Reading the cumulative z-table

◈ 7 cards

The table gives P(Z ≤ z); every other area comes by subtraction and symmetry — and an "area from 0 to z" table gives different cells.

What the cell means

The course's z-table is cumulative: the cell in row $z$ (to one decimal) and column $z$ (the second decimal) is

$Φ (z) = P (Z \leq z)$

the area under the standard normal curve to the left of $z$ . Rows run from $- 3.4$ to $3.4$ in steps of 0.01. Every probability you will ever need from the normal distribution is one of four shapes built from that one cell.

Worked example — the four shapes

Left tail. $P (Z \leq 1.28)$ : row 1.2, column 0.08 → $Φ (1.28) = 0.8997$ . That is the cell itself. (This is why $z_{0.10} = 1.28$ : about 10 % of the area lies to its right.)

Right tail. $P (Z > 0.67)$ : the table gives the left area, $Φ (0.67) = 0.7486$ , so

$P (Z > 0.67) = 1 - 0.7486 = 0.2514$

Between two values. $P (- 1.5 < Z < 0.75) = Φ (0.75) - Φ (- 1.5)$ . Row 0.7, column 0.05 gives $0.7734$ ; row $- 1.5$ , column 0.00 gives $0.0668$ ; the difference is $0.7066$ .

Negative z, by symmetry. If your table stops at 0, use $Φ (- z) = 1 - Φ (z)$ : $Φ (- 2.05) = 1 - Φ (2.05) = 1 - 0.9798 = 0.0202$ . Check it against the row $- 2.0$ , column 0.05 cell if your table has negative rows — same number.

Rounding before the lookup

A z computed from data — $z = (x - μ) / σ$ — is rounded to two decimals before the lookup, because the table has no third decimal. $z = 1.284$ reads row 1.2, column 0.08. Do not interpolate; the paper's answer key was built from the same table.

Three cells to memorise

You will read the table backwards as often as forwards: given an area, find the z. Three critical values come up in every confidence interval and test, and the paper expects you to know them without a lookup:

$z_{0.05} = 1.645$ — the table has $Φ (1.64) = 0.9495$ and $Φ (1.65) = 0.9505$ ; 0.95 sits exactly between, so the convention is the midpoint.
$z_{0.025} = 1.96$ — $Φ (1.96) = 0.9750$ .
$z_{0.005} = 2.576$ — $Φ (2.57) = 0.9949$ , $Φ (2.58) = 0.9951$ ; again the midpoint.

Why other books' tables disagree

Some textbooks print the area from 0 to z instead of the cumulative area: their cell at $z = 1.00$ is 0.3413, ours is 0.8413. The two differ by exactly 0.5 for positive $z$ . A lookup copied from such a book into a cumulative-table answer is wrong by 0.5 — the single most common z-table error on a first midterm.

Rows carry z to one decimal, columns the second decimal; the cell is Φ(z) = P(Z ≤ z). The highlighted pair straddles 0.95, so z₀.₀₅ is taken as 1.645.