How do you solve a normal distribution problem?

State X ~ N(μ, σ²), sketch and shade the region, standardise the boundary values with z = (x − μ)/σ, read the table, and combine the areas using 0.5 and symmetry.

How do I find x when the probability is given?

Turn the probability into the table area A(z), look up z, decide its sign from which tail the value is in, and compute x = μ + zσ.

Is the normal approximation to the binomial in DSE M1?

It is not listed in the learning objectives. What does appear is a normal probability used as p in a binomial distribution over several items.

Applications of the Normal Distribution (DSE M1)

Every normal problem is one of two types. Given values, find a probability: standardise each value, then read the table. Given a probability, find a value: find the z for that area from the table, then undo the standardisation with x = μ + zσ.

Solve Applications of the Normal Distribution questions with AI

Example 2: the cut-off moves right until the shaded upper tail is 10% of the area, at about x = 165.4. Try it yourself →

Normal Distribution: Working Formulas

Standardisez = (x − μ) / σ

Back to xx = μ + zσ

IntervalP(x₁ < X < x₂) = P((x₁ − μ)/σ < Z < (x₂ − μ)/σ)

Strict or notP(X ≤ x) = P(X < x)

Worked Examples — Applications of the Normal Distribution

Example 1: Finding a probability between two values

The weights of apples follow a normal distribution with mean 150 g and standard deviation 12 g. Find the probability that a randomly chosen apple weighs between 144 g and 168 g.

X ~ N(150, 12²)
z₁ = (144 − 150)/12 = −0.5, z₂ = (168 − 150)/12 = 1.5
P(144 < X < 168) = P(−0.5 < Z < 1.5) = A(0.5) + A(1.5)
= 0.1915 + 0.4332 = 0.6247

Answer: 0.6247

Example 2: Finding a value from a probability

For the same apples, the heaviest 10% are graded "large". Find the minimum weight of a large apple.

We need x with P(X > x) = 0.1, so A(z) = 0.5 − 0.1 = 0.4
From the table, z ≈ 1.28
x = 150 + 1.28 × 12 = 165.36

Answer: About 165.4 g

💡 Exam Traps & Secrets

⚠️ Sketch before you look upDraw the bell curve, mark the mean and shade the region. It shows at once whether the answer should be more or less than 0.5 and whether to add or subtract table values.

⚠️ The sign of z in a reverse lookupA value in the lower tail has a negative z. "The lightest 10%" gives z = −1.28, so x = μ − 1.28σ.

✏️ Self-Test: Quick Interactive Practice

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

(Find a probability)X ~ N(70, 10²). Find P(X > 85) and shade the region on the curve.

(Find x)Exam marks are N(62, 9²). The top 5% get a distinction. Find the lowest mark for a distinction.

(Find μ)X ~ N(μ, 5²) and P(X < 40) = 0.2. Find μ.