AI Financial Assistant
BetaAsk questions about your calculation results
3 free questions per session
AI provides general information, not financial advice. Always consult a qualified professional.
What Is a Z-Score?
A z-score tells you how many standard deviations a data point is from the mean. The formula is z = (x - μ) / σ. A z-score of 0 means the value equals the mean. Positive z-scores are above the mean; negative z-scores are below. It standardizes values from different distributions for comparison.
Interpreting Z-Scores
In a normal distribution, a z-score of 1.0 means the value is one standard deviation above the mean (84th percentile). A z-score of 2.0 corresponds to the 97.7th percentile. Values with z-scores beyond ±3 are considered outliers, occurring less than 0.3% of the time.
Z-Scores and Probability
Z-scores connect directly to probability through the standard normal distribution table. The cumulative probability tells you what percentage of the population falls below that z-score. This is essential for hypothesis testing, quality control, and determining statistical significance.
Applications of Z-Scores
Z-scores are used in standardized testing (SAT, IQ scores), financial analysis (identifying unusual stock returns), quality control (Six Sigma), medical research (comparing patient measurements to reference populations), and any field that requires comparing values from different scales.
Frequently Asked Questions
A negative z-score means the value is below the mean. For example, a z-score of -1.5 means the value is 1.5 standard deviations below average.
A z-score of approximately 1.645 corresponds to the 95th percentile (one-tailed), and ±1.96 captures the middle 95% of the distribution (two-tailed).
You can calculate z-scores for any distribution, but the probability interpretations are only accurate for approximately normal (bell-shaped) distributions. For skewed data, consider other methods.
It depends on context. In testing, higher is usually better. In quality control, closer to 0 means closer to target. Generally, z-scores between -2 and +2 are considered typical.