# Normal probability distribution

> IB Mathematics: Analysis and Approaches SL · IB AA SL
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u4-normal-probability-distribution/

This sub-topic covers the properties of the normal distribution, the most widely used continuous probability distribution. You will learn to calculate probabilities, convert values to standard z-scores, and solve inverse problems for real-world applications.

**Prerequisites:** [Continuous probability distributions](https://www.owlsprep.com/study/ib-math-aa-sl-u4-continuous-probability/); [Measures of central tendency and spread](https://www.owlsprep.com/study/ib-math-aa-sl-u3-descriptive-statistics/)

## Learning objectives

- Recall key properties of the normal distribution curve
- Calculate z-scores to standardize any normal distribution
- Find probabilities for normally distributed random variables
- Solve inverse normal problems to find cut-off values from given probabilities

## Properties of the Normal Distribution

**Normal Probability Distribution** — A continuous probability distribution described by a symmetric, bell-shaped curve, with total area under the curve equal to 1. It models many naturally occurring random variables.

*Notation:* X \sim N(\mu, \sigma^2)

*Example:* Heights of adults, measurement errors, standardized test scores

The normal curve is symmetric about its mean $\mu$, so mean = median = mode for any normal distribution. The spread is controlled by the variance $\sigma^2$: larger values produce a wider, flatter curve. The curve is asymptotic to the horizontal axis, meaning it never touches the x-axis.

> **Empirical Rule Mnemonic**
>
> 68-95-99.7: Approximately 68% of data lies within 1 standard deviation of $\mu$, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.

**Worked example:** The masses of loaves of bread from a bakery are normally distributed with mean 500 g and standard deviation 10 g. Use the empirical rule to estimate the interval that contains 68% of loaf masses.

1. Recall 68% of data lies within 1 standard deviation of the mean:
2. $$\mu \pm 1\sigma$$
3. Substitute $\mu = 500$ and $\sigma = 10$:
4. $$500 - 10 = 490, \quad 500 + 10 = 510$$
5. 68% of loaf masses fall between 490 g and 510 g.

> **Exam tip:** The empirical rule is frequently tested on no-calculator Paper 1 questions, so memorize the 68-95-99.7 values.

## Standard Normal Distribution and Z-Scores

**Z-score** — A standardized value that tells you how many standard deviations an observation $x$ is from the mean $\mu$ of the distribution.

*Notation:* z = \frac{x - \mu}{\sigma}

*Example:* An observation 1 standard deviation above the mean has a z-score of 1.

Any normal distribution can be transformed to the **standard normal distribution**, which has mean 0 and standard deviation 1: $Z \sim N(0,1)$. This transformation allows us to use a single reference to find probabilities for any normal distribution.

**Worked example:** Foot lengths of adult men are normally distributed with mean 27 cm and standard deviation 1.5 cm. Find the z-score for a foot length of 29.25 cm.

1. State the z-score formula:
2. $$z = \frac{x - \mu}{\sigma}$$
3. Substitute $x = 29.25$, $\mu = 27$, $\sigma = 1.5$:
4. $$z = \frac{29.25 - 27}{1.5} = \frac{2.25}{1.5} = 1.5$$
5. The z-score is 1.5, meaning 29.25 cm is 1.5 standard deviations above the mean.

**Check your understanding**

1. What is the z-score for a foot length of 24 cm in the same distribution?

   - -2
   - 2
   - -3
   - 3

   *Why:* $z = (24 - 27)/1.5 = -3/1.5 = -2$, which is correct. A negative z-score means the value is below the mean.

## Calculating Normal Probabilities

To find the probability that a normally distributed variable $X$ falls between two values, we first convert the boundary values to z-scores, then find the area under the standard normal curve between the z-scores. For continuous distributions, $P(X = a) = 0$, so $P(a \leq X \leq b) = P(a < X < b)$.

**Exam command terms**

- **Find the probability that X exceeds k** — Calculate $P(X > k) = 1 - P(X \leq k)$

- **Find the proportion of values less than k** — Calculate $P(X < k)$

**Worked example:** Battery lifetimes are normally distributed with mean 50 months and standard deviation 6 months. Find the probability that a randomly selected battery lasts less than 40 months.

1. State the distribution: $X \sim N(50, 6^2)$, we want $P(X < 40)$.
2. Convert $x = 40$ to a z-score:
3. $$z = \frac{40 - 50}{6} \approx -1.667$$
4. Find $P(Z < -1.667)$ using calculator or standard tables:
5. $$P(Z < -1.667) \approx 0.0478$$
6. The probability a battery lasts less than 40 months is approximately 0.048 (3 s.f.).

