Standard Normal Distribution 3 Solvers: Standardization, Z-Table & Area

Z-score Calculator

Standardize raw scores into standard deviation units (Z = (x - μ) / σ), convert bidirectionally between Z-scores and probabilities, and find the area between two bounds.

Modify the values and click the Calculate button to use

• Module 1 (Z-Score Calculator): Computes standard normalization Z = (x - μ) / σ and tail probabilities.

• Module 2 (Z-Table Converter): Enter any 1 of the 6 fields to automatically solve all other Z-score and probability metrics.

• Module 3 (Area Between Two Z-scores): Finds interval probability P(Z₁ ≤ X ≤ Z₂) with an interactive dynamic bell curve.

Z-score Calculator

Use this calculator to compute the z-score of a normal distribution.

Raw Score, x:

Population Mean, μ:

Standard Deviation, σ:

Comprehensive Guide to Z-scores and Standard Normal Distributions

Understanding standard normal distributions, z-table lookups, standardization formulas, and hypothesis testing.

1What is a Z-score?

A Z-score (also known as a standard score) is a dimensionless statistic that quantifies the number of standard deviations an individual observation or raw data point x lies above or below the population mean μ.

Positive Z-score (> 0)Value is ABOVE the mean

Zero Z-score (= 0)Value is EXACTLY EQUAL to the mean

Negative Z-score (< 0)Value is BELOW the mean

2Z-score Formulas for Population and Sample Data

Population Z-Score Formula:

Z = (x - μ) / σ

Used when population mean (μ) and population standard deviation (σ) are known.

Sample Z-Score Formula:

Z = (x - x̄) / s

Used when estimating from sample mean (x̄) and sample standard deviation (s).

3How to Read and Use a Standard Normal Z-Table

A Z-table (standard normal distribution table) maps a specific Z-score to its cumulative probability Φ(Z) = P(X < Z), representing the area under the standard normal curve to the left of Z:

Left-Tail Probability: Area to the left of Z is Φ(Z).
Right-Tail Probability: Area to the right of Z is 1 - Φ(Z).
Symmetry Property: Φ(-Z) = 1 - Φ(Z).
68-95-99.7 Empirical Rule: P(-1 ≤ Z ≤ 1) ≈ 68.27%, P(-2 ≤ Z ≤ 2) ≈ 95.45%, and P(-3 ≤ Z ≤ 3) ≈ 99.73%.

4Practical Applications of Z-Scores

Standardized Test Comparing (SAT vs. ACT)

Because SAT and ACT operate on different numeric scales, converting raw scores to Z-scores reveals which student performed higher relative to their peer group.

Six Sigma & Quality Control

Manufacturing tolerances use Z-scores to evaluate defect rates (e.g. 6-sigma corresponds to fewer than 3.4 defective parts per million).

Frequently Asked Questions

What is considered a "good" or unusual Z-score?

In a standard normal distribution, approximately 95% of data falls within Z = -2 and Z = +2. Any score with |Z| > 2 is relatively unusual (top/bottom 2.5%), and |Z| > 3 is considered a statistical outlier (top/bottom 0.15%).

Can a Z-score be negative?

Yes. A negative Z-score simply indicates that the raw score is less than the population mean (e.g., scoring 45 on an exam where the average was 50).

What is the difference between a Z-test and a T-test?

A Z-test is used when the population standard deviation (σ) is known or the sample size is large (n ≥ 30). A T-test is used when the population standard deviation is unknown and must be estimated from a small sample.

How do you find the probability between two Z-scores?

Compute the cumulative probability for both Z-scores using a Z-table or CDF function, and subtract the smaller area from the larger: P(Z₁ ≤ X ≤ Z₂) = Φ(Z₂) - Φ(Z₁).