Conceptual and mathematical explanation
For a continuous distribution, probability is area under the density curve over an interval. The probability of one exact point is zero; do not confuse density height with probability.
A normal X~N(μ,σ²) model has mean μ and variance σ². Standardising with Z=(X−μ)/σ expresses distance from the mean in SD units; Z~N(0,1).
Tail probabilities use the standard normal cumulative distribution Φ. P(X≥a)=1−Φ((a−μ)/σ). Suitability of a normal model is a separate empirical question.
Key formula
X~N(μ,σ²) Z=(X−μ)/σ P(X≥a)=1−Φ((a−μ)/σ)
Worked example: step by step
- Under fictional X~N(70,100), μ=70 and SD=√100=10.
- For X=80, z=(80−70)/10=1. P(X≥80)=P(Z≥1)≈0.1587.
- P(60≤X≤80)=Φ(1)−Φ(−1)≈0.6827. The exact-point probability P(X=80) is zero in this model.
Independent exercises
- With μ=50 and σ=5, what is z for X=60?
- For standard normal Z, what is P(Z≤0)?
- Write base R code that computes P(X≥80) for X~N(70,100).
Written solutions are private in the instructor guide.
Optional base R check
pnorm(80, mean=70, sd=10, lower.tail=FALSE)
pnorm(80, mean=70, sd=10) - pnorm(60, mean=70, sd=10)
dnorm(80, 70, 10) # density, NOT probability of exactly 80