Z-score calculator with percentiles and p-values

How to use the z-score calculator

1. Under Calculate, choose z from a value, Value from z or z from a percentile.
2. Enter the known number: Value x, z-score or Percentile.
3. Enter the Mean μ and Standard deviation σ of the distribution. For the standard normal table use 0 and 1.
4. Read the z-score, the Value x, and the areas: Percentile (P(X < x)), Above (P(X > x)), Between −|z| and +|z| and the Two-tailed p-value.

What it does and when to use it

A z-score says how many standard deviations a value sits from the mean. It lets you compare results from different scales, such as a maths score and a reading score, and it is the starting point for normal-distribution questions in statistics courses. The reverse modes answer questions like “what score puts someone in the top 5%?”.

If you have raw data, first find its mean and standard deviation with the mean, median and mode calculator. To express a gap between two values as a percentage instead, use the percentage change calculator. Round long answers with the rounding calculator.

How it works

• z from a value: z = (x − μ) ÷ σ.
• Value from z: x = μ + z × σ.
• z from a percentile: z = Φ⁻¹(p ÷ 100), then x = μ + z × σ.

Φ(z) is the area under the standard normal curve to the left of z. The tool computes it with G. West’s 2005 double-precision method, accurate to about 15 decimal places, so you do not need a printed z-table. Φ⁻¹ uses Peter Acklam’s approximation followed by one refinement step. The other areas follow from Φ: above = 1 − Φ(z), between = 2Φ(|z|) − 1, and two-tailed p = 2(1 − Φ(|z|)).

Step by step for an IQ score of 130 (μ = 100, σ = 15):

1. Subtract the mean: 130 − 100 = 30.
2. Divide by the standard deviation: 30 ÷ 15 = 2. So z = 2.
3. Look up Φ(2) = 0.9772498681.
4. Percentile: 97.72%. About 2.28% of scores are higher.

Worked examples

Value to z. x = 130, μ = 100, σ = 15 gives z = 2 and Percentile (P(X < x)): 97.72498681%.

The 95% rule. For the same input, Between −|z| and +|z|: 95.44997361%. This is the “about 95% within 2 standard deviations” rule.

z to value. A class test has mean 70 and σ = 8. Choosing Value from z with z = −1 gives Value x: 62, at the 15.87th percentile.

Percentile to z. The 95th percentile of the standard normal (μ = 0, σ = 1) is z-score: 1.644853627.

Limits and tips

• Percentiles must be strictly between 0 and 100.
• Results show 10 significant figures; the p-value shows 6.
• Calculations run in your browser.