Z-Score & Normal Distribution Calculator
z-scores, normal probabilities and percentiles — with a shaded bell curve and a z-table.
Result
This is the last answer worked out. Fix the input above to update it.
Hover, tap or focus the curve and use the arrow keys to read P(X < x) anywhere on it.
All values
Steps
The 68–95–99.7 rule for this distribution
Standard normal (z) table
Read a z-score as row + column: z = 1.96 is row 1.9, column 0.06.
| z | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 |
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About the Z-Score & Normal Distribution Calculator
Enter a score with the mean μ and standard deviation σ of its distribution to get its z-score — how many standard deviations it lies above or below the mean — together with the area to its left (its percentile) and to its right. Work back from a z-score to the score, find the probability that a normally distributed value falls below, above, between or outside given values, or do an inverse-normal lookup: the value that cuts off a given area, such as the score of the top 10% or the critical value for 95% confidence.
Every answer comes with the working and a shaded bell curve that you can read at any point, the 68–95–99.7 rule for your distribution, and a complete standard normal (z) table that you can search by z-score or by area. Probabilities are worked out directly from the normal distribution function, so they stay precise far into the tails where printed tables stop.
How to use it
- Choose what to find: the z-score of a value, the value from a z-score, a probability, or the value for an area.
- Enter the mean μ and the standard deviation σ — or press Standard normal for μ = 0 and σ = 1, when your numbers are already z-scores.
- Type the value (several scores at once work too), the values that bound the range, or the area and where it lies (left, right, middle or both tails).
- Read the answer, the steps and the shaded curve; hover over, tap or focus the curve to read P(X < x) anywhere.
- To check a textbook, type a z-score or an area into the z-table search: the matching cell is highlighted with its row and column.
Examples
x = 85 in a test with μ = 70 and σ = 10
z = (85 − 70) ÷ 10 = 1.5 P(X < 85) = Φ(1.5) = 0.933193 — the 93.32nd percentile P(X > 85) = 0.0668072
P(60 < X < 80) with μ = 70, σ = 10
z = −1 and z = 1: Φ(1) − Φ(−1) = 0.841345 − 0.158655 = 0.682689 (68.27%)
Area 0.10 to the right, μ = 70, σ = 10
z = Φ⁻¹(0.9) = 1.281552 → x = 70 + 1.281552 × 10 = 82.8155
Middle area 0.95 of the standard normal
Each tail 0.025 → z = ±1.959964 (±1.96)
Common uses
- Comparing scores from tests with different means and spreads by their z-scores.
- Finding what share of a normal population lies above, below or between given values — heights, measurement errors, fill weights.
- Looking up critical values (1.645, 1.96, 2.576) and checking hand calculations with a z-table.
- Setting cut-offs: the score of the top 5%, or the range that holds the middle 90%.
The formulas
- z-score: z = (x − μ) ÷ σ, and back: x = μ + z × σ.
- Area to the left: P(X < x) = Φ(z), where Φ(z) = ½[1 + erf(z ÷ √2)] is the standard normal distribution function.
- Area to the right: P(X > x) = 1 − Φ(z); between two values: P(a < X < b) = Φ(z_b) − Φ(z_a).
- Inverse: the value with area p to its left is x = μ + σ × Φ⁻¹(p); for a central area p, each tail holds (1 − p) ÷ 2.
The normal density and distribution function are those of the NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.1 Normal Distribution, with erf as defined in the NIST Digital Library of Mathematical Functions §7.2. The calculator evaluates Φ to about 15 significant digits, also in the far tails — P(Z > 10) = 7.62 × 10⁻²⁴ — and Φ⁻¹ with Wichura’s algorithm AS 241.
Reading a z-table
A z-table lists Φ(z), the area to the left of z, for z to two decimals: the row gives the first decimal and the column the second, so z = 1.96 is row 1.9, column 0.06, where the table shows 0.9750. For a negative z the row carries the minus sign — z = −1.25 is row −1.2, column 0.05 (0.1056). The area to the right is 1 minus the table value, and some books print the area between 0 and z instead (0.4750 for z = 1.96); choose that layout above to match yours.
Working backwards, look for the area in the body of the table and read off its row and column; when the area falls between two cells, the calculator shows both neighbours and the exact z.
The 68–95–99.7 rule
In every normal distribution about 68.27% of the values lie within one standard deviation of the mean, 95.45% within two and 99.73% within three. So only about 1 value in 22 is more than two standard deviations from the mean, and about 1 in 370 more than three. The table under the result applies the rule to your μ and σ.
When does the normal distribution apply?
A z-score can be worked out for any data, but the probabilities above are only right when the values are approximately normally distributed — a symmetric, bell-shaped histogram. Many measurements are close to normal (heights, measurement errors, test scores scaled to a bell curve); incomes, waiting times and counts of rare events usually are not. Averages of large samples are approximately normal even when the data are not (the central limit theorem). For the mean of a small sample with σ estimated from the same sample, use the t distribution instead.
Limitations
- Probabilities assume a normal distribution; for skewed data they can be far off, even though the z-score itself is still correct.
- For a sample mean with σ estimated from a small sample, the t distribution — not the normal — gives the right probabilities and critical values.
- The z-table shows four decimals and z from −3.99 to 3.99; the calculator above is not limited to that.
- The curve is drawn for up to ±8 standard deviations; areas further out are still computed and shown in the results.
Privacy
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Frequently asked questions
How do I calculate a z-score?
Subtract the mean from the value and divide by the standard deviation: z = (x − μ) ÷ σ. A score of 85 in a test with mean 70 and standard deviation 10 has z = (85 − 70) ÷ 10 = 1.5: one and a half standard deviations above the mean.
How do I turn a z-score into a percentile?
The percentile is the area to the left of z times 100: Φ(z) × 100. For z = 1 that is 84.13, for z = 1.5 it is 93.32 and for z = −1 it is 15.87 — the share of a normal distribution below that value.
What z-score goes with 95% confidence?
For a two-sided interval (2.5% in each tail) z = 1.96 (1.959964); for a one-sided bound (5% in one tail) z = 1.645. For 90% the two-sided value is 1.645 and for 99% it is 2.576. Choose Value for an area and press a critical-value button to see them worked out.
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean: z = −2 is two standard deviations below it. The area to its left is small (Φ(−2) = 0.0228), so only about 2.3% of a normal distribution lies lower.
How do I find the probability between two z-scores?
Subtract the smaller area from the larger: P(z₁ < Z < z₂) = Φ(z₂) − Φ(z₁). Between z = −1 and z = 1 that is 0.841345 − 0.158655 = 0.682689. Choose Probability, then between two values, and type the two values (with μ = 0 and σ = 1 for z-scores).
Is P(X < x) the same as P(X ≤ x)?
For the normal distribution, yes: it is continuous, so the probability of hitting one exact value is 0 and including or excluding the end point changes nothing. It matters for discrete counts, such as the binomial distribution.