Z Score Calculator
Free z score calculator: turn a raw score into a z-score, or a z-score into probability, percentile and area under the normal curve, with formula and steps.
Calculate a z-score or normal probability
z-score: 1.0000
Left-tail area: 0.841345 (84.13% percentile)
Right-tail area: 0.158655
z = (x − μ) ÷ σ = (70 − 60) ÷ 10 = 1. The left-tail area is Φ(1) ≈ 0.841345.
Z-Score Formula
For a raw score x, population mean μ, and standard deviation σ, calculate the z-score with z = (x − μ) / σ. The result tells you how many standard deviations the score is from the mean. A positive z-score is above the mean; a negative z-score is below it.
For example, if a test score is 70, the mean is 60, and the standard deviation is 10, then z = (70 − 60) / 10 = 1. The score is one standard deviation above average. The calculator uses the standard normal distribution to translate that z-score into a left-tail probability, right-tail probability, or percentile.
A standard deviation must be greater than zero. A z-score describes relative position, so it can compare values measured on different scales when the underlying distributions are reasonably modeled by a normal distribution.
How to Find the Probability from a Z-Score
The standard normal cumulative distribution function, written Φ(z), gives the area to the left of z. The area to the right is 1 − Φ(z). For z = 1.96, the left-tail probability is about 0.9750, or 97.5%, while the right-tail probability is about 0.0250, or 2.5%.
To find the area between two values, subtract the lower cumulative probability from the upper one: P(a < Z < b) = Φ(b) − Φ(a). For example, the area between 0 and 1 is approximately 0.8413 − 0.5000 = 0.3413. For a two-tailed probability beyond a positive cutoff z, double the upper-tail area: 2 × [1 − Φ(|z|)]. This is the standard calculation used for a symmetric two-sided normal test.
How to Find a Z-Score on a TI-84
Use normalcdf to find area under the normal curve and invNorm to work backward from a percentile. Press 2nd, then DISTR, and choose the corresponding function.
- For a left-tail probability, enter
normalcdf(−1E99, z, 0, 1). For example,normalcdf(−1E99, 1.96, 0, 1)is approximately 0.975. - For a right-tail probability, enter
normalcdf(z, 1E99, 0, 1). For the area between two z-scores, enternormalcdf(lower z, upper z, 0, 1). - For a percentile-to-z calculation, enter
invNorm(area to the left, 0, 1). For the 97.5th percentile, useinvNorm(.975, 0, 1), which returns about 1.96.
The mean and standard deviation arguments are 0 and 1 because these examples use the standard normal distribution. For a nonstandard normal variable, enter its actual mean and standard deviation in normalcdf.
Z-Table
This compact table shows cumulative left-tail probabilities for common standard-normal z-values. Values are rounded to four decimal places; use the calculator above for other values.
| z-score | Left-tail probability Φ(z) | Percentile |
|---|---|---|
| 0.0 | 0.5000 | 50.00% |
| 0.500 | 0.6915 | 69.15% |
| 1.0 | 0.8413 | 84.13% |
| 1.500 | 0.9332 | 93.32% |
| 1.645 | 0.9500 | 95.00% |
| 1.960 | 0.9750 | 97.50% |
| 2.0 | 0.9772 | 97.72% |
| 2.576 | 0.9950 | 99.50% |
| 3.0 | 0.9987 | 99.87% |
How to read a z-table
Find the z-score in the table and read the cumulative area to its right. For a negative z-score, use symmetry: Φ(−z) = 1 − Φ(z). A table may show a different number of decimal places, but the underlying area is the same.
Related Calculators
Use the normal approximation calculator when a binomial distribution can be approximated by a normal model. For exact discrete probabilities, use the binomial distribution calculator.
Frequently Asked Questions
What is a good z-score?
A z-score close to 0 means a value is near the mean. Whether a score is “good” depends on what the measurement represents; a positive score is above the mean and a negative score is below it.
Can a z-score be negative?
Yes. A negative z-score means the raw score is below the mean. For example, z = −1 means one standard deviation below the mean.
What does a z-score of 1.96 mean?
A z-score of 1.96 is 1.96 standard deviations above the mean. Under the standard normal model, about 97.5% of values lie below it and about 2.5% lie above it.
What is the difference between a z-score and a percentile?
A z-score measures distance from the mean in standard deviations. A percentile is the share of values at or below a score. The percentile depends on the distribution; for a normal distribution, z = 0 is the 50th percentile.