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%. The calculator shows the z-score to four decimal places and the probability as a percentage to two decimal places, so this example displays 1.0000 and 84.13%. 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

In a normal distribution, about 95% of values lie within 2 standard deviations of the mean, and about 99.7% lie within 3. This is called the empirical rule, or the 68-95-99.7 rule. Z-scores beyond +3 or -3 are therefore rare and are often checked as possible 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 uses the population standard deviation, so it assumes that value is known. A t-score is used when the population standard deviation is unknown and the sample standard deviation is used in its place. An older rule of thumb used the t-distribution only for samples of 30 or fewer, but current practice is to use it whenever the sample standard deviation estimates the population value. The t-distribution has thicker tails than the normal curve, which accounts for the extra uncertainty.

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. Abraham de Moivre derived the normal curve in 1733 as an approximation to the binomial distribution. Carl Friedrich Gauss and Pierre-Simon Laplace developed it further.

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 when the population standard deviation is known. A t-score is used when it is not known and the sample standard deviation is used instead, whatever the sample size.

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. OpenStax. Introductory Statistics 2e, 6.1 The Standard Normal Distribution. The standard normal distribution — the z-score formula z = (x − μ)/σ and the 68-95-99.7 empirical rule.
2. NIST/SEMATECH. e-Handbook of Statistical Methods, Cumulative Distribution Function of the Standard Normal Distribution. Standard normal table — area under the normal curve from 0 to z (0.34134 at z = 1).
3. OpenStax. Introductory Statistics 2e, 8.2 A Single Population Mean using the Student t Distribution. Student's t-distribution — when to use t instead of z, and its thicker tails.
4. MacTutor History of Mathematics, University of St Andrews. Abraham de Moivre. Abraham de Moivre biography — de Moivre's 1733 derivation of the normal curve.