WebLab STEM

Mathematics · 10. Normal distribution and statistical inference

Random variables, Gaussian normal distribution and statistical inference

Investigate probability density functions, the 68-95-99.7 rule, Z-score standardisation and predictive data science.

40 minutesContent version: 2.1
Start the interactive pathwayOpen the virtual experiment

Driving question

Why do countless independent natural phenomena—from heights to test scores—converge to the symmetric Gaussian bell curve?

Curriculum coverage

Vietnam 2018 Mathematics 12 (Probability & Statistics); Cambridge AS/A Level 9709 S1; AP Statistics.

Model scope and limitations

Focuses on univariate normal distributions and the central limit theorem; omits high-dimensional multivariate GLMs.

Learning objectives

  • Distinguish discrete and continuous random variables; identify probability density functions with unit area.
  • Apply symmetry and the 68-95-99.7 empirical rule for normal distribution N(μ, σ²).
  • Perform Z-score transformation Z = (X − μ)/σ and evaluate standard normal probabilities N(0, 1).

Prerequisite knowledge

  • Mean (expected value μ), variance σ², and standard deviation σ.
  • Definite integration as the geometric area under a continuous curve.

Core knowledge

Gaussian bell curve

The bell curve f(x) is symmetric about mean μ, peaking at x = μ with inflection points at μ ± σ. Total area beneath the curve equals 1.

Empirical 68-95-99.7 rule

Approximately 68.3% of data lies within 1σ of the mean, 95.4% within 2σ, and 99.7% within 3σ.

Z-score standardisation

Transforming Z = (X − μ)/σ maps any N(μ, σ²) onto standard normal N(0, 1), enabling standardized table lookups.

Worked example

Exam scores follow N(70, 10²). Proportion scoring above 85: Z = (85 − 70)/10 = 1.5. From table, Φ(1.5) = 0.9332. Thus P(X > 85) = 1 − 0.9332 = 0.0668 (6.68%).

Misconceptions to avoid

  • For continuous random variables, exact point probability P(X = c) is identically 0; probability is only meaningful across an interval [a, b].
  • Standard deviation σ measures spread: larger σ widens and flattens the curve, smaller σ sharpens the peak, but total area remains 1.

Virtual experiment procedure: Galton board and normal distribution

  1. Run the Galton board simulator, dropping a large batch of balls (N = 1000) through staggered pegs.
  2. Observe accumulation in collection bins and compare histogram columns to the theoretical continuous Gaussian envelope.
  3. Alter standard deviation or number of rows n to examine variance scaling and distribution dispersion.

Safety and cautions

  • Mathematical simulation is fully safe; note sample privacy and PRNG statistical pseudorandomness in practical data science.

Evidence to collect

  • Grouped frequency table with computed sample mean and sample standard deviation s.
  • Shaded distribution plot verifying empirical areas across ±1σ and ±2σ against theoretical 68%-95% benchmarks.
Open the virtual experiment
📋 Embed this simulation in online lectures (For Teachers & LMS)

Copy the iframe code below to embed this interactive simulation into Canvas, Moodle, Google Sites or educational blogs:

<iframe src="https://visuallab.win/math-lab/?module=galton_board" width="100%" height="560" frameborder="0" allow="fullscreen" loading="lazy"></iframe>

Diagnostic check

1. For continuous normal variable X, what is the exact probability P(X = μ)?

  1. 0
  2. 0.5
  3. 1
Show answer and explanation

Correct answer: 0

The integral of a continuous density function over an interval of width zero is identically 0.

2. If random variable X ~ N(50, 25), what is its standard deviation σ?

  1. 5
  2. 25
  3. 50
Show answer and explanation

Correct answer: 5

In standard notation N(μ, σ²), the second parameter is variance σ² = 25, hence σ = √25 = 5.

Application practice

1. Heights follow N(165, 5²). Approximately 95.4% of students fall within which interval?

  1. [155 cm, 175 cm]
  2. [160 cm, 170 cm]
  3. [150 cm, 180 cm]
Show answer and explanation

Correct answer: [155 cm, 175 cm]

By the 2σ empirical rule: [μ − 2σ, μ + 2σ] = [165 − 10, 165 + 10] = [155, 175].

2. For Z ~ N(0, 1), if P(Z ≤ 1.2) = 0.8849, what is P(Z > 1.2)?

  1. 0.1151
  2. 0.8849
  3. −0.8849
Show answer and explanation

Correct answer: 0.1151

P(Z > 1.2) = 1 − P(Z ≤ 1.2) = 1 − 0.8849 = 0.1151.

3. For X ~ N(100, 16), what is the standardized Z-score for observation x = 108?

  1. +2
  2. +0.5
  3. +8
Show answer and explanation

Correct answer: +2

σ = √16 = 4. Therefore Z = (108 − 100)/4 = 8/4 = +2.

Academic references

  1. Cambridge IGCSE Mathematics 0580 — official syllabus and resources
  2. Cambridge AS & A Level Mathematics 9709 — official syllabus
  3. OpenStax Precalculus 2e — functions, geometry and probability

This lesson is maintained against the cited sources below. Simulations are learning models and do not replace supervised physical-laboratory safety procedures.

Content publisher
WebLab STEM
Technical maintainer
Đức Tiến — Control & Automation Engineer