A z-score of $-2$ means the value is:

2 standard deviations below the mean

The Normal Distribution | HSC Maths Advanced

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 $z = \frac{x - μ}{σ}$ 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:

$z = \frac{x - μ}{σ}$

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

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 11The 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.

Middle band 68%, two bands each side 95%, three bands 99.7%.

Example. Bottles have $μ = 1005$ mL and $σ = 2.5$ 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 $σ = 5$ . If 2.5% are below 100 (so $z = - 2$ ) and 16% are above 130 ( $z = 1$ ), then $μ - 2 σ = 100$ and $μ + σ = 130$ , giving $σ = 10$ , $μ = 120$ .

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 12The standard normal distribution

Any normal variable $X \sim N (μ, σ^{2})$ becomes the standard normal $Z \sim N (0, 1)$ using $z = \frac{x - μ}{σ}$ .

Its density is $f (x) = \frac{1}{σ 2 π} e^{- \frac{( x - μ ) ^{2}}{2 σ ^{2}}}$ . 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 $μ = 170$ cm and $σ = 8$ cm. Find $P (162 < X < 182)$ .

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

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

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

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

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.
$z = \frac{x - μ}{σ}$ 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 $x = μ + z σ$ .

Quiz

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For a symmetric bell-shaped distribution, the mean and median are:
1. A very different
2. B about equal
3. C always 0
4. D always 1
Show answer
B: about equal
Symmetry puts them both in the middle.
A score of 85 has mean 70 and standard deviation 5. Its z-score is:
1. A 15
2. B 3
3. C 17
4. D 0.3
Show answer
B: 3
$\frac{85 - 70}{5} = 3$ .
A z-score of $- 2$ means the value is:
1. A 2 above the mean
2. B 2 standard deviations below the mean
3. C the lowest possible
4. D an error
Show answer
B: 2 standard deviations below the mean
Negative means below.

The Normal Distribution

Starting point: Years 9 and 10

Bell-shaped data

z-scores

Quick summary

Quiz

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