The Sampling Lab

Calculator

Z-score calculator

Turn a value into a z-score and a percentile, or a percentile back into a value, with every step shown.

3 min read · 1 interactive figure

Choose a direction, then type the numbers. The steps below the curve show the arithmetic; the shaded area is the share of a normal population below the value. Drag the blue value along the curve. The top row of the axis is in standard deviations (z), the bottom row in your own units.#
On this page
  1. The formula
  2. Worked examples
  3. The percentile step needs a normal population
  4. z-scores in significance tests

A z-score says how many standard deviations a value lies above or below the mean: z = (x − μ)/σ. A score of 130 on a test with mean 100 and standard deviation 15 has z = (130 − 100)/15 = 2. If the scores are normally distributed, that z also gives a percentile: Φ(2) = 0.9772, so 130 is at about the 98th percentile and 2.3 % of people score higher.

The calculator above runs in three directions: from a value to its z-score and percentile, from a z-score back to a value, and from a percentile to the value that reaches it. Each answer comes with its working, so you can check a homework solution line by line, and with the area drawn on the curve so you can see whether the number is reasonable.

The formula

z = (x − μ) ⁄ σ     x = μ + zσ(1)

Subtracting the mean moves the centre to zero; dividing by the standard deviation rescales the spread to one. The result has no units, which is the point: a child’s height and weight, or two exams marked out of different totals, can be compared once both are expressed in standard deviations from their own means. When the mean and standard deviation come from a sample rather than the whole population, the same formula is written with x and s in place of μ and σ[1]. established

Worked examples

  1. An exam score. The class mean is 70 and the standard deviation 8; you scored 82. Then z = (82 − 70)/8 = 1.5. If the marks are roughly normal, Φ(1.5) = 0.9332, so you beat about 93 % of the class.
  2. A value below the mean. A reading of 91 against a mean of 100 and SD 15 has z = −9/15 = −0.6, and Φ(−0.6) = 0.2743: the 27th percentile.
  3. From a percentile to a value. What score puts someone in the top 10 %? The 90th percentile of the standard normal is z = 1.2816, so on the 100-and-15 scale the cut-off is 100 + 1.2816 × 15 = 119.2.
  4. From a z-score to a value. A laboratory flags results more than 2 standard deviations above a reference mean of 5.0 with SD 0.4. The threshold is 5.0 + 2 × 0.4 = 5.8.

The z table gives the same areas to four decimals, and the chapter on the normal distribution explains where they come from.

The percentile step needs a normal population

A z-score itself assumes nothing about shape: it only counts standard deviations. Turning it into a percentile does. The calculator uses the normal curve for that step, and for data that are skewed or long-tailed the answer can be badly off. In a flat population only 57.7 % of values lie within one standard deviation of the mean, against 68.3 % for a normal one; in a long-tailed population a z of 3 is far more common than the normal’s 0.13 % in each tail. For any distribution at all, Chebyshev’s inequality guarantees that at least 1 − 1/k² of the values lie within k standard deviations, so at least 75 % within two, but that is a floor, not an estimate[2]. established

z-scores in significance tests

Many test statistics are z-scores in disguise: the distance between what you observed and what the null hypothesis predicts, measured in standard errors. Turning that kind of z into a p-value is the job of the p-value calculator, and the chapter on p-values explains what the result does and does not mean.