Normal Distribution Calculator
Convert any raw value to a z-score using a custom mean and standard deviation, then find corresponding normal distribution probabilities. Perfect for standardizing test scores, measurements, or survey results.
About this calculator
The z-score formula standardizes a raw value x onto the standard normal scale: z = (x − μ) / σ, where μ is the mean and σ is the standard deviation. A z-score of 0 means the value equals the mean; a z-score of +1 means it is one standard deviation above. Once you have z, you can find percentiles and probabilities using the standard normal cumulative distribution function (CDF). For example, z = 1.96 corresponds to the 97.5th percentile. The empirical rule states that 68%, 95%, and 99.7% of data fall within 1, 2, and 3 standard deviations of the mean respectively. Z-scores are widely used in grading, clinical reference ranges, finance, and quality control.
How to use
A student scores 78 on a test where the class mean is 65 and the standard deviation is 10. Step 1 — Apply the formula: z = (78 − 65) / 10 = 13 / 10 = 1.3. Step 2 — A z-score of 1.3 means the student scored 1.3 standard deviations above average. Step 3 — Using the CDF, P(X ≤ 78) ≈ 0.9032, meaning the student outperformed roughly 90% of the class. Enter value = 78, mean = 65, and standard deviation = 10 to get z = 1.3.
Frequently asked questions
What does a negative z-score mean in a normal distribution?
A negative z-score means the raw value is below the mean. For example, z = −1.5 means the value is 1.5 standard deviations below average. The corresponding cumulative probability will be less than 0.5, meaning fewer than 50% of values fall below that point. Negative z-scores are just as informative as positive ones — they indicate how far below average a measurement sits, which is important in grading, risk assessment, and clinical testing.
How do I use z-scores to compare values from different normal distributions?
Z-scores put values from different distributions on the same standardized scale, making direct comparisons meaningful. If a student scores 80 on a math test (μ=70, σ=8) and 75 on a English test (μ=60, σ=10), their z-scores are (80−70)/8 = 1.25 and (75−60)/10 = 1.50 respectively. Despite the higher raw score in math, the student performed relatively better in English. This standardization technique is essential whenever comparing measurements with different units or scales.
What is the difference between a z-score calculator and a normal distribution probability calculator?
A z-score calculator converts a raw value x into a standardized score using z = (x − μ) / σ, telling you how many standard deviations x is from the mean. A normal distribution probability calculator goes one step further, converting that z-score into an area under the bell curve — a probability or percentile. Many calculators, including this one, combine both steps: you input x, μ, and σ, and it returns both the z-score and the cumulative probability P(X ≤ x). Understanding both outputs gives you the full picture of where a value sits in its distribution.