Z-Score Calculator - Standard Score & Probability Tool

What Is a Z-Score?

A z-score, also called a standard score, measures how many standard deviations a data point is from the mean of a dataset. A z-score of 0 means the value equals the mean. A positive z-score means the value is above the mean, and a negative z-score means it is below the mean.

Z-Score Formula

The z-score of a data point is calculated with this formula:

$$z = \frac{x - \mu}{\sigma}$$

Where:

• $z$ is the z-score
• $x$ is the data point
• $\mu$ is the mean of the dataset
• $\sigma$ is the standard deviation of the dataset

The standard deviation cannot be zero, because dividing by zero has no defined result. It also cannot be negative, since a standard deviation is always zero or positive by definition.

How to Calculate a Z-Score

1. Find the mean of the dataset by adding all the values and dividing by the number of values, $n$.

$$\mu = \frac{\sum_{i=1}^{n} x_i}{n}$$

2. Find the standard deviation. First calculate the variance, the average of the squared differences from the mean, then take its square root.

$$\sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n}}$$

3. Subtract the mean from the data point and divide by the standard deviation.

$$z = \frac{x - \mu}{\sigma}$$

Worked Example

A student scores 85 on a test. The class mean is 75, and the standard deviation of the class scores is 10.

• Data point: $x = 85$
• Mean: $\mu = 75$
• Standard deviation: $\sigma = 10$

$$z = \frac{85 - 75}{10} = 1.0$$

The student's z-score is 1.0. Their score is one standard deviation above the class average.

Z-Score and Cumulative Probability

A z-score can be converted into a cumulative probability, the chance that a randomly chosen value from a normal distribution falls at or below that point. This is the area under the standard normal curve to the left of $z$.

$$P(Z \leq z) = \Phi(z)$$

For the student's z-score of 1.0, the cumulative probability is about 0.8413, or 84.1%. This means the student scored better than roughly 84% of the class, assuming the scores follow a normal distribution.

A few reference points:

Interpreting a Z-Score

Most values in a normal distribution fall within 2 standard deviations of the mean. Z-scores beyond +3 or -3 are rare and often treated as outliers, values that differ sharply from the rest of the data.

Z-scores also let people compare values measured on different scales. A test score and a height measurement cannot be compared directly, but their z-scores can, because both describe distance from a mean in the same standard-deviation units.

Z-Score vs. T-Score

A z-score assumes the population standard deviation is known and works best with large samples. A t-score is used instead when the population standard deviation is unknown and the sample is small, typically under 30 observations. Both describe distance from a mean, but a t-score comes from the t-distribution, which has heavier tails to account for the extra uncertainty in a small sample.

Common Uses of Z-Scores

• Standardized testing. Comparing a student's raw score to the overall distribution of scores.
• Outlier detection. Flagging data points far from the mean in a dataset.
• Quality control. Checking whether a manufactured item falls within an acceptable range.
• Hypothesis testing. Z-tests use z-scores to judge whether a sample mean differs meaningfully from a known population mean.

History

The z-score builds on the normal distribution, described by Abraham de Moivre in 1733 and developed further by Carl Friedrich Gauss and Pierre-Simon Laplace in the early 19th century. Standardizing values into units of standard deviation became common practice in psychological testing and quality control during the 20th century.

Frequently Asked Questions

What is a z-score in statistics? A z-score is the number of standard deviations a data point sits from the mean of its dataset. It is found with the formula z = (x − μ) / σ.

How do I interpret a z-score result? A z-score of 0 means the value equals the mean. A positive value is above the mean, and a negative value is below it. Values beyond +3 or -3 are unusual.

Can a z-score be negative? Yes. A negative z-score simply means the data point is below the mean. A z-score of -2 means the value is 2 standard deviations below the mean.

What is the difference between a z-score and a t-score? A z-score is used with large samples when the population standard deviation is known. A t-score is used with small samples, generally under 30, when it is not known.

What does cumulative probability mean for a z-score? It is the probability that a randomly selected value from a normal distribution is less than or equal to the given z-score. It equals the area under the normal curve to the left of that point.

What happens if the standard deviation is zero? The z-score is undefined, because the formula would require dividing by zero. A standard deviation of zero means every value in the dataset is identical.

References

1. Standard score, Wikipedia: https://en.wikipedia.org/wiki/Standard_score
2. Normal distribution and z-scores, Khan Academy: https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/z-scores/a/z-scores-review