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Z calculator icon showing a bell curve

Z Calculator

Z Calculator

Z-score, z-table lookup, z-test hypothesis testing, and confidence intervals

Calculate Z-Score

Z → Cumulative Probability

Probability → Z (inverse)

Standard Normal (Z) Table

Rows = z to first decimal, columns = second decimal. Cell = P(Z < z). Matching row/col highlighted after a lookup.

Test Type

Sample

Test Settings

Confidence Interval for a Mean

Calculation History

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For educational and statistical reference purposes. Verify critical results independently. — calculator24x7.com

Four related statistics problems, standardizing a value, reading the normal table, testing a hypothesis, and building a confidence interval, all revolve around the same z-distribution. This calculator handles all four in one tool instead of requiring separate lookups and formulas for each.

How to use this Z calculator

  1. Z-Score tab: enter a raw value, population mean, and standard deviation to standardize it into a z-score, percentile, and probability.
  2. Z-Table tab: look up the cumulative probability for a given z-score, or work backward from a probability to find the corresponding z-score.
  3. Z-Test tab: choose one-sample or two-sample testing, enter your sample statistics, significance level, and tail type, then run the test.
  4. Confidence Interval tab: enter sample mean, population standard deviation, sample size, and confidence level to compute the interval.

Z-Score: standardizing a raw value

Z = (x − μ) ÷ σ

The z-score tells you how many standard deviations a raw value sits from the population mean. This tab converts your raw value into that z-score, then reports the corresponding percentile and both tail probabilities, P(X < x) and P(X > x).

Z-Table: forward and inverse lookup

The Z-Table tab covers both directions of the classic standard normal table lookup: given a z-score, find the cumulative probability P(Z < z); or given a target cumulative probability, find the z-score that produces it (the inverse lookup, useful for finding critical values). The full standard normal table is also displayed for reference, with the matching row and column highlighted after a lookup.

Z-Test: one-sample and two-sample hypothesis testing

The one-sample z-test checks whether a sample mean differs significantly from a hypothesized population mean, when the population standard deviation is known. The two-sample z-test compares the means of two independent samples. Both support your choice of significance level (α = 0.10, 0.05, or 0.01) and tail type (two-tailed, left-tailed, or right-tailed), and return the z-statistic, critical value(s), p-value, and a reject/fail-to-reject decision.

One-sample: Z = (x̄ − μ₀) ÷ (σ ÷ √n)

Confidence Interval: estimating a population mean

Given a sample mean, known population standard deviation, sample size, and desired confidence level, this tab computes the margin of error and the resulting interval, the range within which the true population mean is estimated to fall at your chosen confidence level.

Margin of Error = Z(critical) × (σ ÷ √n)
Confidence Interval = x̄ ± Margin of Error

Frequently asked questions

When should I use a z-test instead of a t-test?

A z-test is appropriate when the population standard deviation is known and your sample size is reasonably large; when the population standard deviation is unknown and must be estimated from the sample, a t-test is typically the more appropriate choice instead.

What does the z-table’s cumulative probability actually represent?

P(Z < z) is the probability that a standard normal random variable falls below your specified z-score, this is the core value the standard normal table reports and what most z-score-based calculations ultimately rely on.

What’s the difference between one-tailed and two-tailed tests?

A two-tailed test checks for a difference in either direction from the hypothesized value, while a one-tailed (left- or right-tailed) test checks for a difference in one specific direction only, which changes both the critical value and how the p-value is calculated.