Z-Table Calculator

A z-score (also called a standard score) measures how many standard deviations a data point is from the mean of its distribution. It is calculated as z = (X − μ) / σ, where X is the raw value, μ is the mean, and σ is the standard deviation. Positive z-scores indicate values above the mean; negative z-scores indicate values below.

P(Z < z) is the cumulative probability that a randomly selected value from a standard normal distribution is less than or equal to your z-score. It represents the area under the normal curve to the left of z. For example, P(Z < 1.96) ≈ 0.975, meaning about 97.5% of values fall below z = 1.96.

The left-tail probability P(Z < z) gives the area to the left of your z-score under the normal curve. The right-tail probability P(Z > z) gives the area to the right. These two always sum to 1. Right-tail values are commonly used in hypothesis testing to find p-values for upper-tail tests.

Use the formula z = (X − μ) / σ, where X is your raw score, μ is the population mean, and σ is the population standard deviation. For example, if a test score is 85, the mean is 75, and the standard deviation is 10, then z = (85 − 75) / 10 = 1.0. Switch to 'Raw Score → Z-Score & Probability' mode in this calculator to do this automatically.

The two-sided probability P(−|z| < Z < |z|) represents the area within |z| standard deviations of the mean — the middle region of the bell curve. Its complement P(Z < −|z| or Z > |z|) is the combined area in both tails. This is commonly used in two-tailed hypothesis tests and confidence interval calculations.

For a 95% confidence interval, the critical z-score is approximately ±1.96. This means that 95% of the area under the standard normal curve falls between z = −1.96 and z = 1.96. For 99% confidence, the critical z-score is approximately ±2.576.

Theoretically, z-scores can range from negative infinity to positive infinity. In practice, z-scores beyond ±4 correspond to probabilities so close to 0 or 1 that they are rarely meaningful. Most standard z-tables cover values from −3.4 to +3.4, which covers more than 99.93% of the distribution.

In hypothesis testing, you compute a test statistic (z-score) and then look up the corresponding p-value from the z-table. For a one-tailed test, the p-value is P(Z > z) or P(Z < z). For a two-tailed test, the p-value is P(Z < −|z| or Z > |z|). If the p-value is below your significance level (e.g., 0.05), you reject the null hypothesis.