- Z-score
- 1.5
- Normal percentile
- 93.32%
How it works
Central coverage and percentile are different
Under a standard normal model, z = 1.96 is about the 97.5th percentile, while the interval from −1.96 to 1.96 covers about 95%. A one-sided percentile and a central interval describe different shaded regions.
Relating a value to its mean and spread
A z-score expresses how far a value is from the entered mean in standard-deviation units. Its sign shows the direction of that difference. A probability interpretation additionally relies on the selected distribution assumption; standardizing a value does not by itself make the data normally distributed. Use the interval or inverse mode only when that is the probability question you intend to answer.
Z-score: formula and assumptions
z = (x − mean) / standard deviation.
The percentile assumes a normal distribution, not merely a dataset with this mean and spread.
Normal probabilities & inverse: formula and assumptions
Left=Φ(z); right=SF(z); central=1−2SF(|z|); outside=2SF(|z|).
Probabilities use the interval 0–1, not percentages. A central probability determines the magnitude of z; both signs are shown.
Probability between two z-scores: formula and assumptions
Probability=Φ(upper)−Φ(lower), evaluated with direct tails when appropriate.
The standard normal distribution has mean zero and standard deviation one.
Worked example
Using the Z-score example:
- Value
- 85
- Mean
- 70
- Standard deviation
- 10
z = (x − mean) / standard deviation.
Example result
Z-score: 1.5
Normal percentile: 93.32%
These are illustrative inputs. Change the values to explore a different scenario.
Formula references
- OpenStax: Algebra and Trigonometry (opens in a new tab)
- NIST statistical methods handbook (opens in a new tab)
References explain the underlying methods; they do not imply endorsement or individual professional review of Finflo.
Common questions
Does calculating a z-score prove the data are normal?
No. Standardization uses a mean and standard deviation. Interpreting the result as a normal percentile adds a normal-distribution assumption.
Should a probability of 95% be entered as 95 or 0.95?
Enter 0.95 in probability fields. Those modes use probabilities from 0 to 1 rather than percentages.
Prepared by Finflo. See our methodology and editorial policy.