Difficulty: Introductory | Prerequisites: Basic probability, z-scores
Big Picture
The Normal (Gaussian) distribution is the single most important continuous probability distribution in introductory statistics. Most real-world measurement data, from light bulb lifetimes to exam scores, cluster around a central value with symmetric tails. If you can standardise a value into a z-score, you can look up or compute the probability of any outcome. This topic sits at the heart of everything that follows in the course: confidence intervals, hypothesis tests, and regression all lean on the Normal distribution.
If a random variable follows a Normal distribution, you can calculate the probability of any range of values using the cumulative distribution function (CDF), and you can work backwards from a probability to find the corresponding value using the quantile (inverse CDF) function. In R, pnorm handles probabilities and qnorm handles quantiles. These two functions, plus the rules for sums of independent Normals, cover the majority of introductory Normal distribution problems.
Normal distribution (Gaussian distribution)
A continuous probability distribution defined by two parameters: the mean (μ) and the standard deviation (σ). Its density curve is the familiar symmetric bell shape. Think of it as the default model for data that clusters around a centre with roughly equal spread in both directions.
Mean (μ)
The centre of the Normal distribution, locating the peak of the bell curve. In simple terms, it is the expected or average value of the variable.
Standard deviation (σ)
The spread parameter. A larger σ makes the bell curve wider and flatter; a smaller σ makes it taller and narrower. Think of it as a ruler that measures how far typical observations fall from the mean.
Z-score (standard score)
The number of standard deviations a value sits from the mean: z = (X − μ) / σ. It converts any Normal variable to the Standard Normal (μ = 0, σ = 1). In simple terms, it answers "how unusual is this value?"
CDF / pnorm
The cumulative distribution function gives P(X ≤ x), the probability that the variable falls at or below a given value. In R, pnorm(x, mean, sd) computes this directly. Think of it as "area to the left" under the bell curve.
Quantile function / qnorm
The inverse of the CDF. Given a probability p, it returns the value x such that P(X ≤ x) = p. In R, qnorm(p, mean, sd) does this. Think of it as "what value marks the boundary for this percentile?"
Continuity correction
An adjustment of ±0.5 applied when using a continuous distribution (like the Normal) to approximate a discrete one. For P(X = 2450), you compute P(2449.5 < X < 2450.5) instead. Think of it as giving a single integer its own thin slice of area under the curve.
pnorm(x, μ, σ) returns P(X ≤ x), the left-tail area.
For P(X > x), compute 1 − pnorm(x, μ, σ).
For P(a < X < b), compute pnorm(b, μ, σ) − pnorm(a, μ, σ).
You can also standardise first: pnorm((x − μ) / σ) uses the Standard Normal and gives the same result.
Worked example (light bulbs, μ = 2400, σ = 200):
P(X ≤ 2554): pnorm(2554, 2400, 200) = 0.7794. About 78% of bulbs last 2554 hours or fewer.
P(X = 2450) using continuity correction: diff(pnorm(c(2449.5, 2450.5), 2400, 200)) = 0.00193. Essentially zero for a continuous distribution, as expected.
P(2259 < X < 2371): pnorm(2371, 2400, 200) − pnorm(2259, 2400, 200) = 0.2020.
P(both of two independent bulbs last > 2404): first find P(X > 2404) = 1 − pnorm(2404, 2400, 200) ≈ 0.4920, then square it: 0.4920² ≈ 0.2421.
qnorm(p, μ, σ) returns the value at the p-th quantile (left-tail area = p).
qnorm(p, μ, σ, lower.tail = FALSE) returns the value where the upper-tail area equals p.
Worked examples:
First quartile (Q1) of bulb lifetimes: qnorm(0.25, 2400, 200) = 2265.1 hours.
Middle 99.7% boundaries: qnorm(0.0015, 2400, 200) = 1806.5 and qnorm(0.9985, 2400, 200) = 2993.5. This aligns with the three-sigma rule (μ ± 3σ).
Top 9% cutoff: qnorm(0.09, 2400, 200, lower.tail = FALSE) = 2668.2 hours. The best 9% of bulbs last at least this long.
If X₁, X₂, ..., Xₙ are independent and each follows N(μ, σ²), then their sum S = X₁ + ... + Xₙ follows N(nμ, nσ²). The mean of the sum is nμ. The standard deviation of the sum is σ√n (not nσ).
Worked example:
Four independent bulbs, each N(2400, 200²). What is P(total lifetime < 9000 hours)?
