# Normal distribution and applications

> IB Mathematics AI SL · Statistics and Probability
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u4-normal-distribution-and-applications/

This sub-topic covers properties of the normal distribution, a widely used symmetric continuous probability distribution for modeling real-world phenomena. You will learn to calculate probabilities and solve applied problems with inverse normal techniques.

**Prerequisites:** [Continuous probability distributions](https://www.owlsprep.com/study/ib-math-ai-sl-u4-continuous-probability/); [Z-scores and standardization](https://www.owlsprep.com/study/ib-math-ai-sl-u4-z-scores/)

## Learning objectives

- Identify key properties of the normal distribution curve
- Calculate cumulative probabilities for normally distributed variables using GDC
- Use inverse normal to find percentiles for real-world problems
- Apply normal distribution to solve applied probability problems

## Properties of the Normal Distribution

**Normal Distribution** — A symmetric, bell-shaped continuous probability distribution centered at the population mean $\mu$, with spread determined by the variance $\sigma^2$. The total area under the curve equals 1.

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

*Example:* Heights of adults, IQ scores, manufacturing measurement errors

The normal distribution is symmetric around its center, so **mean = median = mode** for a perfectly normal distribution. The empirical rule (68-95-99.7 rule) describes how much data falls within a given number of standard deviations from the mean.

> **Empirical Rule Mnemonic**
>
> Remember: 68% within 1σ, 95% within 2σ, 99.7% within 3σ. Just repeat 68-95-99.7 to remember the values easily.

**Worked example:** A variable $X$ follows a normal distribution $X \sim N(50, 10^2)$. What percentage of data lies between 40 and 60?

1. Identify the parameters of the distribution:
2. $$\mu = 50, \quad \sigma = 10$$
3. Rewrite the bounds in terms of mean and standard deviation:
4. $$40 = 50 - 10 = \mu - \sigma, \quad 60 = 50 + 10 = \mu + \sigma$$
5. By the empirical rule, 68% of all data lies within 1 standard deviation of the mean.
6. Final answer: $68\%$

## Calculating Normal Probabilities

For IB AI SL, you will always use your graphing display calculator (GDC) to find normal probabilities, rather than manual calculation or z-tables. We find the area under the curve between any two bounds, which corresponds to the probability of the variable falling in that interval.

**Cumulative Normal Probability** — The probability that a normally distributed variable $X$ is less than or equal to a given value $x$, written $P(X \leq x)$.

For any interval, we use the following rules:
- $P(X > a) = 1 - P(X \leq a)$ (complement rule for upper tails)
- $P(a < X < b) = P(X < b) - P(X < a)$ for intervals between two values

**Worked example:** Given $X \sim N(25, 3^2)$, find $P(X < 28)$.

1. Extract parameters: mean $\mu = 25$, standard deviation $\sigma = 3$ (note the input to GDC is standard deviation, not variance)
2. We need the cumulative probability from the lower bound (usually set to -1E99 for negative infinity on GDC) up to 28.
3. Input the values into the GDC normal cumulative distribution function.
4. Final result:
5. $$P(X < 28) \approx 0.841$$

*Calculator:* allowed

## Inverse Normal Calculations

Inverse normal is the reverse process of calculating a probability: we know the cumulative probability (percentile) and need to find the corresponding value of $X$. This is a very common exam question, especially in applied contexts.

**Inverse Normal** — Given a cumulative probability $p = P(X \leq x)$, inverse normal returns the value $x$ that is the $p$-th percentile of the distribution.

> **tip**
>
> Always draw a quick sketch of the normal curve to confirm which tail you need, this avoids mistakes with cumulative probabilities.

**Worked example:** The masses of apples are normally distributed with mean 150g and standard deviation 10g. Find the minimum mass of the heaviest 10% of apples.

