How do I read a z-table, and why does this replace it?

A z-table lists Φ(z), the area to the left of z, for z in steps of 0.01. Enter any z here and the page gives that area to six decimals, plus the right tail, the central band and the two-tailed probability, which tables make you work out by hand.

What is the probability of z greater than 2?

1 − Φ(2) = 1 − 0.9772 = 0.0228, about 2.3%. Beyond ±2 in both directions it is 4.6%, the '95% within two standard deviations' of the empirical rule read from the other side.

What z-score gives 95%, 99% or 90%?

For a central 95% band, ±1.96; for 99%, ±2.576; for 90%, ±1.645. For a single tail — 95% below — z = 1.645, and 99% below is 2.326. Enter those z values here to see the areas come out.

How is the CDF computed without a table?

By the Abramowitz & Stegun 26.2.17 polynomial: a rational function of |z| multiplied by the normal density, accurate to 7.5 × 10⁻⁸. Statistical software uses more precise routines, but for anything a table would answer this is exact to every shown digit.

How do I find the area between two z-scores?

Subtract the smaller Φ from the larger. Between z = −1 and z = 2: 0.9772 − 0.1587 = 0.8185, about 82% of a normal distribution.

How do I get a p-value from a z-score?

For a two-tailed test, read the two-tailed line: z = 2.1 gives P(|Z| > 2.1) ≈ 0.0357, significant at the 5% level but not at 1%. For a one-tailed test, use the right tail.

Normal Distribution Calculator (Z-Table, CDF and Two-Tailed Probability)

The probability below, above, between and beyond a z-score on the standard normal curve — a z-table without the table.

Inputs

Z-score

-1010

Include CalculateEveryday mark

P(Z ≤ z)

0.975002

Z: 1.96
P(Z > z), the right tail: 0.024998
P(−1.96 < Z < 1.96): 0.950004
Two-tailed: P(|Z| > 1.96): 0.049996
Percentile: 97.50%

Use this calculator on your own site

Free embed

Two lines of HTML, unlimited views
Shows a small CalculateEveryday bar, with a credit link under the calculator

What each Normal Distribution input means

Z-score: A z-score — any number, negative for below the mean. To use a raw value, convert it first with z = (x − μ) ÷ σ, or use the z-score calculator.

Normal Distribution formula

P(Z ≤ z) = Φ(z), the area under the standard normal curve to the left of z · P(Z > z) = 1 − Φ(z) · P(−z < Z < z) = Φ(z) − Φ(−z) · two-tailed P(|Z| > z) = 2 × (1 − Φ(z))

\Phi(z) = \int_{-\infty}^{z} \frac{1}{\sqrt{2\pi}} e^{-t^{2}/2}\, dt

The standard normal distribution is the bell curve with mean 0 and standard deviation 1, and its cumulative distribution function Φ(z) is the area under it to the left of z — the probability that a standard normal value is at most z, which is what a z-table lists. There is no closed formula for that area, so the page uses the Abramowitz & Stegun polynomial approximation, accurate to seven decimal places, well beyond any printed table. From the left-tail area everything else follows: the right tail is 1 − Φ(z), the central band between −z and z is Φ(z) − Φ(−z), and the two-tailed probability of being further out than ±z is what remains. Those are the numbers behind hypothesis tests: a two-tailed 5% test rejects at |z| > 1.96 because the area beyond ±1.96 is 0.05.

Normal Distribution worked example

z = 1.96: Φ(1.96) = 0.975, so 97.5% of standard normal values are below 1.96 and 2.5% are above. Between −1.96 and +1.96 lies 0.975 − 0.025 = 0.95, which is why ±1.96 standard errors is a 95% confidence interval and why a two-tailed test at the 5% level uses 1.96 as its cut-off. For z = 1, Φ = 0.8413, and the central band is 0.6827 — the 68% of the empirical rule.

More Normal Distribution examples

z = −1.5: Φ(−1.5) = 0.066807, so about 6.7% of a normal distribution lies more than 1.5 standard deviations below the mean, and 86.6% lies within ±1.5.

z = 2.576: Φ = 0.995002, leaving 0.5% in each tail — which is why ±2.576 bounds a 99% confidence interval.

Where the Normal Distribution calculator can give the wrong answer

This is the standard normal only. For a value from a normal distribution with mean μ and SD σ, convert to z = (x − μ) ÷ σ first — the z-score calculator does that — and then read the probability here.
The approximation's error is under 7.5 × 10⁻⁸, so the six decimals shown are reliable; far in the tails (|z| > 6) the true probabilities are so small that the last digits are meaningless anyway.
The area is a probability for a continuous variable, so P(Z = z) is zero and 'at most z' and 'less than z' are the same number.
The two-tailed line is for symmetric tests. A one-tailed test at 5% uses z = 1.645, not 1.96; the right-tail line gives that directly.
Data that are not normal — skewed, bounded, discrete — do not follow these probabilities even after standardising.

Spotted a mistake? Tell us

Normal Distribution reference table

Standard normal probabilities for common z-values
z	P(Z ≤ z)	P(Z > z)	P(\|Z\| > z), two-tailed
0	0.5000	0.5000	1.0000
0.5	0.6915	0.3085	0.6171
1	0.8413	0.1587	0.3173
1.28	0.8997	0.1003	0.2005
1.5	0.9332	0.0668	0.1336
1.645	0.9500	0.0500	0.1000
1.96	0.9750	0.0250	0.0500
2	0.9772	0.0228	0.0455
2.326	0.9900	0.0100	0.0200
2.576	0.9950	0.0050	0.0100
3	0.9987	0.0013	0.0027

Source: The standard normal distribution, Φ(z), to four decimal places

Normal Distribution FAQ

: A z-table lists Φ(z), the area to the left of z, for z in steps of 0.01. Enter any z here and the page gives that area to six decimals, plus the right tail, the central band and the two-tailed probability, which tables make you work out by hand.
: 1 − Φ(2) = 1 − 0.9772 = 0.0228, about 2.3%. Beyond ±2 in both directions it is 4.6%, the '95% within two standard deviations' of the empirical rule read from the other side.
: For a central 95% band, ±1.96; for 99%, ±2.576; for 90%, ±1.645. For a single tail — 95% below — z = 1.645, and 99% below is 2.326. Enter those z values here to see the areas come out.
: By the Abramowitz & Stegun 26.2.17 polynomial: a rational function of |z| multiplied by the normal density, accurate to 7.5 × 10⁻⁸. Statistical software uses more precise routines, but for anything a table would answer this is exact to every shown digit.
: Subtract the smaller Φ from the larger. Between z = −1 and z = 2: 0.9772 − 0.1587 = 0.8185, about 82% of a normal distribution.
: For a two-tailed test, read the two-tailed line: z = 2.1 gives P(|Z| > 2.1) ≈ 0.0357, significant at the 5% level but not at 1%. For a one-tailed test, use the right tail.