A Z-score (standard score) measures how many standard deviations a data point is from the mean of a normal distribution. A Z-score of 0 is at the mean; 1.96 is 1.96 standard deviations above the mean.

What does left-tail probability mean?

Left-tail probability P(Z ≤ z) is the area under the standard normal curve to the left of the given Z-score. For Z = 0, this equals 0.5000 (50%). For Z = 1.96, it equals approximately 0.9750.

How do I find the p-value for a one-tailed test?

For a right-tailed test, the p-value = right-tail probability (P(Z > z)). For a left-tailed test, the p-value = left-tail probability (P(Z ≤ z)). For a two-tailed test, multiply the smaller tail by 2.

Why does P(Z ≤ 1.96) ≈ 0.9750?

1.96 is a critical value corresponding to the 97.5th percentile of the standard normal distribution, commonly used for 95% confidence intervals (leaving 2.5% in each tail).

What is the range of valid Z-scores?

Technically Z-scores can be any real number, but values beyond ±4 have probabilities extremely close to 0 or 1 (within 0.003%). This tool accepts values from −4 to 4, covering 99.994% of practical cases.

How accurate are the results?

Results are computed using the Hart rational approximation of the error function (erfc), which matches standard Z-tables to at least 4 decimal places across the entire range.

Z Table Lookup

Name: Z Table Lookup
Availability: InStock
Author: Nham Vu

Enter a Z-score to instantly look up the cumulative probability (left-tail and right-tail area) from the standard normal distribution.

Enter Z-Score

Z-Score value

Range: −4.00 to 4.00

Common Z-Scores

Enter a Z-score to see the cumulative probability.

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Summary

Enter a Z-score to instantly look up the cumulative probability (left-tail and right-tail area) from the standard normal distribution.

How it works

Enter a Z-score value in the input field (e.g. 1.96 or -2.33).
The tool computes the cumulative distribution function (CDF) of the standard normal distribution.
The left-tail probability P(Z ≤ z) is displayed — this is the shaded area to the left of the Z-score.
The right-tail probability P(Z > z) = 1 − P(Z ≤ z) is shown for one-tailed tests.
A bell curve visualization highlights the shaded area corresponding to your input.
Use the two-tailed section to find P(−|z| ≤ Z ≤ |z|) for two-sided hypothesis tests.

Use cases

Look up p-values for hypothesis testing in statistics courses.
Find critical Z-scores for confidence intervals (e.g. 95% → ±1.96).
Check cumulative probability for quality control (Six Sigma, process capability).
Verify textbook Z-table values quickly without a printed table.
Compute tail probabilities for one-tailed and two-tailed statistical tests.
Understand standard normal probabilities for finance and risk analysis.

Frequently Asked Questions

Last updated: 2026-06-09 · Reviewed by Nham Vu