Mathematics 9709 · Probability & Statistics 1

The standard normal & z-table

A normal distribution is like a histogram that has become a smooth bell curve.
Probability is area under the curve.
The total area is $1$ .
Every normal curve can be converted to the master curve $Z\sim N(0,1)$ .
Then every area comes from one table: $\Phi$ .

Φ(z) = area to the LEFT of z

Φ(z) is always the area to the LEFT. Right tail, negative z, and “between” are all this one number plus a little arithmetic.

The curve is symmetric about the mean, so $P(Z<0)=0.5$ . The table gives $\Phi(z)$ , the area to the left of $z$ ; for a right tail use $1-\Phi(z)$ , and for a negative boundary use symmetry, $\Phi(-z)=1-\Phi(z)$ . That is the whole table.

Drag $z$ : the shaded area to its left is $\Phi(z)=P(Z<z)$ , and the right tail is the rest.

z = 1.00·Φ(z) = P(Z < z) = 0.8413·right tail 0.1587

Turn any area into a single table lookup:

Sketch the bell, mark 0, and shade the region you want.
Left of a positive $z$ → read $\Phi(z)$ straight off.
Right tail → $1-\Phi(z)$ .
Negative $z$ → use symmetry, $\Phi(-z)=1-\Phi(z)$ .
Between $a$ and $b$ → $\Phi(z_b)-\Phi(z_a)$ .

You want	z is	Φ expression
$P(Z<z),\ z>0$	positive	$\Phi(z)$
$P(Z>z),\ z>0$	positive	$1-\Phi(z)$
$P(Z<z),\ z<0$	negative	$1-\Phi(\|z\|)$
$P(a<Z<b)$	either	$\Phi(z_b)-\Phi(z_a)$

Worked example

Simplest rung: one tail, straight off the table

$Z\sim N(0,1)$ . Find $P(Z<1.23)$ and $P(Z\ge1.23)$ — the most basic lookup there is.

Left of a positive $z$ : $P(Z<1.23)=\Phi(1.23)=0.8907$ .
The right tail is the leftover: $P(Z\ge1.23)=1-0.8907=0.1093$ .

Answer

0.8907\quad\text{and}\quad0.1093

Worked example

A negative z, by symmetry

$P(Z<-1.2)=1-\Phi(1.2)=1-0.8849$ ; the other side mirrors it, $P(Z>-1.2)=\Phi(1.2)$ .

Answer

0.1151\quad\text{and}\quad0.8849

Worked example

Between, straddling zero

$P(-0.5<Z<1.8)=\Phi(1.8)-\Phi(-0.5)=\Phi(1.8)+\Phi(0.5)-1=0.9641+0.6915-1$ . This is exactly the shape a “between” question collapses to after standardising. (9709/52/M/J/23 Q5)

Answer

0.6556

Run the table backwards: a probability fixes a z

The same table also works backwards.
Given an area, hunt the body of the table for the closest $\Phi$ value.
Then read off the matching $z$ .
This reverse lookup is $\Phi^{-1}$ .
It is the gateway to every “find $\mu$ or $\sigma$ ” question in §8.3.

Worked example

Inverse: read the table backwards

Find $a$ with $P(Z<a)=0.9072$ . The closest body value is $0.9066=\Phi(1.32)$ ; the leftover $0.0006$ is “ADD 6”, a 4th figure of 4, so $a=\Phi^{-1}(0.9072)=1.324$ .

Answer

a=1.324

Worked example

Inverse with a NEGATIVE answer

Find $b$ with $P(Z\ge b)=0.7713$ . The tail to the right is bigger than 0.5, so the boundary sits below 0 and $b$ is negative. Work with the mirror: $\Phi^{-1}(0.7713)=0.743$ , then put the sign back: $b=-0.743$ .

Answer

b=-0.743

Examiner note

Always sketch and shade before looking up — the sketch is what tells you (and the examiner) whether you need

\Phi

1-\Phi

, and whether an inverse

z

is positive or negative. Keep

z

to 4 significant figures and round only the final probability.

Common mistake

N(\mu,\sigma^2)

holds the variance second:

N(20,9)

means

\sigma=3

, not 9 — the biggest silent error in the chapter. And

\Phi

is always the area to the left.

Common mistake

Treating

P(X<7)

and

P(X\le7)

as different and “adjusting” the boundary. For a continuous variable a single value has zero probability — they are equal. (The

\pm0.5

nudge belongs only to the discrete approximation in §8.4.)