> **Exam tip:** Always give your final probability answer to 3 significant figures unless the question specifies another accuracy.

*Calculator:* allowed

## Inverse Normal Calculations

When we know a probability and need to find the corresponding value of $x$, we use an inverse normal calculation. Common problems include finding the cut-off score for the top 10% of a distribution, or finding the value of the mean or standard deviation given a probability.

**Worked example:** Exam marks are normally distributed with mean 65 and standard deviation 12. Find the minimum mark required to be in the top 15% of candidates.

1. Let $X =$ exam mark, so $X \sim N(65, 12^2)$. We need $x$ such that $P(X > x) = 0.15$.
2. Rewrite to get a cumulative probability: $P(X \leq x) = 1 - 0.15 = 0.85$.
3. Use inverse normal to find the z-score for cumulative probability 0.85: $z \approx 1.036$.
4. Rearrange the z-score formula to solve for $x$:
5. $$x = \mu + z\sigma = 65 + (1.036)(12) \approx 77.43$$
6. The minimum mark required is 78 (to the nearest whole mark) or 77.4 (3 s.f.).

> **Exam tip:** Always confirm your cumulative probability for inverse normal: top p% means cumulative probability = 1 - p/100, not p/100.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Applying continuity correction to a continuous normal distribution probability calculation
  - Why it fails: Continuity correction is only used when approximating a discrete binomial distribution with the normal distribution, not for already continuous normal variables.
  - Correct: For a continuous normal variable, $P(X < 10) = P(X \leq 10)$, no correction is needed.
- **Wrong:** Confusing variance and standard deviation in the notation $N(\mu, \sigma^2)$
  - Why it fails: The second parameter is variance, but most calculators require input of standard deviation, so mixing them up leads to wrong answers.
  - Correct: If given standard deviation $\sigma$, the variance is $\sigma^2$ for notation, but input $\sigma$ directly into your calculator.
- **Wrong:** Using the given tail probability directly for inverse normal
  - Why it fails: If you need the top 10%, the cumulative probability for inverse normal is 0.9, not 0.1. Using 0.1 gives a negative z-score and wrong cut-off.
  - Correct: Always write $P(X < x) = \text{cumulative probability}$ before calculating inverse normal.
- **Wrong:** Rounding z-scores early before calculating the final value of $x$
  - Why it fails: Rounding z to one decimal place (e.g. 1.3 instead of 1.282) introduces inaccuracies that cost marks.
  - Correct: Keep full precision of z from your calculator until the final step, then round the final answer.
- **Wrong:** Forgetting empirical rule percentages for Paper 1 (no calculator) questions
  - Why it fails: No normal tables are provided for Paper 1, so empirical rule questions rely on memorization.
  - Correct: Memorize 68% for 1σ, 95% for 2σ, 99.7% for 3σ before exam day.

## Cheatsheet

| Concept | Formula/Rule | Key Note |
| --- | --- | --- |
| General Normal | $X \sim N(\mu, \sigma^2)$ | Second parameter is variance |
| Z-score | $z = \frac{x - \mu}{\sigma}$ | Standardizes any normal to $N(0,1)$ |
| Empirical Rule | 68% within 1σ, 95% within 2σ, 99.7% within 3σ | For Paper 1 no-calculator |
| Find $P(X < x)$ | Convert x to z, find cumulative probability | Give answer to 3 s.f. |
| Inverse Normal: find x for $P(X < x) = p$ | $x = \mu + z_p\sigma$ | $z_p$ = inverse CDF at p |
| Top 10% cut-off | Cumulative probability = 0.9 | Always use cumulative probability |

## What's next

The normal distribution is a core probability topic, and mastery of this sub-topic is essential for solving combined probability questions on the IB AA SL exam. You will often see the normal distribution combined with other topics like conditional probability. Note that Bayes' theorem is AA HL only and is not required at SL, and that the normal approximation to the binomial, confidence intervals and hypothesis testing belong to Math AI, not AA, so they are not part of AA SL. Practice both probability and inverse normal calculations to ensure you can tackle any exam question on this frequently tested topic.

- [Calculus](https://www.owlsprep.com/study/ib-math-aa-sl-u5-overview/)
- [Introduction to limits and differentiation](https://www.owlsprep.com/study/ib-math-aa-sl-u5-introduction-to-limits-and-differentiation/)
- [Basic differentiation rules: power, product, quotient, chain rule](https://www.owlsprep.com/study/ib-math-aa-sl-u5-basic-differentiation-rules-power-product/)

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ib-math-aa-sl-u4-normal-probability-distribution/