Z-score Calculator

What this calculator does

A z-score, also called a standard score, expresses a raw value as the number of standard deviations it lies above or below the mean. This conversion puts measurements from different scales on a common footing. A score of 75 on one test and a score of 620 on another are not directly comparable, but their z-scores are.

This calculator converts a raw score x, a population mean μ, and a population standard deviation σ into a z-score. It also estimates three normal probabilities: the chance of a value below x, the chance of a value above x, and the chance of a value between the mean and x. The probability estimates use the standard normal distribution.

How to use this calculator

1. Enter the raw score you want to standardize.
2. Enter the mean of the reference population.
3. Enter the population standard deviation.
4. Click Calculate.

The headline shows the z-score. The rows below show the input values and the three probability estimates. A positive z-score means the value is above the mean; a negative z-score means it is below.

Formulas

z = (x − μ) / σ

P(X < x) = Φ(z)
P(X > x) = 1 − Φ(z)
P(mean < X < x) = |Φ(z) − 0.5|

Φ is the cumulative distribution function of the standard normal distribution. The calculator evaluates it with a numerical approximation that is accurate to several decimal places. The probability between the mean and x is the absolute distance from 0.5 because the normal curve is symmetric.

Default example

The default inputs are a raw score of 5, a mean of 3, and a standard deviation of 2. The z-score is (5 − 3) / 2 = 1, meaning the value lies one standard deviation above the mean.

For a standard normal distribution, the probability of a value below 5 is about 0.841345. The probability above 5 is about 0.158655. The probability between the mean of 3 and the value 5 is about 0.341345. These three values describe the same position from different directions.

Assumptions and limitations

• The standard deviation must be greater than zero. A zero standard deviation would make the division undefined and would mean every observation is identical.
• The z-score itself only requires a mean and a standard deviation. The probability estimates additionally assume that the underlying distribution is normal.
• The probabilities are estimates from a continuous model. Real data may be discrete, bounded, or skewed, so the normal approximation may not fit.
• A sample standard deviation can be used when the sample is large, but for small samples with an unknown population standard deviation a t distribution is more appropriate.
• Very extreme z-scores produce probabilities that round to zero or one at the displayed precision.
• The calculator does not perform a hypothesis test or return a p-value for a specific test design.

Frequently asked questions

What does a z-score of zero mean?

A z-score of zero means the raw value is exactly equal to the mean. It sits at the center of the standard normal curve, and the probability of observing a value below it is 0.5.

Is a z-score the same as a percentile?

They are related but not identical. The percentile is the percentage of the distribution at or below the value. For a normal distribution, the percentile is Φ(z) × 100. A z-score of 1 corresponds to roughly the 84th percentile, not the 100th.

Why does a negative z-score matter?

A negative z-score simply shows that the value is below the mean. Its magnitude still measures distance from the mean in standard deviation units, and the probability below the value is less than 0.5. The sign carries direction, not importance.

Source

Reference: Calculator.net — Z-score Calculator.