How Far Is This Value From Average? Z-Score Calculator
How far is this value from average?
Z-Score Calculator
Percentile, Probability & Normal Distribution
Enter any value, mean, and standard deviation — get z-score, percentile, and full probability breakdown in real time.
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A z-score answers one of the most fundamental questions in statistics: relative to the average, how unusual is this particular value? Z-scores express distance from the mean in units of standard deviations — making them the universal currency for comparing measurements across completely different scales. A student who scored 85 on a test with mean 75 and standard deviation 8 has a z-score of 1.25 — meaning their score is 1.25 standard deviations above average. That z-score can be looked up in a standard normal table (or calculated directly) to find that roughly 89.4% of students scored lower. The same z-score framework applies to any normally distributed data: heights, IQ scores, manufacturing tolerances, investment returns, blood pressure readings, and standardized test scores. Z-scores are the foundation of statistical inference. They power confidence intervals, hypothesis tests, quality control limits, and the calculation of p-values. Understanding z-scores means understanding how to answer: "Is this result surprising, or within normal variation?" — which is the central question of statistics. This calculator computes the z-score from any value, mean, and standard deviation, then converts it to a percentile (cumulative probability) and one-tailed probability — answering how common or rare the observation is.
- →Finding how many standard deviations a test score, measurement, or result is from the mean
- →Converting a raw score to a percentile rank
- →Checking whether a value is within normal range or an outlier
- →Comparing scores from different tests or scales on equal footing
- →Statistics homework involving normal distribution, hypothesis testing, or confidence intervals
A pediatrician measures a 6-year-old boy's height as 46.5 inches. The population mean for 6-year-old boys is 45.5 inches with SD = 2.1 inches. Z-score = (46.5 − 45.5) / 2.1 = 0.476. Percentile: 68th percentile — this child is taller than 68% of boys his age. The doctor notes the child is in a completely normal range (z-score between −2 and +2 covers 95% of children).
What is a z-score?
A z-score (also called standard score) measures how many standard deviations a data point is from the mean of its distribution. Formula: z = (x − μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. Positive z-scores are above the mean; negative z-scores are below.
What is a 'normal' z-score?
For normally distributed data, 68.27% of values have z-scores between −1 and +1, 95.45% between −2 and +2, and 99.73% between −3 and +3. Values with |z| > 2 are often considered unusual; |z| > 3 is considered a statistical outlier. In many fields, |z| > 1.96 corresponds to statistical significance at the 5% level.
How do I convert a z-score to a percentile?
The percentile (cumulative probability) corresponds to the area under the standard normal curve to the left of your z-score. For z = 1.0, the percentile is 84.13% — meaning 84.13% of values fall below this point. This calculator computes this conversion automatically using the standard normal CDF approximation.
What is the difference between z-score and t-score?
Both measure distance from the mean in standard deviation units, but t-scores are used when the population standard deviation is unknown and estimated from a sample (which is most of the time in practice). T-scores follow a t-distribution rather than normal distribution, and the shape depends on sample size (degrees of freedom). Z-scores assume you know the true population σ.
Can z-scores be negative?
Yes. A negative z-score simply means the value is below the mean. A z-score of −1.5 means the value is 1.5 standard deviations below average. There's no limit to how negative or positive a z-score can be, though values beyond ±4 are extremely rare in practice.