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Z-Score Calculator

Find how many standard deviations a value is from the mean, and what share of a normal distribution lies below it. You can also go backwards from a z-score or a percentile to the raw value, or find the probability between two values, with the area shaded on a bell curve.

What do you want to find?

Z-score from a valueValue from a z-scoreZ-score and value from a percentileProbability between two values

Raw score (x)

Z-score (z)

Percentile (area to the left)

Lower value (a)

Upper value (b)

Mean (μ)

Standard deviation (σ)

Use μ = 0 and σ = 1 for the standard normal distribution

Enter your values to see the result.

How to use the Z-Score Calculator

Choose what to find: a z-score, a value, a percentile cutoff or a probability between two values.
Enter the mean (μ) and standard deviation (σ). Use 0 and 1 for the standard normal distribution.
Enter the value, z-score or percentile.
Read the result with its percentile and tail probabilities.
Look at the shaded curve and the reference table, or copy the results.

Z-score formulas

Find	Formula	Example (μ = 70, σ = 10)
Z-score	z = (x − μ) ÷ σ	x = 85 → z = 1.5
Raw score	x = μ + z × σ	z = −0.8 → x = 62
Percentile	Φ(z), the area left of z	z = 1.5 → 93.32%
Area to the right	1 − Φ(z)	z = 1.5 → 6.68%
Two-tailed p-value	2 × (1 − Φ(\|z\|))	z = 1.5 → 0.1336
Between a and b	Φ(z_b) − Φ(z_a)	60 to 80 → 68.27%

Common z-scores and percentiles

z	Area to the left (percentile)	Typical use
−2.326	1%	Bottom 1% cutoff
−1.645	5%	One-tailed 5% (lower)
−1	15.87%	One SD below the mean
0	50%	The mean / median
1	84.13%	One SD above the mean
1.282	90%	Top 10% cutoff
1.645	95%	One-tailed 5% (upper), 90% two-sided interval
1.96	97.5%	95% two-sided interval
2.326	99%	Top 1% cutoff
2.576	99.5%	99% two-sided interval
3	99.87%	Three SD above the mean

Probabilities come from the standard normal cumulative distribution function Φ, computed with a high-accuracy approximation of the error function (relative error below about 10⁻⁷). Percentile cutoffs use an inverse normal algorithm refined to match it.

Frequently asked questions

What is a z-score?

A z-score (standard score) tells you how many standard deviations a value is above or below the mean: z = (x − μ) ÷ σ. A z of 0 is exactly average, +2 is two standard deviations above and −1 is one below.

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

Look up the cumulative normal probability Φ(z), the area under the bell curve to the left of z. For example Φ(1) ≈ 0.8413, so z = 1 is about the 84th percentile. This calculator computes Φ(z) directly, so you do not need a z-table.

How do I find the value for a given percentile?

Find the z-score whose left-tail area equals the percentile (the inverse normal), then convert: x = μ + z × σ. The 90th percentile is z ≈ 1.2816; with mean 100 and SD 15 that is about 119.22.

What is a two-tailed p-value?

It is the probability of getting a z-score at least as far from 0 as yours in either direction: 2 × (1 − Φ(|z|)). For z = 1.96 it is about 0.05, the usual cutoff for statistical significance at the 5% level.

What z-scores cover 68%, 95% and 99.7% of values?

In a normal distribution about 68.27% of values lie within ±1 standard deviation, 95.45% within ±2 and 99.73% within ±3. The exact z for the middle 95% is ±1.96 and for 99% it is ±2.576.

Does my data have to be normally distributed?

The z-score itself can be calculated for any data, but turning it into a percentile or probability assumes the data follows a normal (bell-shaped) distribution. For strongly skewed data those percentages can be misleading.

Is a negative z-score bad?

No. It simply means the value is below the mean. Whether that is good depends on the context: a negative z for a race time means faster than average.

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