
- Practice reading the z-table in both directions (z → probability and probability → z)
- Draw pictures! Sketch the normal curve and shade the area you're looking for
- Memorize the Empirical Rule (68-95-99.7) for quick estimates
- Memorize common z-values: 1.645 (90%), 1.96 (95%), 2.576 (99%)
- Check your answers: Probabilities must be between 0 and 1
- Interpret in context: Always relate back to the real-world meaning
Quick Decision Tree
| Question Type | What to Do |
|---|---|
| Given x, find probability | x → z → look up in table → answer |
| Given probability, find x | Look up prob in table → z → x = μ + zσ |
| Find P(X < a) | Calculate z, look up in table |
| Find P(X > a) | Calculate z, look up in table, subtract from 1 |
| Find P(a < X < b) | Two z-scores, subtract probabilities |
| Find percentile | Probability → z (from table) → x = μ + zσ |
Module 4: The Normal Distribution
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