Z-Score Calculator
Calculate standard scores and probabilities for normal distributions.
Formulas Used
What Is a Z-Score?
A Z-score (also called a standard score) gives you an idea of how far a data point is from the mean of a dataset. Technically, it's a measure of how many standard deviations below or above the population mean a raw score is.
Z-scores are a way to compare results from a test to a "normal" population. Results from tests or surveys have thousands of possible results and units. However, those results can often seem meaningless. For example, knowing that someone's weight is 150 pounds might be good information, but if you want to compare it to the "average" person's weight, looking at a vast table of data can be overwhelming. A z-score can tell you where that person's weight is compared to the average population's mean weight.
Interpreting Z-Scores
- Z-score = 0: The raw score is exactly equal to the mean.
- Positive Z-score: The raw score is higher than the mean average. For example, a z-score of +1.5 means the score is 1.5 standard deviations above the mean.
- Negative Z-score: The raw score is lower than the mean average. A z-score of -2 means the score is 2 standard deviations below the mean.
P-Values and Normal Distribution
Once you calculate the z-score, you can use a Standard Normal Table (Z-table) or a calculator to find the probability of observing that score (or a more extreme one). This is often called the p-value in hypothesis testing.
- Left-tailed P(Z < z): The probability of getting a score less than the one observed.
- Right-tailed P(Z > z): The probability of getting a score greater than the one observed. (Also known as 1 - Left-tailed).
- Two-tailed P(|Z| > |z|): The probability of getting a score more extreme in either direction. This is exactly double the smaller of the left/right tailed probabilities.
Frequently Asked Questions
A Z-score of 2 indicates that the raw score is exactly two standard deviations above the population mean. In a normal distribution, only about 2.28% of the population will score higher than a Z-score of +2.
Yes, any raw score that is below the mean will result in a negative Z-score. It simply means the value is smaller than the average.
There isn't a universally "good" Z-score; it depends on context. If it's a test score, a higher positive Z-score is better. If it's a race time, a lower negative Z-score is better. Generally, scores between -2 and +2 are considered "usual," while scores beyond those are considered "unusual" or "outliers."
Once you have the Z-score, you use the Standard Normal Distribution (Z-table) to find the corresponding percentile. The left-tailed probability P(Z < z) multiplied by 100 gives the percentile. For example, a Z-score of 1.28 corresponds to approximately the 90th percentile, meaning 90% of the population scores below that value.
In finance, the Z-score is best known from the Altman Z-Score model, which predicts the probability of a company going bankrupt within two years. It uses five financial ratios, weighted and summed, to produce a single score. A score below 1.81 indicates financial distress, 1.81–2.99 is a grey zone, and above 2.99 suggests financial health.
A Z-score is used when you know the population mean and standard deviation, or when you have a large sample (n > 30). A T-score (from the Student's t-distribution) is used when you are working with small samples and the population standard deviation is unknown, estimating it from the sample instead. The T-distribution has heavier tails than the normal distribution, reflecting greater uncertainty with smaller samples.