What are the properties of a normal distribution?

The curve is bell-shaped and symmetric about the mean; the mean, median and mode coincide; the standard deviation determines how flat it is; and the total area under the curve is 1.

Do I need to integrate the normal density function?

No. Finding expectations or variances of continuous distributions by integration is not required. Normal probabilities come from the standard normal table.

What is the difference between a discrete and a continuous random variable?

A discrete variable takes separate values, each with its own probability. A continuous variable can take any value in an interval; single values have probability 0 and probabilities are areas under a density curve.

The Normal Distribution (DSE M1): Properties of the Bell Curve

A normal distribution X ~ N(μ, σ²) is a continuous distribution whose curve is bell-shaped and symmetric about the mean μ. The mean, median and mode are equal, the standard deviation σ controls how flat the curve is, and the total area under the curve is 1. A probability is an area under the curve.

Solve The Normal Distribution: Definition and Properties questions with AI

N(50, σ²) as σ grows from 4 to 16. The curve stays centred on 50 and keeps area 1, so it must get lower as it gets wider. Try it yourself →

Properties of the Normal Distribution

NotationX ~ N(μ, σ²): mean μ, standard deviation σ

SymmetryP(X < μ − a) = P(X > μ + a), P(X < μ) = 0.5

Total areaArea under the curve = 1

A single valueP(X = x₁) = 0, so P(X ≤ x₁) = P(X < x₁)

Density functionf(x) = (1 / (σ√(2π))) e^{−(x − μ)² / (2σ²)}

Worked Examples — The Normal Distribution: Definition and Properties

Example 1: Using symmetry

X ~ N(50, 8²) and P(X < 58) = 0.8413. Find P(X > 58), P(X < 42) and P(42 < X < 58).

P(X > 58) = 1 − 0.8413 = 0.1587
42 and 58 are both 8 from the mean, so by symmetry P(X < 42) = P(X > 58) = 0.1587
P(42 < X < 58) = 1 − 2(0.1587) = 0.6826

Answer: 0.1587, 0.1587, 0.6826

Example 2: Comparing two normal curves

Two normal distributions have the same mean 50. One has standard deviation 4 and the other 16. Describe how their curves differ.

Both are symmetric about x = 50 and both have area 1.
The curve with σ = 4 is taller and narrower: values cluster near 50.
The curve with σ = 16 is flatter and wider: values are more spread out.

Answer: Same centre; smaller σ gives a taller, narrower curve.

💡 Exam Traps & Secrets

⚠️ N(μ, σ²) is written with the varianceN(50, 64) has standard deviation 8, not 64. Take the square root before standardising.

⚠️ Probability is area, not heightThe height of the curve at a point is not a probability. For a continuous variable P(X = x) is 0; only intervals have probability.

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Effect of σ)Draw the normal curves N(60, 5²) and N(60, 10²) on the same axes and compare them.

(Symmetry)X ~ N(100, 15²) and P(X > 120) = 0.0912. Find P(80 < X < 120).

(Continuous variable)Explain why P(X ≤ 70) and P(X < 70) are equal for a normal variable.