Why one table works for every bell

Subtracting $\mu$ slides any bell so its centre sits at 0; dividing by $\sigma$ rescales its width to 1. That maps any $N(\mu,\sigma^2)$ exactly onto $N(0,1)$ — so one area table $\Phi$ serves them all. Standardising (§8.2) is just that map.

PreviousGeometric mean & mode NextStandardising: z = (x − μ)/σ

The standard normal & z-table

A normal distribution is like a histogram that has become a smooth bell curve.
Probability is area under the curve.
The total area is $1$ .
Every normal curve can be converted to the master curve $Z\sim N(0,1)$ .
Then every area comes from one table: $\Phi$ .

Φ(z) = area to the LEFT of z

Φ(z) is always the area to the LEFT. Right tail, negative z, and “between” are all this one number plus a little arithmetic.

Drag $z$ : the shaded area to its left is $\Phi(z)=P(Z<z)$ , and the right tail is the rest.

z = 1.00·Φ(z) = P(Z < z) = 0.8413·right tail 0.1587

Turn any area into a single table lookup:

Sketch the bell, mark 0, and shade the region you want.
Left of a positive $z$ → read $\Phi(z)$ straight off.
Right tail → $1-\Phi(z)$ .
Negative $z$ → use symmetry, $\Phi(-z)=1-\Phi(z)$ .
Between $a$ and $b$ → $\Phi(z_b)-\Phi(z_a)$ .

You want	z is	Φ expression
$P(Z<z),\ z>0$	positive	$\Phi(z)$
$P(Z>z),\ z>0$	positive	$1-\Phi(z)$
$P(Z<z),\ z<0$	negative	$1-\Phi(\|z\|)$
$P(a<Z<b)$	either	$\Phi(z_b)-\Phi(z_a)$

Worked example

Simplest rung: one tail, straight off the table

$Z\sim N(0,1)$ . Find $P(Z<1.23)$ and $P(Z\ge1.23)$ — the most basic lookup there is.

Left of a positive $z$ : $P(Z<1.23)=\Phi(1.23)=0.8907$ .
The right tail is the leftover: $P(Z\ge1.23)=1-0.8907=0.1093$ .

Answer

0.8907\quad\text{and}\quad0.1093

Worked example

A negative z, by symmetry

$P(Z<-1.2)=1-\Phi(1.2)=1-0.8849$ ; the other side mirrors it, $P(Z>-1.2)=\Phi(1.2)$ .

Answer

0.1151\quad\text{and}\quad0.8849

Worked example

Between, straddling zero

$P(-0.5<Z<1.8)=\Phi(1.8)-\Phi(-0.5)=\Phi(1.8)+\Phi(0.5)-1=0.9641+0.6915-1$ . This is exactly the shape a “between” question collapses to after standardising. (9709/52/M/J/23 Q5)

Answer

0.6556

Run the table backwards: a probability fixes a z

The same table also works backwards.
Given an area, hunt the body of the table for the closest $\Phi$ value.
Then read off the matching $z$ .
This reverse lookup is $\Phi^{-1}$ .
It is the gateway to every “find $\mu$ or $\sigma$ ” question in §8.3.

Worked example

Inverse: read the table backwards

Find $a$ with $P(Z<a)=0.9072$ . The closest body value is $0.9066=\Phi(1.32)$ ; the leftover $0.0006$ is “ADD 6”, a 4th figure of 4, so $a=\Phi^{-1}(0.9072)=1.324$ .

Answer

a=1.324

Worked example

Inverse with a NEGATIVE answer

Answer

b=-0.743

Examiner note

Always sketch and shade before looking up — the sketch is what tells you (and the examiner) whether you need

\Phi

1-\Phi

, and whether an inverse

z

is positive or negative. Keep

z

to 4 significant figures and round only the final probability.

Common mistake

N(\mu,\sigma^2)

holds the variance second:

N(20,9)

means

\sigma=3

, not 9 — the biggest silent error in the chapter. And

\Phi

is always the area to the left.

Common mistake

Treating

P(X<7)

and

P(X\le7)

as different and “adjusting” the boundary. For a continuous variable a single value has zero probability — they are equal. (The

\pm0.5

nudge belongs only to the discrete approximation in §8.4.)

Why one table works for every bell