Z Score Calculator
What a Z Score Calculator reveals
A z score states exactly how far a value sits from the mean, measured in standard deviations. This Z Score Calculator covers both directions: raw scores or probabilities in, z scores out, plus the matching normal probability. Teachers use it for grading curves, analysts for spotting outliers, and quality engineers for tracking process drift.
The formula and a worked example
The core formula is z = (x - mean) / SD. Take an IQ of 130 with mean 100 and SD 15: z = (130 - 100) / 15 = 2, two standard deviations above average, roughly the top 2.3 percent. Run the steps in reverse and any probability returns its matching z score.
A manufacturing example exercises both modes. Bearings are stamped while a shaft line averages 50 mm with SD 0.4 mm; enter Raw score 51.2, Mean 50, and Standard deviation 0.4, and Calculate returns z = 3 with a matching probability of 0.9987, meaning just 0.13 percent of shafts run that wide. Flip to Z score from P, type Probability 0.975, and the answer is z = 1.96, the classic two sided critical value. That beats reading a table, and nudging Precision upward settles ties when two decimals sit close together.
How to run the calculation
First choose the mode: Z score & P from raw score, or Z score from P. For the first, enter the Raw score, the Mean, and the Standard deviation, then press Calculate. For the reverse, type the Probability instead and adjust Precision for extra decimals. Sanity check the sign: values below the average must return negative.
Frequently Asked Questions
What does a negative z score mean?
The raw value sits below the mean. The magnitude shows how far below, so z = -1.5 means one and a half standard deviations under the distribution average.
What percentile is a z of 2?
About the 97.7th percentile, since 97.7 percent of a normal distribution falls below z = 2. Roughly 2.3 percent of values exceed it, an elite but reachable band.
Can a probability produce the z score?
Yes. Choose the Z score from P mode, enter the cumulative Probability, and the calculator inverts the normal curve. It is the fastest route to critical values for any alpha.
Why do z scores enable comparisons?
They strip away units. A 130 IQ test result and a 650 SAT score both become standard deviations from their own means, so different scales sit on one comparable ladder.
Is the mean always zero here?
No. The formula accepts any mean and any positive standard deviation. Mean 0 and SD 1 belong only to the standardized normal distribution, where z scores are the values themselves.