Z-Score Calculator
Calculate the Z-score (standard score) from a raw score, population mean, and standard deviation. Instantly find probabilities and percentiles under the normal distribution curve.
What is a Z-Score?
A Z-score (or standard score) indicates how many standard deviations a data point is from the mean (average) of the distribution. It is a key concept in statistics for standardizing and comparing data from different distributions.
- Z = 0: The score is exactly at the mean.
- Positive Z: The score is above the mean.
- Negative Z: The score is below the mean.
The Z-Score Formula
The formula for calculating a Z-score is:
z = (x - μ) / σ
Where:
- x is the raw score you want to standardize.
- μ (mu) is the population mean.
- σ (sigma) is the population standard deviation.
Understanding Probabilities
Once you calculate a Z-score, you can use a Standard Normal Table (Z-table) to find the probability associated with that score. This calculator automates that process:
- P(Z < z): The percentile rank. The percentage of data falling below this score.
- P(Z > z): The percentage of data falling above this score.
? Frequently Asked Questions
There is no 'good' or 'bad' Z-score inherently; it describes position. However, in many contexts (like test scores), a positive Z-score means you performed better than average. A Z-score of +2.0 puts you in the top 2.3% of the population.
Yes. A negative Z-score simply means the raw score is below the mean. For example, if the average height is 70 inches and you are 68 inches, your Z-score will be negative.
A Z-score can be directly converted to a percentile using the standard normal distribution. Z = 0 corresponds to the 50th percentile. Z = +1 is roughly the 84th percentile.