Normal distribution: practice problems with step-by-step solutions

Also called: Gaussian distribution · z-scores

Free Normal distribution practice problems with diagrams, hints and full worked solutions. New values every time, at three levels: area in one tail, area between two values, percentiles.

Free practice problemEasy
μ = 6567.5P = ?

The exam scores are normally distributed with mean 65 points and standard deviation 10 points. What fraction is above 67.5 points? Find P(X > x) as a decimal.

μ = 65σ = 10x = 67.5
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What you practise

  1. Easyarea in one tail
  2. Mediumarea between two values
  3. Hardpercentiles

How to solve Normal distribution problems

  1. Convert the value to a z-score: z = (x − μ)/σ.
  2. Look up Φ(z), the area to the left of z.
  3. For the area to the right use 1 − Φ(z); for an interval subtract the two left areas.
  4. For a percentile, work backwards: find z for the area, then x = μ + z·σ.

Common mistakes

  • Reading the table for the wrong tail.
  • Forgetting to divide by σ.
  • Using the sign of z the wrong way round for percentiles below 50%.

Key formulas

z-score
z = (x − μ)/σ
Below x
P(X < x) = Φ(z)
Above x
P(X > x) = 1 − Φ(z)
Between
Φ(z2) − Φ(z1)
Percentile
x = μ + z·σ

Learn Normal distribution from scratch

Before this: Descriptive statistics

1The bell curve

Many quantities, such as heights, IQ scores or the weight of packets from a machine, follow a normal distribution. Most values lie near the middle, and values far away are rare.

The graph is a symmetric bell. It has two numbers: the mean μ, where the top is, and the standard deviation σ, which sets the width.

7085100μ115130

The key idea: a probability is an area under the curve. The total area is 1. The shaded left half has area 0.5, so half of all values lie below the mean.

About 68% of values lie within one σ of the mean, and about 95% within two σ.

Self-check Packets have a normally distributed weight with mean 500 g. What fraction of packets weighs less than 500 g?
  1. 0.5
  2. 0.68
  3. 0.95
  4. It depends on σ

Answer: 0.5. The curve is symmetric around the mean, so exactly half of the area lies below μ, whatever σ is. 0.68 is the area within one σ on both sides.

2z-scores and the table

Every normal distribution becomes the same standard normal curve (mean 0, σ = 1) when you convert a value x to a z-score:

z = (x − μ) / σ

The table gives Φ(z): the area to the left of z. For example Φ(1.2) = 0.8849, so 88.49% of values lie below z = 1.2.

  • Below x: P(X < x) = Φ(z).
  • Above x: P(X > x) = 1 − Φ(z), because the total area is 1.
  • For a negative z: Φ(−z) = 1 − Φ(z), by symmetry. So Φ(−1.2) = 0.1151.

Values you will often meet: Φ(1) = 0.8413, Φ(1.5) = 0.9332, Φ(2) = 0.9772.

Self-check Φ(1.2) = 0.8849. What is P(Z > 1.2)?
  1. 0.1151
  2. 0.3849
  3. 0.8849
  4. 1.2

Answer: 0.1151. The table gives the area to the left. The area to the right is 1 − 0.8849 = 0.1151. Using 0.8849 directly reads the wrong tail.

3Between two values and percentiles

For the area between two values, subtract the two left areas: everything below the upper value, minus everything below the lower value.

P(x₁ < X < x₂) = Φ(z₂) − Φ(z₁)

A percentile works backwards. The 90th percentile is the value with 90% of the area to its left.

  • Find the z with Φ(z) equal to the area. For 90%, z = 1.2816. For 80%, z = 0.8416.
  • Convert back to x: x = μ + z · σ.

For a percentile below 50%, z is negative. The 10th percentile has z = −1.2816.

Self-check IQ scores are normal with μ = 100 and σ = 15. The z for the 80th percentile is 0.8416. What is the 80th percentile?
  1. 100.8
  2. 112.6
  3. 115
  4. 87.4

Answer: 112.6. x = 100 + 0.8416 · 15 = 112.6. Using the minus sign gives 87.4, the 20th percentile; 100.8 forgets to multiply by σ.

4The method step by step

  1. Sketch the bell curve, mark μ and the value(s), and shade the area that is asked.
  2. Convert each value to a z-score: z = (x − μ)/σ.
  3. Look up Φ(z), the area to the left.
  4. Below: Φ(z). Above: 1 − Φ(z). Between: Φ(z₂) − Φ(z₁).
  5. For a percentile, go backwards: area → z → x = μ + zσ.
  6. Compare with your sketch: a shaded area smaller than half must give a number below 0.5.

5A worked example

Follow every step. Try to predict the next one before you open it.

Worked example

IQ scores are normally distributed with mean 100 and standard deviation 15. What fraction P lies between 91 and 121?

1

Sketch the area

91x₁100121x₂

The shaded area covers the middle and more of the right side. Expect a number a bit above 0.5.

2

Lower z-score

z₁ = (91 − 100) / 15 = −9/15 = −0.6
3

Upper z-score

z₂ = (121 − 100) / 15 = 21/15 = 1.4
4

Look up both areas

Φ(1.4) = 0.9192
Φ(−0.6) = 1 − Φ(0.6) = 1 − 0.7257 = 0.2743
5

Subtract

P = 0.9192 − 0.2743 = 0.6449

This is a bit above 0.5, as the sketch predicted.

Answer

P ≈ 0.645: about 64.5% of scores lie between 91 and 121.