Sum has mean = 4 × 2400 = 9600 and SD = 200 × √4 = 400.
pnorm(9000, 9600, 400) = 0.0668, so about a 6.7% chance the four bulbs together last fewer than 9000 hours.
Z-score: z = (X − μ) / σ
CDF (left-tail probability): P(X ≤ x) = pnorm(x, μ, σ)
Right-tail probability: P(X > x) = 1 − pnorm(x, μ, σ)
Interval probability: P(a < X < b) = pnorm(b, μ, σ) − pnorm(a, μ, σ)
Sum of n independent Normals: S ~ N(nμ, nσ²), so SD(S) = σ√n
Continuity correction for a single discrete value: P(X = k) ≈ P(k − 0.5 < X < k + 0.5)
Students often think P(X = exact value) is meaningful for a continuous Normal. It is not. The probability of any single point is technically zero; you need a range, or a continuity correction if approximating a discrete count.
Students sometimes multiply the standard deviation by n instead of by √n when summing independent Normals. The variance adds linearly; the standard deviation does not.
Confusing pnorm (probability from a value) with qnorm (value from a probability) is extremely common. If the question gives you a number on the measurement scale and asks for a probability, you want pnorm. If it gives you a percentile or probability and asks for a cutoff, you want qnorm.
Students forget that pnorm gives the left tail by default. For "more than" questions, you need 1 − pnorm(...) or lower.tail = FALSE.
⚠️ "Find the probability that X is between a and b" is a standard exam question. Know the subtraction pattern: pnorm(b) − pnorm(a).
⚠️ "Find the value such that p% of observations fall below it" is a quantile question. Use qnorm(p, μ, σ).
⚠️ The sum-of-Normals rule appears on exams as "four light bulbs" or "a sample of n items." The key detail: SD of the sum is σ√n, not nσ.
⚠️ "Both bulbs last longer than..." requires computing P(X > value) first, then squaring it (assuming independence). This tests whether you recognise the independence multiplication rule.
True or false: P(X = 2400) is approximately 0.5 for X ~ N(2400, 200).
Fill in the blank: qnorm(0.5, 100, 15) returns ___ because the median of a Normal equals the ___.
True or false: The standard deviation of the sum of 9 independent N(50, 10²) variables is 90.
Fill in the blank: To find P(X > 300) for X ~ N(250, 25), compute 1 − pnorm(___, ___, ___).
True or false: pnorm(2554, 2400, 200) and pnorm((2554 − 2400) / 200) give the same answer.
Answers: 1. False (point probability is 0 for continuous distributions). 2. 100; mean. 3. False (SD = 10 × √9 = 30, not 90). 4. 300, 250, 25. 5. True.
Q: X ~ N(2400, 200²). What is the probability that a randomly chosen bulb lasts at most 2554 hours?
A: P(X ≤ 2554) = pnorm(2554, 2400, 200) ≈ 0.7794.
Q: What is the first quartile (Q1) of bulb lifetimes?
A: qnorm(0.25, 2400, 200) ≈ 2265.1 hours. 25% of bulbs last fewer than about 2265 hours.
Q: Four independent bulbs are used. What is the probability their combined lifetime is less than 9000 hours?
A: Sum ~ N(9600, 400²). P(S < 9000) = pnorm(9000, 9600, 400) ≈ 0.0668.
Q: Two bulbs are chosen independently. What is the probability both last longer than 2404 hours?
A: P(X > 2404) = 1 − pnorm(2404, 2400, 200) ≈ 0.4920. P(both) = 0.4920² ≈ 0.2421.
Q: What range of lifetimes captures the middle 99.7% of all bulbs?
A: qnorm(0.0015, 2400, 200) ≈ 1806.5 to qnorm(0.9985, 2400, 200) ≈ 2993.5 hours. This is approximately μ ± 3σ.
This material connects directly to the Normal approximation to the Binomial (covered in the companion study notes), where pnorm and qnorm are reused with mean = np and SD = √(np(1−p)). It also lays the groundwork for sampling distributions and the Central Limit Theorem: once you know how sums of independent Normals behave, the CLT generalises that idea to sums of non-Normal variables.
Normal distribution, Gaussian distribution, bell curve, pnorm, qnorm, z-score, standard score, CDF, cumulative distribution function, quantile function, inverse CDF, percentile, continuity correction, sum of normals, independent normal variables, σ√n, three-sigma rule, 68-95-99.7 rule, R statistics, light bulb lifetime problem, Purdue STAT