# The Normal Distribution

> The bell curve, z-scores, the 68-95-99.7 rule, standard normal tables and finding probabilities.

From HSC Maths Advanced on WhizzNotes, free study notes for NSW students: https://whizznotes.com/mathematics-advanced/normal-distribution/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The milk bottle machine
A dairy machine fills bottles labelled 1 litre. It does not put in exactly 1000 mL every time. Most bottles have close to 1005 mL, a few have a bit more or less, and very few are far off. If you graph thousands of bottles, you get a smooth, symmetrical hill: the **bell curve**.

Heights, test scores, measurement errors and the weights of apples all form this shape. It is called the **normal distribution**, and knowing its rules lets the dairy say how many bottles will be under 1 litre without checking every single one.

**In short:** The normal distribution is a symmetric bell-shaped curve where the mean, median and mode are equal. About 68% of values lie within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3, and $z = \dfrac{x - \mu}{\sigma}$ counts how many standard deviations a value is from the mean.

## Starting point: Years 9 and 10

### Bell-shaped data

Data that is **symmetric** and **bell-shaped** has most values near the middle and fewer further out. For this shape, the mean, median and mode are all about equal.

### z-scores

A **z-score** tells you how many standard deviations a value is from the mean:

$$z = \frac{x - \mu}{\sigma}$$

- $z = 0$: exactly at the mean.
- $z = 2$: two standard deviations **above** the mean.
- $z = -1$: one standard deviation **below**.

### Worked example: Which test was better?
Mia scored 72 in Maths (mean 60, standard deviation 8) and 80 in English (mean 74, standard deviation 6).

Maths: $z = \dfrac{72 - 60}{8} = 1.5$. English: $z = \dfrac{80 - 74}{6} = 1$.

Compared with her class, her Maths result was better, even though the mark was lower.

### Key idea: z-scores compare different tests
A z-score removes the units and the size of the test. It tells you how unusual a result is.

### Watch out: Subtract the mean first
$z = \dfrac{x - \mu}{\sigma}$, not $\dfrac{x}{\sigma} - \mu$.

### Calculator tip: Brackets in z-scores
Type $z = \dfrac{x - \mu}{\sigma}$ as `( 78 - 65 ) ÷ 8`. Without the brackets the calculator divides 65 by 8 first. Keep the sign: a negative $z$ means below the mean.

**Mind map**

```text
Normal distribution
  Shape
    Bell, symmetric
    Mean = median = mode
  z-scores
    z = (x - μ)/σ
    Compare results
  Empirical rule
    68% within 1σ
    95% within 2σ
    99.7% within 3σ
  Standard normal
    Z with mean 0, sd 1
    Tables give P(Z ≤ z)
  Uses
    Probabilities
    Find a cut-off
```

### Year 11: The 68-95-99.7 rule
For a normal distribution:
- about **68%** of values are within 1 standard deviation of the mean,
- about **95%** are within 2,
- about **99.7%** are within 3.

*Diagram: Normal curve with the bands within 1, 2 and 3 standard deviations labelled 68, 95 and 99.7 percent*
*Middle band 68%, two bands each side 95%, three bands 99.7%.*

**Example.** Bottles have $\mu = 1005$ mL and $\sigma = 2.5$ mL. 1000 mL is $z = -2$. About 95% of bottles are within 2 standard deviations, so 5% are outside, and half of that, **2.5%**, are below 1000 mL.

**Uneven intervals.** Split at the mean and use halves of the rule: 34% from the mean to 1 sd, 47.5% to 2 sd, 49.85% to 3 sd. With mean 50 and sd 5, the percentage between 45 and 65 ($-1$ to $+3$ sd) is $34 + 49.85 = 83.85\%$.

**Working backwards.** If 95% of values lie between 40 and 60, the mean is the midpoint 50 and 10 is two standard deviations, so $\sigma = 5$. If 2.5% are below 100 (so $z = -2$) and 16% are above 130 ($z = 1$), then $\mu - 2\sigma = 100$ and $\mu + \sigma = 130$, giving $\sigma = 10$, $\mu = 120$.

