Sample Notes: The Normal Distribution
A2 Level Mathematics – Topic 5.5: The Normal Distribution
Definition and Properties
- The normal distribution is a continuous probability distribution for a random variable.
- It has a characteristic bell-shaped curve, symmetrical about the mean.
- The distribution is described by two parameters:
- μ (mu): the mean
- σ (sigma): the standard deviation
- Standard notation: X ~ N(μ, σ²) where:
- X is the normally distributed variable
- μ is the expected value (mean)
- σ² is the variance
Characteristics of the Normal Curve
- Symmetrical about the vertical line x = μ
- Mean = Median = Mode
- Area under the curve = 1
- 68% of values lie within ±1σ of the mean
- 95% lie within ±2σ
- 99.7% lie within ±3σ
- The curve approaches but never touches the x-axis
- The curve is unimodal (single peak)
Standard Normal Distribution
- When μ = 0 and σ = 1, we get the standard normal distribution
- Denoted by Z ~ N(0, 1)
- Any normal variable X can be standardized to Z using:
Z = (X − μ) / σ
- This process is called standardization
- Used to find probabilities using standard normal tables
Using Normal Distribution Tables
- Tables typically show P(Z < z) for values of z
- To find P(Z > z), use: 1 − P(Z < z)
- To find P(a < Z < b), use: P(Z < b) − P(Z < a)
- For negative z-values: P(Z < −z) = 1 − P(Z < z) due to symmetry
Finding Probabilities for X ~ N(μ, σ²)
- Step 1: Convert the X value to Z using Z = (X − μ) / σ
- Step 2: Use standard normal tables to find the area under the curve
- Examples:
- P(X < x₁) → Convert to Z and lookup
- P(X > x₁) = 1 − P(X < x₁)
- P(x₁ < X < x₂) = P(X < x₂) − P(X < x₁)
Finding X Given a Probability
- Given a probability value p, we find Z = zₚ such that:
- P(Z < zₚ) = p
- Then use inverse standardization to find X:
- X = μ + zₚ * σ
- Useful for confidence intervals, percentiles, cut-offs
Sketching Normal Distribution Curves
- Mark the mean μ on the horizontal axis
- Show values at μ − σ, μ + σ, μ − 2σ, μ + 2σ, etc.
- Indicate the required area under the curve (e.g., shade P(X > 85))
- Label the axis with values and relevant probabilities
Conditions for Using Normal Distribution
- The variable must be continuous
- The data must be symmetrical and bell-shaped
- The mean and standard deviation must be known
- Large enough sample size to apply Central Limit Theorem (if from sample mean)
Approximating Binomial Distribution Using Normal Distribution
- Binomial: X ~ B(n, p)
- Normal approximation: X ~ N(np, npq)
- where q = 1 − p
- Conditions:
- np > 5 and nq > 5
- Apply continuity correction:
- To find P(X = x), use P(x − 0.5 < X < x + 0.5)
- P(X ≤ x) → P(X < x + 0.5)
- P(X ≥ x) → P(X > x − 0.5)
- Then standardize using Z = (X − np) / √(npq)
Examples of Continuity Correction
- P(X = 5): Use P(4.5 < X < 5.5)
- P(X < 6): Use P(X < 5.5)
- P(X > 6): Use P(X > 6.5)
- P(4 ≤ X ≤ 7): Use P(3.5 < X < 7.5)
Solving Probability Questions – Stepwise Method
- Identify the distribution (either X ~ N(μ, σ²) or binomial approximated as normal)
- Apply continuity correction if needed (only for binomial approx)
- Standardize using Z = (X − μ) / σ
- Use normal distribution tables to find the probability
- Subtract/add as needed for greater than or between values
- If asked to find a value given probability, reverse the Z formula
Key Formulae
- Z = (X − μ) / σ
- X = μ + Zσ
- Binomial mean = np
- Binomial variance = npq
- If X ~ N(μ, σ²), then:
- P(X < x₁) = area to the left
- P(X > x₁) = 1 − P(X < x₁)
- P(x₁ < X < x₂) = P(X < x₂) − P(X < x₁)
Common Problem Types
- Find P(X < x)
- Find P(X > x)
- Find P(x₁ < X < x₂)
- Find x such that P(X < x) = p
- Use Z to calculate percentiles and boundaries
- Approximate binomial probabilities with normal and correct continuity
Assumptions and Limitations
- Normal distribution assumes symmetry — not always true in real data
- Extreme values in skewed distributions will give incorrect probabilities
- For binomial approximation:
- Must meet np > 5 and nq > 5
- Use continuity correction for discrete to continuous transition
Special Notes on Standardization
- Always use 4 decimal places for Z values
- Don’t forget to reverse standardization when finding raw scores
- Practice standardizing both directions (X to Z and Z to X)
Graphical Interpretation
- Always shade areas correctly
- Label mean and σ multiples clearly
- Sketching helps visualize tail areas or between intervals
Real-World Applications
- Standardized test scores (e.g., SAT, IQ)
- Measurement errors in physics
- Manufacturing quality control
- Financial modeling (returns, risks)
- Medical statistics (cholesterol, blood pressure distributions)