1. Define the distribution: Let $M$ = mass of an apple, so $M \sim N(150, 10^2)$
2. We need the mass $m$ such that $P(M > m) = 0.10$. Convert this to a cumulative probability (area to the left):
3. $$P(M \leq m) = 1 - 0.10 = 0.90$$
4. Use the inverse normal function on your GDC with $p=0.90$, $\mu=150$, $\sigma=10$
5. Result: $m \approx 162.8$g. Rounded to a reasonable value, the minimum mass is 163g.

*Calculator:* allowed

## Real-World Applications

The normal distribution is one of the most widely used probability distributions in statistics. It models many naturally occurring continuous variables, as well as measurement errors, test scores, and variation in manufacturing processes. For exam questions, you will always be told if a variable can be modeled with a normal distribution.

**Worked example:** Scores on an IQ test are normally distributed with mean 100 and standard deviation 15. What is the probability that a randomly selected person has an IQ between 90 and 110?

1. Define the distribution: $X \sim N(100, 15^2)$
2. We need $P(90 < X < 110) = P(X < 110) - P(X < 90)$
3. Calculate each cumulative probability with GDC: $P(X < 110) \approx 0.7475$, $P(X < 90) \approx 0.2525$
4. Subtract to find the interval probability:
5. $$P(90 < X < 110) = 0.7475 - 0.2525 = 0.495$$

## Common pitfalls

- **Wrong:** Inputting variance instead of standard deviation into the GDC.
  - Why it fails: Normal distribution notation $N(\mu, \sigma^2)$ lists variance as the second parameter, but most GDCs ask for standard deviation.
  - Correct: Always take the square root of the variance to get standard deviation before inputting to your calculator.
- **Wrong:** Using the given percentage directly as the cumulative probability for upper tail inverse normal problems.
  - Why it fails: If asked for the top 10% of values, the cumulative area to the left is 90%, not 10%.
  - Correct: Always draw a sketch of the curve, subtract the upper tail percentage from 1 to get the correct cumulative probability for inverse normal.
- **Wrong:** Assuming any continuous variable is normally distributed.
  - Why it fails: Only symmetric, bell-shaped variables are normal. Skewed variables cannot be modeled with the normal distribution.
  - Correct: Never use normal distribution methods unless the question explicitly states the variable is normally distributed.
- **Wrong:** Mixing up $P(X < a)$ and $P(X > a)$ when calculating probabilities.
  - Why it fails: It is easy to forget the complement rule for upper tails.
  - Correct: Shade the required area on a quick sketch, then use $P(X > a) = 1 - P(X \leq a)$ for upper tail probabilities.

## Cheatsheet

| Concept | Notation/Rule | Use Case |
| --- | --- | --- |
| Normal distribution | $X \sim N(\mu, \sigma^2)$ | Model symmetric bell-shaped continuous data |
| Empirical rule | 68% within $\mu \pm 1\sigma$, 95% within $\mu \pm 2\sigma$, 99.7% within $\mu \pm 3\sigma$ | Quick probability estimates |
| Cumulative probability | $P(a < X < b) = P(X < b) - P(X < a)$ | Find probability given a value of $X$ |
| Inverse normal | Given $P(X \leq x) = p$, find $x$ | Find value given a probability/percentile |
| Upper tail probability | $P(X > a) = 1 - P(X \leq a)$ | Calculate area above a given value |

## What's next

Normal distribution is the foundation of inferential statistics, which is the largest remaining topic in the IB AI SL statistics unit. The skills you learned here for calculating probabilities and using inverse normal will be directly used when you construct confidence intervals and run hypothesis tests for population means. These topics are heavily assessed in both Paper 1 and Paper 2, so mastering normal distribution now will make these more advanced topics much easier to understand. You can also connect what you learned here to binomial distributions, which is another key probability distribution you have studied in this unit.

- [Binomial Distribution](https://www.owlsprep.com/study/ib-math-ai-sl-u4-binomial-distribution/)
- [Calculus](https://www.owlsprep.com/study/ib-math-ai-sl-u5-overview/)
- [Derivative as gradient of tangent and rate of change](https://www.owlsprep.com/study/ib-math-ai-sl-u5-derivative-as-gradient-of-tangent/)

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