Statistics

Z-Score Calculator

Calculate z = (x − μ) / σ from an observation, mean and positive standard deviation, or compare the standardized positions of two observations. Percentile estimates assume a normal distribution.

Z-Score1.5000 View results

Mode*

Raw Value (x)*

Mean (μ)*

Standard Deviation (σ)*

Raw Value 2 (x₂)

Mean 2 (μ₂)

Standard Deviation 2 (σ₂)

Real-time Results

Z-Score

1.5000

The standardized score (number of standard deviations from the mean)

Percentile

93.32%

The percentile rank of the value

Percentage Above

6.68%

Percentage of values above this point

Why Choose Z-Score Calculator

Calculate z = (x − μ) / σ from an observation, mean and positive standard deviation, or compare the standardized positions of two observations. Percentile estimates assume a normal distribution.

How to Calculate

1Choose a single value or the comparison mode.
2Enter the observation, mean and standard deviation. For x = 75, μ = 60 and σ = 10, z = 1.5.
3Read the z-score and estimated normal percentile. In comparison mode, enter the second set of values.

Frequently Asked Questions (FAQ)

What does z = 1.5 mean?

The observation is 1.5 standard deviations above its mean. For a normal distribution, about 93.32% of values lie below that point.

Can I use a zero standard deviation?

No. The formula divides by σ, which must be positive. A distribution with no variation cannot be standardized using this formula.

Are the percentiles valid for any distribution?

No. A z-score standardizes a value without requiring normality, but this tool estimates percentiles using the normal cumulative distribution function.

Does the tool calculate confidence intervals?

No. This page takes observations, means and standard deviations. It does not estimate a confidence interval or perform a significance test.

Authoritative References & Sources

NIST — Normal distribution

View Source

Calculation Principles & Mathematical Formula

Z-Score Formula:
z = (x - μ) / σ

Where:
- x is the data point value
- μ is the mean of the population or sample
- σ is the standard deviation of the population or sample

The z-score represents how many standard deviations a data point is from the mean.
A positive z-score indicates the data point is above the mean, while a negative z-score indicates it is below the mean.

Finding a value from a z-score:
x = μ + (z × σ)

The standard normal distribution has a mean of 0 and a standard deviation of 1.

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