Simulic
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Normal distribution – probabilities, z-scores and inverse normal

MathematicsProbabilityAges 17–18

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Drag the mean μ and standard deviation σ of X ~ N(μ, σ²), shade P(a < X < b) and read the probability, z-scores, percentiles, the inflection points at μ ± σ and the 68–95–99.7 rule at once. Other screens find x from a probability (inverse normal), compare two values by standardising to N(0, 1), and fit a normal curve to a data histogram and to a binomial distribution B(n, p).

Lesson: The normal distribution, z-scores and the 68–95–99.7 rule

What it shows

The normal model X ~ N(μ, σ²) is a symmetric bell-shaped curve whose total area is 1; a probability is the area under it. Standardising with z = (x − μ)/σ turns any normal variable into Z ~ N(0, 1), so P(a < X < b) = Φ(z_b) − Φ(z_a); the sim computes Φ numerically instead of using a table. The inverse normal reverses this: x = μ + zσ. The data sets are simulated, and the sample standard deviation divides by n − 1. The binomial screen compares exact binomial probabilities with the normal approximation, with or without a continuity correction.

How to use

On Probability, drag the top of the curve to change μ, the green dot at μ + σ to change σ, and the lines a and b; pick the type under Find and tick 68–95–99.7 rule. On Inverse, choose a question and move p. On Compare, set both distributions and press Standardise to N(0, 1). On Data, change Data set and Sample size n, or press New sample. On Binomial, move n, p and k and switch Continuity correction.

Parameters you can change

  • Screen Probability, Inverse normal, Compare two distributions, Normal model for data, Binomial approximation
  • Mean μ -100–300
  • Standard deviation σ 0.1–100
  • Probability to find P(a < X < b), P(X < b), P(X > a), P(X < a or X > b)
  • Lower bound a -1000–1000
  • Upper bound b -1000–1000
  • Show the 68–95–99.7 rule
  • Show the z axis
  • Probability p (Inverse screen) 0.001–0.999
  • Inverse question P(X < x) = p, P(X > x) = p, P(μ − d < X < μ + d) = p
  • Mean of distribution B -100–300
  • Standard deviation of distribution B 0.1–100
  • Value in distribution A -1000–1000
  • Value in distribution B -1000–1000
  • Data set (Data screen) Heights (simulated), Reaction times (simulated), Waiting times for a bus (simulated), Time between geyser eruptions (simulated)
  • Sample size n 10–2000
  • Random seed (same number, same data) 1–999
  • Number of trials n of B(n, p) 1–200
  • Probability of success p of B(n, p) 0.01–0.99
  • Value k 0–200
  • Binomial probability to find P(X ≤ k), P(X ≥ k), P(X = k)
  • Continuity correction

Questions to explore

  1. Why does P(μ − σ < X < μ + σ) stay 0.6827 whatever values of μ and σ you choose?
  2. Which score is better relative to its group, 65 in test A or 68 in test B?
  3. For which values of n and p does the normal curve fit the binomial bars well?

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