MathIsimple

Normal Distributions & Sampling

Master the normal distribution, z-scores, Central Limit Theorem, and sampling distributions. Learn to calculate probabilities, construct confidence intervals, and understand statistical inference.

12th Grade
Statistics
~90 min
🎮 Interactive Activity: Z-Score Calculator

Calculate the z-score for a given value!

Given:

x = 75
μ = 70
σ = 5

Calculate z:

🎮 Interactive Activity: Central Limit Theorem - Standard Error

Calculate the standard error of the sample mean!

Given:

Population mean μ = 50
Population std dev σ = 12
Sample size n = 36

Calculate standard error:

1. Normal Distribution Fundamentals

The Bell Curve

The normal distribution is a symmetric, bell-shaped distribution characterized by its mean (μ) and standard deviation (σ). Many natural phenomena follow this distribution.

Probability Density Function

Formula: f(x)=frac1sigmasqrt2pie−frac12left(fracx−musigmaright)2f(x) = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{1}{2}\\left(\\frac{x-\\mu}{\\sigma}\\right)^2}

Parameters: μ (mean) determines center, σ (standard deviation) determines spread

Properties: Symmetric about μ, total area under curve = 1

Notation: X ~ N(μ, σ²) means X follows normal distribution

Example 1: Height Distribution

Scenario: Adult heights are normally distributed with μ = 70 inches, σ = 3 inches

Interpretation: Most people are around 70 inches, with fewer people at extreme heights

Probability: P(68 ≤ X ≤ 72) ≈ 0.50 (about 50% of people are within 2 inches of mean)

Example 2: 68-95-99.7 Rule

Rule: For any normal distribution:

• ~68% of values within 1σ of mean (μ ± σ)

• ~95% of values within 2σ of mean (μ ± 2σ)

• ~99.7% of values within 3σ of mean (μ ± 3σ)

Application: Quick probability estimates without calculations

2. Z-Scores and Standardization
3. Central Limit Theorem
4. t-Distribution
5. Confidence Intervals
6. Confidence Intervals for Proportions
7. Real-World Applications
Frequently Asked Questions

Practice Time!

Practice Quiz
10
Questions
0
Correct
0%
Score
1
What is a z-score?
○
2
If X ~ N(70, 9), what is the z-score for x = 76?
○
3
What does the Central Limit Theorem state?
○
4
What is the standard error of the sample mean?
○
5
For a normal distribution, approximately what percentage of values fall within 1 standard deviation of the mean?
○
6
When should you use a t-distribution instead of a normal distribution?
○
7
What is the mean of the standard normal distribution?
○
8
If a 95% confidence interval for μ is (72, 88), what can we conclude?
○
9
What is the relationship between sample size and standard error?
○
10
For a normal distribution with mean 50 and standard deviation 10, what is P(40 ≤ X ≤ 60)?
○