**Rescaling marks.** Keep the z-score. A mark of 70 on a test with mean 60 and sd 5 has $z = 2$. Rescaled to mean 65 and sd 10, it becomes $65 + 2 \times 10 = 85$.

### Year 12: The standard normal distribution
Any normal variable $X \sim N(\mu, \sigma^2)$ becomes the **standard normal** $Z \sim N(0, 1)$ using $z = \dfrac{x - \mu}{\sigma}$.

Its density is $f(x) = \dfrac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x - \mu)^2}{2\sigma^2}}$. You never integrate this by hand. Instead use a **table** of $P(Z \leq z)$, or a calculator.

**Using the table.** Some values: $P(Z \leq 1) \approx 0.8413$, $P(Z \leq 1.5) \approx 0.9332$, $P(Z \leq 2) \approx 0.9772$.

- $P(Z > z) = 1 - P(Z \leq z)$
- $P(Z \leq -z) = 1 - P(Z \leq z)$ by symmetry
- $P(a \leq Z \leq b) = P(Z \leq b) - P(Z \leq a)$

**Example.** Heights are normal with $\mu = 170$ cm and $\sigma = 8$ cm. Find $P(162 < X < 182)$.

$z$-values: $\dfrac{162 - 170}{8} = -1$ and $\dfrac{182 - 170}{8} = 1.5$.

$$P(-1 < Z < 1.5) = 0.9332 - (1 - 0.8413) = 0.7745$$

**Working backwards.** The top 2.28% of heights are above what value? $P(Z > 2) \approx 0.0228$, so $x = 170 + 2 \times 8 = 186$ cm.

**Symmetry tricks.** If $P(Z < N) = 0.7$, then $P(Z > N) = 0.3$, and by symmetry $P(Z < -N) = 0.3$. So $P(-N < Z < N) = 0.7 - 0.3 = 0.4$.

**A transformed normal.** If $Y = 100 - 3X$ and you need $P(Y > 40)$, rewrite it in $X$: $100 - 3X > 40$ means $X < 20$. Then find $P(X < 20)$.

**Conditional normal probabilities.** $P(X > 26 \mid X > 22) = \dfrac{P(X > 26)}{P(X > 22)}$. With mean 18 and sd 4 these are $z > 2$ (2.5%) and $z > 1$ (16%), giving about $\dfrac{2.5}{16} \approx 16\%$.

**From a proportion to a count.** Multiply by the population. If 13.5% of 20 million people lie between two ages, that is about 2.7 million people.

## Quick summary

- Normal data is bell-shaped and symmetric. Mean, median and mode are equal.
- $z = \dfrac{x - \mu}{\sigma}$ counts standard deviations from the mean.
- 68% within 1 sd, 95% within 2 sd, 99.7% within 3 sd.
- Convert to $Z \sim N(0, 1)$ and use $P(Z \leq z)$ tables, with symmetry and $1 -$ for the other side.
- Work backwards from a probability to $z$, then $x = \mu + z\sigma$.

## Quiz

Try these questions from the chapter quiz.

1. For a symmetric bell-shaped distribution, the mean and median are:
   - A. very different
   - B. about equal
   - C. always 0
   - D. always 1
   - Answer: B. about equal. Symmetry puts them both in the middle.

2. A score of 85 has mean 70 and standard deviation 5. Its z-score is:
   - A. 15
   - B. 3
   - C. 17
   - D. 0.3
   - Answer: B. 3. $\dfrac{85 - 70}{5} = 3$.

3. A z-score of $-2$ means the value is:
   - A. 2 above the mean
   - B. 2 standard deviations below the mean
   - C. the lowest possible
   - D. an error
   - Answer: B. 2 standard deviations below the mean. Negative means below.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/mathematics-advanced/normal-distribution/#quiz