Z-Score Calculator: Percentile, Steps and Graph

How to use the z-score calculator

Choose Z-score from a value to standardize a score, or Value from a z-score to work backward to the original units.
Enter the raw score (or z-score), the mean and the standard deviation. If you only have raw data, find the mean and standard deviation first with the standard deviation calculator.
Pick which area to shade: below z (the percentile), above z, both tails, or the middle.
Click Calculate. You'll get the z-score, the percentile and tail probabilities, a shaded normal curve and the full working. Download the graph as a PNG if you need it.

The z-score formula

A z-score standardizes a value by subtracting the mean and dividing by the standard deviation:

z = x − μσ

When you're working from a sample rather than a whole population, the same idea uses the sample statistics:

z = x − xs

And to go back from a z-score to the original units, rearrange it:

x = μ + zσ

z: The z-score (standard score): distance from the mean in standard deviations
x: The raw score, the value you want to standardize
μ, x: The population mean, or the sample mean
σ, s: The population standard deviation, or the sample standard deviation

To turn a z-score into a percentile, find the area under the standard normal curve to the left of it, written Φ(z). This calculator works that out exactly, so you don't need a z-table.

Worked example

Suppose newborn birth weights in a hospital have a mean of 7.5 lb and a standard deviation of 1.1 lb. How unusual is a baby who weighs 9.2 lb?

Write the formula
z = x − μσ
Substitute x = 9.2, μ = 7.5, σ = 1.1
z = 9.2 − 7.51.1 = 1.71.1
Divide and round to two decimals
z = 1.545… ≈ 1.55
Find the area to the left of z
Φ(1.55) = 0.9394

What it means: the baby is about 1.55 standard deviations heavier than average, at roughly the 94th percentile. About 6% of babies in this hospital weigh more. That's on the heavy side, but not unusual by the ±2 rule.

How to interpret a z-score

The sign of z tells you the direction: positive is above the mean and negative is below. The size tells you how far. For data that roughly follow a normal (bell-shaped) distribution, the 68-95-99.7 rule gives a quick feel for how common a z-score is:

Z-score range	Share of values inside	How common outside it
−1 to +1	About 68%	Common (about 1 in 3)
−2 to +2	About 95%	Unusual (about 1 in 20)
−3 to +3	About 99.7%	Rare (about 1 in 370)

Z-scores are especially handy for comparing results on different scales. If you scored 1,250 on the SAT and a friend scored 27 on the ACT, converting both to z-scores with each test's mean and standard deviation shows who did better relative to other test-takers.

One caution: percentiles from z-scores assume the data are roughly normal. For strongly skewed data, such as incomes, a z-score still measures distance from the mean, but the percentile it implies may be off.

Frequently asked questions

What does a z-score tell you?

A z-score tells you how many standard deviations a value is from the mean. A z-score of 2 means the value is 2 standard deviations above average; −1 means 1 standard deviation below. Because it puts everything on the same scale, you can compare values measured in different units, such as an SAT score and an ACT score.

Can a z-score be negative?

Yes. A negative z-score means the value is below the mean, a positive one means it is above, and a z-score of 0 means it is exactly average. The sign tells you the direction; the size tells you how far.

What is considered an unusual z-score?

A common rule is that values with a z-score beyond ±2 are unusual (only about 5% of a normal distribution is that far out) and values beyond ±3 are very unusual (about 0.3%). Some fields use these cutoffs to flag possible outliers.

How do I convert a z-score to a percentile?

Look up the area to the left of the z-score under the standard normal curve, using a z-table or this calculator. Multiply by 100 to get the percentile. For example, z = 1 has about 0.8413 of the area to its left, so it is at roughly the 84th percentile.

Should I use the population or sample standard deviation?

Use the population mean and standard deviation (μ and σ) when they are known, for example from published test norms. If you only have a sample, use the sample mean and sample standard deviation in their place. The calculation is the same. Our standard deviation calculator can find them for you.

What is the difference between a z-score and a t-score?

A z-score uses a known population standard deviation and the normal distribution. A t-score is used in hypothesis tests when the standard deviation is estimated from a small sample, and it follows the t distribution, which has heavier tails. For describing where a single value sits, the z-score is the standard choice.