Skip to main content
Simple Useful Tools
Simple Useful Tools · Statistics

Z-Score Calculator

Calculate a z-score from a value, mean, and standard deviation, or switch to Dataset mode to standardize multiple values.

Inputs

Result

Z-score

1.0000

Percentile

84.13th percentile

Below

84.13%

Above

15.87%

Interpretation

This value is 1 standard deviation above the mean.

Standard normal curve

Standard normal distribution with z = 1.0000. Approximately 84.13% of values fall below this z-score.

Calculation steps

1. Start with the formula

z = (X - μ) / σ

2. Substitute the values

z = (115 - 100) / 15

3. Find the difference

z = 15 / 15

4. Divide

z = 1.0000

How to use the Z-Score Calculator

Use Single value mode when you already know a raw score, mean, and standard deviation. Enter those three values, press Calculate, and the calculator returns the z-score, percentile, below and above probabilities, interpretation, standard normal curve, and calculation steps.

Use Dataset mode when you want to standardize a list of values. Paste values separated by commas, spaces, tabs, or new lines. You can also paste a column directly from Excel or Google Sheets. Choose Population or Sample, then calculate z-scores to see the count, mean, standard deviation, and z-score for each value.

What is a z-score?

A z-score, also called a standard score, measures how far a value is from the mean in standard deviation units. It describes position relative to a mean and standard deviation.

  • z = 0 means the value equals the mean.
  • z > 0 means the value is above the mean.
  • z < 0 means the value is below the mean.
  • z = 1 means one standard deviation above the mean.
  • z = -2 means two standard deviations below the mean.

How to calculate a z-score

The standard z-score formula is:

z = (X − μ) / σ
  • z = z-score
  • X = raw score
  • μ = mean
  • σ = standard deviation

In plain language, subtract the mean from the value, then divide by the standard deviation.

How to interpret a z-score

The sign tells you direction. Positive z-scores are above the mean, negative z-scores are below the mean, and zero is exactly at the mean.

The size tells you distance. A z-score of 0.5 is 0.5 standard deviations above the mean, while a z-score of -1.5 is 1.5 standard deviations below the mean. The farther the z-score is from zero, the farther the value is from the mean.

Interpretation depends on context. For a normal distribution, values within about ±2 standard deviations contain roughly 95% of observations, but that is a descriptive rule of thumb, not an automatic outlier test.

Z-scores and percentiles

For a standard normal distribution, a z-score can be converted to the proportion of values below it using the normal cumulative distribution. The calculator's Percentile and Below values represent this left-tail probability.

  • z = 0 is approximately the 50th percentile.
  • z = 1 is approximately the 84.13th percentile.
  • z = -1 is approximately the 15.87th percentile.

This percentile conversion assumes that the normal distribution is an appropriate model for the data.

Z-score example

Suppose the raw score is 115, the mean is 100, and the standard deviation is 15.

z = (115 − 100) / 15
z = 1

The value 115 is one standard deviation above the mean. Under a normal distribution, approximately 84.13% of values fall below it.

How to calculate z-scores for a dataset

Dataset standardization uses the same idea for every value in a list:

  1. Calculate the dataset mean.
  2. Calculate the dataset standard deviation.
  3. Apply the z-score formula to each value.

Z-scores are useful because they put values on a common standardized scale. This calculator preserves the original order of the entered values.

Population vs. sample standard deviation

Choose Population when the entered values represent the entire group you are analyzing. Choose Sample when the entered values are a sample drawn from a larger population.

Population standard deviation uses n in the variance calculation, while sample standard deviation uses n − 1 to estimate variation in the larger population. The calculator keeps both options available because the right choice depends on how the data was collected.

Z-score vs. normal distribution

A z-score describes a standardized position relative to a mean and standard deviation. A standard normal distribution is a specific distribution with mean 0 and standard deviation 1.

Converting values to z-scores lets them be interpreted on that standardized scale. To explore normal probabilities directly, use the Normal Distribution Calculator.

Frequently asked questions

What is a z-score?
A z-score, also called a standard score, measures how far a value is from the mean in standard deviation units. A z-score of 0 means the value equals the mean, a positive z-score is above the mean, and a negative z-score is below the mean.
How do you calculate a z-score?
Use the formula z = (X - μ) / σ. Subtract the mean from the raw score, then divide that difference by the standard deviation.
What does a positive or negative z-score mean?
A positive z-score means the value is above the mean. A negative z-score means the value is below the mean. The sign shows direction, not whether the value is good or bad.
What does a high z-score tell you?
A high z-score tells you that a value is far above the mean compared with the standard deviation. Whether that is unusual, expected, good, or bad depends on the context and the distribution being analyzed.
What is considered a good or normal z-score?
There is no universally good z-score. Interpretation depends on context. For normally distributed data, values closer to 0 are closer to the mean, while values farther from 0 are less common; ±2 standard deviations is often used as a descriptive reference, not a universal rule.
How do you convert a z-score to a percentile?
For a standard normal distribution, convert the z-score using the normal cumulative distribution. The result is the proportion of values expected to fall below that z-score, which is the left-tail percentile.
What percentile is a z-score of 1?
A z-score of 1 is approximately the 84.13th percentile under the standard normal distribution. That means about 84.13% of values fall below it in that model.
How do you calculate z-scores for a dataset?
First calculate the dataset mean and standard deviation. Then apply the z-score formula to each value. This puts each value on the same standardized scale while preserving the original order of the data.