6Common mistakes

  • Reading the wrong tail: using Φ(z) when the question asks for the area above x.
  • Forgetting to divide by σ in the z-score.
  • Adding the two left areas for an interval instead of subtracting them.
  • Using the wrong sign of z for a percentile below 50%.
  • Mixing up σ and the variance σ². Divide by σ, the standard deviation.
Self-check Exam scores are normal with μ = 65 and σ = 10. Φ(−1.5) = 0.0668. What fraction scores above 50?
  1. 0.0668
  2. 0.4332
  3. 0.5668
  4. 0.9332

Answer: 0.9332. z = (50 − 65)/10 = −1.5, and P(X > 50) = 1 − Φ(−1.5) = 0.9332. 0.0668 is the area below 50: the wrong tail. Most students score above 50, so the answer must be large.

7How exam questions look

  • Easy: the area in one tail. Find P(X < x) or P(X > x) with one z-score.
  • Medium: the area between two values, Φ(z₂) − Φ(z₁), as in the example.
  • Hard: percentiles. Find the value below which 80% or 90% of the data lie.

Always sketch first. A ten-second sketch catches most tail mistakes.

Learn it step by step in the appLearn it step by step in the app

Worked examples

Example 1Easy
μ = 3236.8P = ?

The delivery times are normally distributed with mean 32 min and standard deviation 6 min. What fraction is above 36.8 min? Find P(X > x) as a decimal.

μ = 32σ = 6x = 36.8

Find P(X > x)

Show hints
  1. Convert to a z-score: z = (x − μ)/σ.
  2. z = (36.8 − 32)/6 = 0.8.
  3. The area to the right is 1 − Φ(z).
Show the answer and the step-by-step solution

Normal distribution: convert a value x to a z-score, z = (x − μ)/σ, which counts standard deviations from the mean. A z-table (or Φ(z)) gives the area to the left of z, which is P(X < x). The area to the right is 1 − Φ(z), and the area between two values is the difference of their two areas.

z-score

z = (36.8 − 32)/6 = 0.8

Probability

P = 1 − Φ(0.8) = 1 − 0.7881 = 0.2119

Answer

P = 0.2119

That is 21.19%.

Example 2Medium
750 g769.7 gX + Y?

X is normal with mean 250 g and standard deviation 4 g; Y is independent and normal with mean 500 g and standard deviation 9 g. Find the probability P that X + Y is more than 769.7 g. Round z to two decimals.

μX = 250, σX = 4μY = 500, σY = 9X + Y > 769.7 g

Find P

Show hints
  1. E(X + Y) = 250 + 500 = 750 g.
  2. σ = √(4² + 9²) = 9.849 g.
  3. z = (769.7 − 750)/9.849.
Show the answer and the step-by-step solution

Sums and differences of normal variables: if X and Y are independent and normal, X + Y and X − Y are normal too. Means add or subtract: E(X ± Y) = μX ± μY. Variances always add, also for a difference: Var(X ± Y) = σX² + σY², so σ = √(σX² + σY²). Standard deviations never add directly. The total of n independent items with the same distribution has mean n·μ and standard deviation σ·√n. Then standardise as usual: z = (x − mean)/σ and read Φ(z).

Distribution of the sum

μ = 750 g, σ = √(4² + 9²) = 9.849 g

Standardise

z = (769.7 − 750)/9.849 = 2

Upper tail

P = 1 − Φ(2) = 1 − 0.9772 = 0.02275

Answer

P = 0.02275
Example 3Hard
μ = 178x = ?20%

The heights of adult men are normally distributed with mean 178 cm and standard deviation 7 cm. Find the value x below which 20% of them lie (the 20th percentile).

μ = 178σ = 7P = 20%

Find x (cm)

Show hints
  1. Work backwards: find the z with Φ(z) = P/100 in the table.
  2. z = −0.8416.
  3. Then x = μ + z·σ.
Show the answer and the step-by-step solution

Normal distribution: convert a value x to a z-score, z = (x − μ)/σ, which counts standard deviations from the mean. A z-table (or Φ(z)) gives the area to the left of z, which is P(X < x). The area to the right is 1 − Φ(z), and the area between two values is the difference of their two areas.

z for the percentile

Φ(z) = 0.2 → z = −0.8416

Back to x

x = 178 + −0.8416·7 = 172.11 cm

Answer

x = 172.11 cm

Questions and answers

What is Normal distribution?

Many quantities, such as heights, IQ scores or the weight of packets from a machine, follow a normal distribution. Most values lie near the middle, and values far away are rare.

The graph is a symmetric bell. It has two numbers: the mean μ, where the top is, and the standard deviation σ, which sets the width.

How do I solve Normal distribution problems step by step?
  1. Convert the value to a z-score: z = (x − μ)/σ.
  2. Look up Φ(z), the area to the left of z.
  3. For the area to the right use 1 − Φ(z); for an interval subtract the two left areas.
  4. For a percentile, work backwards: find z for the area, then x = μ + z·σ.
What are common mistakes in Normal distribution?
  • Reading the table for the wrong tail.
  • Forgetting to divide by σ.
  • Using the sign of z the wrong way round for percentiles below 50%.
What do the Normal distribution exercises cover?
  • Easy: area in one tail
  • Medium: area between two values
  • Hard: percentiles

Every problem has new values, so you learn the method instead of an answer. Stuck? Open a small hint first, and the full explanation only when you need it.

How practising works

  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Your progress, mistakes and saved problems are kept, so you can review before the exam.

Topics before and after this one

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Descriptive statistics

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Confidence intervals

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Printable worksheet

Normal distribution worksheet with answers
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