Z-score calculator: value to z to percentile

By Jude Wallis · Published

A z-score calculator standardizes a raw value: it computes z = (x - mu) / sigma, where mu is the mean and sigma is the standard deviation, then reports the percentile (the area to the left of z). Enter a value of 82 with mean 70 and SD 8 to get z = 1.5 at about the 93.32nd percentile.

z-score

1.5

Percentile: 93.32% of values fall below x = 82. Area to the right: 0.0668.

Steps

  1. 1.z = (x - mean) / SD = (82 - 70) / 8
  2. 2.z = 12 / 8 = 1.5
  3. 3.area to the left of z (from the standard normal table): 0.9332

AP Statistics: Unit 1 (topics 1.9 Comparisons of the Distributions for One Quantitative Variable, 2.11 The Normal Distribution). In the Fall 2026 AP Statistics course, standardizing a value with a z-score is Unit 1, topic 1.9. LO 1.9.D calculates z-scores from population parameters using z = (x - mu) / sigma, and LO 1.9.E compares z-scores as measures of relative position. Reading the standard normal curve to turn a z-score into a percentile is Unit 2, topic 2.11 (The Normal Distribution), where LO 2.11.E uses z-scores with a standard normal table to find areas and percentiles.

What a z-score is

A z-score tells you how many standard deviations a value sits above or below the mean. A z-score of 1.5 means the value is 1.5 standard deviations above the mean. A z-score of 0 means the value equals the mean, and a negative z-score means the value is below the mean. The sign tells you the direction, and the size tells you how unusual the value is, so larger absolute z-scores sit farther out in the tails.

Z-scores put values from different distributions on one common scale. A raw score of 82 on a quiz and a raw score of 640 on a college-entrance test mean nothing side by side, because they come from different means and spreads. Once you convert both to z-scores, you can compare their relative positions directly.

The z-score formula

To standardize a value, subtract the mean and divide by the standard deviation:

z=x−μσz = \frac{x - \mu}{\sigma}

Here xx is the raw value, μ\mu (mu) is the population mean, and σ\sigma (sigma) is the population standard deviation. When you only have sample data, use the sample mean xˉ\bar{x} (x-bar) and the sample standard deviation ss in their place, and the arithmetic stays the same.

The percentile comes from the standard normal curve. The area to the left of your z-score is the proportion of the distribution that falls below it, so multiplying that area by 100 gives the percentile. You can read this area from a z-table, or let the calculator return it for you.

How to use the z-score calculator

The calculator runs the formula in both directions.

  • To go from a raw value to a z-score, enter your value xx, the mean, and the standard deviation. The tool returns the z-score and the percentile.
  • To go from a z-score or a percentile back to a raw value, enter the z-score (or the percentile) along with the mean and standard deviation. The tool solves x=μ+zσx = \mu + z\sigma for the raw value.

Read the two outputs as a pair. The z-score is the standardized position, and the percentile is the share of a normal distribution that falls below your value. This z-score to percentile calculator reads that area from the standard normal curve, so treat the percentile as exact only when your data is approximately normal.

When z-scores apply

A z-score is defined for any distribution, because it only measures distance from the mean in standard-deviation units. The percentile the calculator reports is the part that assumes a normal distribution, since it reads the area from the standard normal curve.

If your data is strongly skewed or has heavy outliers, the z-score is still correct, but its normal-curve percentile can be misleading. Check the shape with a histogram or the empirical rule before you trust a percentile that assumes normality.

Two common settings fit the normal model well. Standardized test scores are often designed to be roughly normal, and sampling distributions of the mean become approximately normal through the central limit theorem. For the reverse problem, finding a raw value from a target area, use the normal distribution calculator.

Standardizing a quiz score of 82

A quiz has a population mean of 70 and a standard deviation of 8. A student scores 82. Find the z-score and the percentile.

  1. Identify the values: x=82x = 82, μ=70\mu = 70, σ=8\sigma = 8.

  2. Subtract the mean: 82−70=1282 - 70 = 12.

  3. Divide by the standard deviation: z=12/8=1.5z = 12 / 8 = 1.5.

  4. Look up z=1.5z = 1.5 in the z-table: row 1.5, column 0.00 gives 0.9332.

  5. Convert to a percentile: 0.9332×100=93.32%0.9332 \times 100 = 93.32\%.

z = 1.5, which places the score at about the 93.32nd percentile (about 93.32% of scores fall below 82).

Frequently asked questions

How do I turn a z-score into a percentile?

Look up the z-score in a standard normal table and read the area to its left, then multiply by 100. For z=1.5z = 1.5 the area is 0.9332, so the percentile is about 93.32. The calculator does this lookup for you and reports the percentile directly.

What is a good z-score?

There is no universally good z-score, because it depends on the context. A larger positive z-score means a value is farther above the mean, which is good for a test score but bad for something you want low, like an error rate. Read a z-score as a position, not as a quality rating.

Can a z-score be negative or greater than 3?

Yes to both. A negative z-score marks a value below the mean, and a z-score above 3 marks a value more than 3 standard deviations above it. Values beyond plus or minus 3 are rare in a normal distribution (about 0.3% of the area lies past that range), but they are valid.