Normal Distribution
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Direct answer
The normal curve with mean μ and standard deviation σ has density f(x) = (1/(σ√(2π))) e^(-(x - μ)^2/(2σ^2)), a bell symmetric about μ where mean, median and mode coincide. Every normal probability question reduces to standardisation: Z = (X - μ)/σ converts any normal variable to the standard normal Z ~ N(0, 1), and areas under the standard curve (supplied in tables) give probabilities via P(a < X < b) = Φ((b - μ)/σ) - Φ((a - μ)/σ). The empirical rule fixes the headline areas: about 68.27 percent within μ ± σ, 95.44 percent within μ ± 2σ, and 99.73 percent within μ ± 3σ. The curve's points of inflexion sit at μ ± σ — the geometric fingerprint of σ — and symmetry gives Φ(-z) = 1 - Φ(z), which is how negative Z-values are handled without a second table.
What you must remember
- Density: f(x) = (1/(σ√(2π))) e^(-(x - μ)^2/(2σ^2)); the prefactor makes total area 1, and σ enters both the exponent and the normalising constant.
- Standardisation: Z = (X - μ)/σ transforms N(μ, σ^2) to N(0, 1); read all probabilities from the standard table, never integrate the density.
- Empirical rule: 68.27, 95.44, 99.73 percent within 1, 2, 3 standard deviations — the three numbers JEE expects without a table.
- Symmetry arithmetic: Φ(-z) = 1 - Φ(z), so P(Z < -1) = 1 - 0.8413 = 0.1587; the tails are mirror images about 0.5.
- Inflexion points: the curve changes concavity at μ ± σ; given inflexion at x = 2 and x = 8, the distribution is N(5, 9) — mean 5, σ = 3, variance 9.
- Central trimmings: quartiles sit at μ ± 0.6745σ (so QD = 2σ/3), and mean deviation is 0.7979σ ≈ (4/5)σ; for a normal curve the chain 4 QD ≈ 5 MD ≈ 6 SD holds.
- Linear combinations: aX + b with X normal is normal with mean aμ + b and SD |a|σ; sums of independent normals stay normal.
Reading a question off the bell
Let X ~ N(50, 100), so μ = 50 and σ = 10 (the second parameter is the variance in this notation). Find P(X > 60): standardise to z = (60 - 50)/10 = 1, and P(Z > 1) = 1 - Φ(1) = 1 - 0.8413 = 0.1587, about 16 percent. Find P(40 < X < 60): this is the central band μ ± σ, and the empirical rule supplies 0.6827 directly — or via the table, Φ(1) - Φ(-1) = 2Φ(1) - 1 = 0.6826. The same band arithmetic answers reverse questions: if the strongest 2.28 percent of wires exceed a threshold, note 1 - 0.9772 = 0.0228 corresponds to z = 2, so the cutoff is μ + 2σ. Notice the workflow's economy — every question, forward or backward, travels through z = (x - μ)/σ, and the phrase "symmetric about" or an inflexion point hands you μ and σ before any probability is computed.
Where students slip
JEE Main examines exactly the machinery above: standardise, look up or use 68-95-99.7, report a four-decimal probability or a threshold. Numerical answers cluster around 0.1587, 0.8413, 0.0228 and 0.9772 (the z = 1 and z = 2 neighbourhoods) — recognising them speeds option elimination. Advanced rarely goes beyond the same computations but hides the distribution: a binomial with large np and nq invites the de Moivre-Laplace (normal) approximation, or symmetric density conditions (mean = median = mode, inflexion data) identify normality implicitly. The recurring errors: reading N(μ, σ^2) as N(mean, SD) and using σ = 100 in the example above (variance, not SD, is the second parameter); computing P(X > 60) as Φ(1) rather than 1 - Φ(1), choosing the fat side of the tail; and forgetting Φ(-z) = 1 - Φ(z) when the table lists only positive arguments. A quiet conceptual trap: normality of X does not transfer to X^2 or e^X — only linear functions stay normal. Statistics units of both syllabi list the normal distribution explicitly.
Frequently asked questions
How is any normal variable converted to the standard normal?
By Z = (X - μ)/σ; then X < b corresponds to Z < (b - μ)/σ, and standard-normal tables supply the probability.
What percentages do the 1, 2 and 3 sigma bands contain?
Approximately 68.27, 95.44 and 99.73 percent — the empirical rule; each band leaves half the remainder in either tail.
What do the inflexion points of the normal curve reveal?
They sit at μ ± σ, so their positions hand you both the mean (midway) and the standard deviation (half the distance), fixing the distribution completely.
In the notation X ~ N(50, 100), what are the mean and standard deviation?
Mean 50 and standard deviation 10 — the second parameter is the variance, a distinction worth two marks in every other paper.
Why is Φ(-z) = 1 - Φ(z)?
Symmetry of the bell about zero: the area left of -z mirrors the area right of z, and the two together make the total area 1.