A z-table lists probabilities for the standard normal distribution at tabulated z-values. This page provides four such tables: cumulative for positive z, cumulative for negative z, complementary cumulative for positive z, and complementary cumulative for negative z, each accurate to four decimal places.

How do I look up a probability in a z-table?

Locate the row matching the first decimal of your z-score, then move across to the column matching the second decimal. The cell value is the probability. For z = 1.96 in the cumulative-positive table, row 1.9 and the column for the second decimal 6 give about 0.9750.

What is the difference between cumulative and complementary cumulative?

Cumulative tables give P(Z less than or equal to z), the area to the left of z. Complementary cumulative tables give P(Z greater than z), the area to the right. The two are exact complements: P(Z greater than z) equals 1 minus P(Z less than or equal to z).

How do I find P(Z greater than z) without a complementary table?

Use the relationship P(Z greater than z) equals 1 minus P(Z less than or equal to z). Look up P(Z less than or equal to z) in the cumulative-positive table and subtract from 1. The complementary cumulative tables on this page do that subtraction for you.

Normal Distribution Z-Score Table

Getting Started with the Table

This page hosts four standard-normal z-tables selected from a dropdown above. The table area is empty until you pick a type.

Quick orientation:

Type of Z-table dropdown to choose between four configurations
$z$ -values across one half of the standard-normal range in $0.1$ row increments
$\mu = 0$ and standard deviation $\sigma = 1$

Choosing a Z-Table Type

The dropdown offers four configurations covering the two probability directions and the two halves of the

z

-axis:

Cumulative (less than Z) Positive Values — Gives $P(Z \leq z)$ for $z$ from $0$ to $4.1$
Cumulative (less than Z) Negative Values — Gives $P(Z \leq z)$ for $z$ from $-4.0$ to $0$
Complementary Cumulative (greater than Z) for Positive Values — Gives $P(Z > z)$ for $z$ from $0$ to $4.1$
Complementary Cumulative (greater than Z) for Negative Values — Gives $P(Z > z)$ for $z$ from $-4.1$ to $0$

Pick the table that matches the direction of the inequality in your question and the sign of your

z

-value.

Looking Up a Probability

Standard z-tables organize values by the first decimal of

z

along the row axis and the second decimal along the columns. To look up

P(Z \leq 1.96)

in the cumulative-positive table:

$1.9$
$6$
$0.9750$

The result is the cumulative probability — the area to the left of

z

under the standard normal curve. For

z

-values not exactly tabulated, interpolate between the two nearest entries or fall back to a calculator for exact precision.

Working with Negative Z-Values

For negative

z

-values, use the dedicated negative tables rather than computing from the positive tables each time. The Cumulative Negative Values table gives

P(Z \leq -z)

directly for

z

from

-4.0

to

0

.

You can also derive negative-value probabilities from positive ones using the symmetry of the standard normal:

P(Z \leq -z) = 1 - P(Z \leq z)

Both approaches give the same numerical result. The negative table is faster for lookups; the symmetry relation is handy when you only have a positive-value table on hand or want to double-check a value.

Cumulative vs Complementary Cumulative

The four tables split along two axes — direction of inequality and sign of

z

:

Cumulative ( $P(Z \leq z)$ ) gives the area to the left of $z$
Complementary cumulative ( $P(Z > z)$ ) gives the area to the right of $z$

The two are exact complements:

P(Z > z) = 1 - P(Z \leq z)

. Use cumulative tables for "at most" or "less than" questions, and complementary cumulative tables for "more than" or "exceeds" questions.

For two-sided questions like

P(|Z| < z)

, combine the two:

P(|Z| < z) = 2 \cdot P(Z \leq z) - 1

.

What Is the Standard Normal Distribution?

The standard normal distribution is the normal distribution with mean

\mu = 0

and standard deviation

\sigma = 1

. It is symmetric around

0

with the classic bell shape, and the total area under its density curve is

1

.

Its probability density function is:

\varphi(z) = \dfrac{1}{\sqrt{2\pi}}\, e^{-z^2/2}

and its cumulative distribution function

\Phi(z) = P(Z \leq z)

is exactly what the cumulative z-tables on this page list to four decimal places.

Any normal distribution

N(\mu, \sigma^2)

can be converted to standard normal via the

z

-score transformation, so a single set of standard tables answers questions for every normal distribution.

For a full treatment, see the normal distribution page.

Understanding Z-Scores

A $z$-score measures how many standard deviations a value

x

lies from the mean

\mu

of a normal distribution with standard deviation

\sigma

:

z = \dfrac{x - \mu}{\sigma}

Positive

z

-scores are above the mean, negative below. A

z

-score of

0

is exactly at the mean.

Key reference values from the cumulative table:

$z = 1.00$ gives $P(Z \leq z) \approx 0.8413$
$z = 1.645$ gives $P(Z \leq z) \approx 0.9500$ (the $95$ th percentile)
$z = 1.96$ gives $P(Z \leq z) \approx 0.9750$ (basis of $95\%$ confidence intervals)
$z = 2.33$ gives $P(Z \leq z) \approx 0.9901$

For more on standardization, see the z-score page.

The Empirical Rule

For any normal distribution, the empirical rule (also called the

68

-

95

-

99.7

rule) says:

$68\%$ of values lie within $1$ standard deviation of the mean: $\mu \pm \sigma$
$95\%$ lie within $2$ standard deviations: $\mu \pm 2\sigma$
$99.7\%$ lie within $3$ standard deviations: $\mu \pm 3\sigma$

Expressed via

z

-scores:

P(-1 \leq Z \leq 1) \approx 0.68

,

P(-2 \leq Z \leq 2) \approx 0.95

, and

P(-3 \leq Z \leq 3) \approx 0.997

.

Most observed data points fall within

\pm 3\sigma

, which is why z-tables typically only need to cover

z

-values up to about

\pm 4

.

For more, see the empirical rule page.

Common Applications

Z-tables are central to:

Hypothesis testing — Computing $p$ -values and critical values for $z$ -tests
Confidence intervals — Looking up critical $z$ -values like $1.96$ for $95\%$ confidence
Percentile calculations — Converting a $z$ -score to a percentile rank, or vice versa
Quality control — Determining process capability and defect rates from $z$ -scores
Standardized testing — Converting raw scores to percentile ranks
Risk modeling — Computing tail probabilities for returns assumed normal

Wherever a continuous variable is well-approximated by a normal distribution and a probability question arises, a z-table gives the answer to four decimal places without computation.

Related Tables and Tools

Related Tables:

Binomial Distribution Table - Cumulative binomial probabilities; the binomial converges to a normal distribution for large

n

.

Poisson Distribution Table - Cumulative Poisson probabilities; approximated by the normal for large means.

t Distribution Table - Critical values for small-sample inference; converges to standard normal as degrees of freedom grow.

Related Tools:

Normal Distribution Calculator - Returns exact

P(Z \leq z)

for any

z

without lookup or interpolation.

Z-Score Calculator - Converts a raw value

x

to a

z

-score given

\mu

and

\sigma

.

Related Concepts:

Central Limit Theorem - Why the normal distribution shows up so widely across applied statistics.

Empirical Rule - The

68

-

95

-

99.7

summary of normal-distribution spread.

Normal Distribution (Z-Score) Table