Reading the Z-Table

The area to the left

Areas under the standard normal curve have no simple formula, so they are printed in a table. A z-table gives, for each z, the area under the curve to the left of z: the probability that Z is less than z. It is written $\Phi \left(z\right)$Φ(z), so $\Phi \left(z\right)=P\left(Z<z\right)$Φ(z) = P(Z < z).

The whole area is 1 and the curve is symmetric about 0, so $\Phi \left(0\right)=0.5$Φ(0) = 0.5. As z grows, more of the curve lies to its left, and $\Phi \left(z\right)$Φ(z) climbs toward 1.

The area to the left of z = 1.96 is shaded. It is $\Phi \left(1.96\right)=0.9750$Φ(1.96) = 0.9750, everything below 1.96.

Rows and columns

A printed table gives z to two decimal places. The rows give z to one decimal place, and the columns give its second decimal place. To find $\Phi \left(1.96\right)$Φ(1.96), go down to the row for 1.9, then across to the column for 0.06, since 1.9 + 0.06 = 1.96. The entry there is 0.9750.

The same way, $\Phi \left(1.5\right)$Φ(1.5) is in the row for 1.5 and the column for 0.00: 0.9332. And $\Phi \left(1\right)=0.8413$Φ(1) = 0.8413, so about 84% of the area lies to the left of z = 1.

Part of a z-table. The row for 1.9 and the column for 0.06 meet at 0.9750, the area to the left of z = 1.96.

Reading the table backward

Often the area is known and z is wanted. Which z has 5% of the area above it, in the upper tail? The table gives areas to the left, so first subtract the tail from 1: 95% lies to the left, and the area to look up is 0.9500.

Look for 0.9500 among the entries. It is not there exactly: in the row for 1.6, the column for 0.04 gives 0.9495 and the column for 0.05 gives 0.9505. 0.9500 is halfway between them, so z is halfway between 1.64 and 1.65: z = 1.645.

The usual mistake is to look up the tail itself, 0.0500. That is the area to the left of a negative z, far down the lower tail.

In the row for 1.6, the entries for 1.64 and 1.65 are 0.9495 and 0.9505. The area 0.9500 is halfway between them, at z = 1.645.

The upper 5% of the area, beyond z = 1.645. The other 95% lies to its left.

Where the familiar values come from

The values used again and again in statistics are all read from the table this way. For 2.5% in the upper tail, look up 1 − 0.025 = 0.9750; it is in the table exactly, at z = 1.96. A two-tailed 5% level puts 2.5% in each tail, so it uses 1.96 as well, not 1.645.

For 10% in the upper tail, look up 0.9000. The entry for 1.28 is 0.8997 and the entry for 1.29 is 0.9015, so 0.9000 is just past 1.28: z = 1.28 to two decimal places, and 1.2816 more exactly. For 1%, look up 0.9900: 2.32 gives 0.9898 and 2.33 gives 0.9901, so z = 2.33, more exactly 2.3263. For 0.5%, look up 0.9950: 2.57 gives 0.9949 and 2.58 gives 0.9951, so z is about 2.575, written 2.58, and more exactly 2.5758.

Negative z by symmetry

Most tables list only positive z. The curve is symmetric about 0, so the area to the left of −z is the same as the area to the right of +z, which is $1-\Phi \left(z\right)$1 − Φ(z). So $\Phi \left(-z\right)=1-\Phi \left(z\right)$Φ(−z) = 1 − Φ(z).

For example, $\Phi \left(-1\right)=1-\Phi \left(1\right)=1-0.8413=0.1587$Φ(−1) = 1 − Φ(1) = 1 − 0.8413 = 0.1587. Reading $\Phi \left(-1\right)$Φ(−1) as 0.8413 is the slip to avoid: 0.8413 is the area to the left of +1. And $\Phi \left(-1.645\right)=1-0.9500=0.05$Φ(−1.645) = 1 − 0.9500 = 0.05, so −1.645 cuts off the lowest 5% of the area, just as 1.645 cuts off the highest 5%.

The lowest 5% of the area, to the left of z = −1.645: the mirror image of the upper tail beyond 1.645.