Z Score Calculator

Z-Score Calculator – Standard Scores & Normal Distribution

A Z-Score Calculator allows you to find how far a raw data point is from the mean of a dataset, measured in standard deviations. Z-scores, also known as standard scores, are widely used in statistics to compare values from different normal distributions, identify outliers, and calculate probabilities under the normal curve.

On this page, you can learn what z-scores mean, how to calculate them manually, and how to interpret results with our free online calculator.

What Is a Z-Score?

A z-score tells you how many standard deviations away from the mean a data point lies.

• A positive z-score means the value is above the mean.
• A negative z-score means the value is below the mean.
• A z-score of 0 means the value equals the mean.

This standardization makes it easier to compare data across different scales or distributions.

Z-Score Formula

The formula for calculating a z-score is:

z = frac{x - mu}{sigma}

Where:

• = raw score
• = population mean
• = population standard deviation

If using a sample instead of a population, replace with (sample standard deviation).

Step-by-Step Example

Suppose you scored 85 on a test. The class mean was 75, with a standard deviation of 5.

z = frac{85 - 75}{5} = frac{10}{5} = 2

Your score is 2 standard deviations above the mean. This places you in approximately the top 2.5% of scores, assuming a normal distribution.

Interpreting Z-Scores

• z < -3 or z > +3 → extreme outliers.
• z between -2 and +2 → typical range for ~95% of data (68–95–99.7 rule).
• z between -1 and +1 → within ~68% of data.

This interpretation is critical in quality control, psychology testing, academic grading, and probability problems.

Using Z-Scores to Find Probabilities

Z-scores map directly to probabilities in the standard normal distribution.

Example: What is the probability of scoring below z = 1.0?

Looking up in the standard normal table (or calculator) gives:

P(Z < 1.0) approx 0.8413

Applications of Z-Scores

1. Statistics & Research – Compare data points across studies.
2. Education – Standardized test scoring (SAT, GRE, etc.).
3. Finance – Detect unusual returns in stock markets.
4. Manufacturing – Quality control using Six Sigma.
5. Health & Psychology – Standardized patient test results.

Z-Score vs T-Score

• Z-score: Used when the population standard deviation is known or the sample size is large ().
• T-score: Used when the population standard deviation is unknown and sample size is small.

Both standardize scores but differ in the distribution they follow.

How to Use the Z-Score Calculator

1. Enter your raw score (x).
2. Provide the mean (µ) of the dataset.
3. Enter the standard deviation (σ).
4. The calculator instantly shows the z-score.
5. Optionally, view the probability (area under the normal curve) for your z-value.

Frequently Asked Questions (FAQ)

1) What does a z-score of 0 mean?

It means the data point is exactly equal to the mean of the distribution.

2) What is considered a high z-score?

Any z-score above +2 or below -2 is considered unusual, and beyond ±3 is an extreme outlier.

3) Can z-scores be negative?

Yes. Negative z-scores indicate values below the mean.

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

Use the standard normal distribution table or calculator. For example, z = 1.0 corresponds to the 84th percentile.

5) What’s the difference between z-score and standard deviation?

A standard deviation is a measure of spread. A z-score tells you how many standard deviations away from the mean a particular value lies.

6) Do z-scores always assume a normal distribution?

Yes, probabilities derived from z-scores are only accurate under the assumption of a normal distribution.

7) Can z-scores be used with skewed data?

You can calculate them, but interpretation may not be accurate because the normal distribution assumption is violated.

Try the Free Z-Score Calculator

Use our Z-Score Calculator above to instantly standardize raw scores, find probabilities under the normal curve, and understand how far your value is from the mean.