What is the z-score?
Fill in any three of the value, mean, standard deviation, and z-score to get the fourth, with its percentile.
- Z-score
- 1.5
A value of 85 with mean 70 and standard deviation 10 has a z-score of 1.5; 93.32% of a normal population is below it.
+1 to +2 SD
- Below −3 SD
- −3 to −2 SD
- −2 to −1 SD
- Within ±1 SD
- +1 to +2 SD
- +2 to +3 SD
- Above +3 SD
- Percentile
- 93.32%
- P(Z < z), area to the left
- 0.933193
- P(Z > z), area to the right
- 0.066807
- P(−|z| < Z < |z|), area between
- 0.866386
- P(|Z| > |z|), two-tailed
- 0.133614
Z-score: 1.5. A value of 85 with mean 70 and standard deviation 10 has a z-score of 1.5; 93.32% of a normal population is below it.
Where is it on the normal curve?
How to calculate
Computes the z-score of a value from the mean and standard deviation, or any one of the four from the other three, with the percentile and tail probabilities of the standard normal distribution.
Example with the default inputs (Value (x) 85, Mean (μ) 70, Standard deviation (σ) 10): A value of 85 with mean 70 and standard deviation 10 has a z-score of 1.5; 93.32% of a normal population is below it.
Formula: z = (x − μ) ÷ σ; Φ(z) is the standard normal cumulative distribution function.
- The mean and standard deviation are those of the population the value comes from.
- The percentile and the probabilities assume the values follow a normal distribution. The z-score itself does not.
- The standard deviation must be more than 0.
Worked examples
Each example is checked against the calculator on every build.
- Value (x) 85, Mean (μ) 70, Standard deviation (σ) 10 gives Z-score 1.5, P(Z < z), area to the left 0.933193, P(Z > z), area to the right 0.066807, Percentile 93.31928%, P(|Z| > |z|), two-tailed 0.133614.Source: hand calculation in content.mdx: (85 − 70) ÷ 10 = 1.5; NIST e-Handbook 1.3.6.7.1 table, 0.5 + 0.43319 = 0.93319; Python NormalDist().cdf(1.5)
- Z-score -2, Mean (μ) 100, Standard deviation (σ) 15 gives Value (x) 70, P(Z < z), area to the left 0.02275, P(−|z| < Z < |z|), area between 0.9545.Source: hand calculation in content.mdx: 100 + (−2) × 15 = 70; NIST e-Handbook 1.3.6.7.1 table, 0.5 − 0.47725 = 0.02275; Python NormalDist().cdf(-2)
- Value (x) 130, Mean (μ) 100, Z-score 2 gives Standard deviation (σ) 15.Source: hand calculation in content.mdx: (130 − 100) ÷ 2 = 15
- Value (x) 60, Z-score -1, Standard deviation (σ) 5 gives Mean (μ) 65, P(−|z| < Z < |z|), area between 0.682689.Source: hand calculation in content.mdx: 60 − (−1) × 5 = 65; NIST e-Handbook 1.3.6.7.1 table, 2 × 0.34134 = 0.68268; Python 1 − 2 × NormalDist().cdf(-1)
- Value (x) 70, Mean (μ) 70, Standard deviation (σ) 12 gives Z-score 0, P(Z < z), area to the left 0.5, P(Z > z), area to the right 0.5, P(|Z| > |z|), two-tailed 1, P(−|z| < Z < |z|), area between 0.Source: hand calculation in content.mdx: (70 − 70) ÷ 12 = 0, and Φ(0) = 0.5 by symmetry
How it works
The z-score (standard score) of a value x from a population with mean μ and standard deviation σ is
z = (x − μ) ÷ σ
Fill in any three and the calculator finds the fourth:
- z = (x − μ) ÷ σ
- x = μ + z × σ
- μ = x − z × σ
- σ = (x − μ) ÷ z, which has an answer only when it comes out above 0 (x above the mean for a positive z, below it for a negative z)
From z, it also shows these standard normal probabilities, where Φ is the cumulative distribution function of the standard normal distribution (mean 0, standard deviation 1, NIST e-Handbook 1.3.6.6.1):
- P(Z < z), the area to the left: Φ(z).
- Percentile: 100 × Φ(z), in percent.
- P(Z > z), the area to the right: 1 − Φ(z), worked out as Φ(−z).
- P(|Z| > |z|), both tails: 2 × Φ(−|z|).
- P(−|z| < Z < |z|), the area between: 1 − 2 × Φ(−|z|).
The band under the result says how many standard deviations (SD) the value is from the mean. Each band includes its lower end and excludes its upper end:
| Band | z |
|---|---|
| Below −3 SD | z < −3 |
| −3 to −2 SD | −3 ≤ z < −2 |
| −2 to −1 SD | −2 ≤ z < −1 |
| Within ±1 SD | −1 ≤ z < 1 |
| +1 to +2 SD | 1 ≤ z < 2 |
| +2 to +3 SD | 2 ≤ z < 3 |
| Above +3 SD | z ≥ 3 |
Assumptions
- μ and σ are the mean and standard deviation of the population the value comes from, and σ is more than 0.
- The probabilities and the percentile assume the values follow a normal distribution. The z-score itself is valid for any distribution.
- Φ is computed to full double precision, not read from a table, so it can differ from a printed table in the fifth decimal place.
- The chart draws the normal curve with mean μ and standard deviation σ from μ − 4σ to μ + 4σ and shades the area below x, which is the percentile.
Worked examples by hand
A score of 85, mean 70, standard deviation 10 (the default). z = (85 − 70) ÷ 10 = 15 ÷ 10 = 1.5. NIST's table gives 0.43319 for the area from 0 to 1.5, so Φ(1.5) = 0.5 + 0.43319 = 0.93319, the 93.32% percentile. The area to the right is 1 − 0.93319 = 0.06681, and both tails together are 2 × 0.06681 = 0.13361.
z = −2, mean 100, standard deviation 15. x = 100 + (−2) × 15 = 70. NIST's table gives 0.47725 for the area from 0 to 2, so by symmetry Φ(−2) = 0.5 − 0.47725 = 0.02275. The area within 2 standard deviations is 1 − 2 × 0.02275 = 0.9545.
Value 130, mean 100, z = 2. σ = (130 − 100) ÷ 2 = 15.
Value 60, z = −1, standard deviation 5. μ = 60 − (−1) × 5 = 65. NIST's table gives 0.34134 for the area from 0 to 1, so the area within 1 standard deviation is 2 × 0.34134 = 0.68268.
Value 70, mean 70, standard deviation 12. z = 0 ÷ 12 = 0, and Φ(0) = 0.5 because the normal curve is symmetric about its mean.
Other questions people ask
How do I calculate a z-score?
Subtract the mean from the value, then divide by the standard deviation: z = (x − μ) ÷ σ. A test score of 85 in a class with mean 70 and standard deviation 10 has z = (85 − 70) ÷ 10 = 1.5.
What does a z-score tell me?
It tells you how many standard deviations a value is from the mean. A positive z is above the mean, a negative z is below it, and z = 0 is exactly the mean. Because it has no units, you can compare values from different scales, such as an SAT score and an ACT score.
How do I turn a z-score into a percentile?
Look up Φ(z), the area to the left of z under the standard normal curve, and multiply by 100. For z = 1.5, NIST’s table gives an area of 0.43319 from 0 to 1.5, so Φ(1.5) = 0.5 + 0.43319 = 0.93319, the 93rd percentile. This assumes the values follow a normal distribution.
What is a good or unusual z-score?
No z-score is good or bad on its own. In a normal distribution, about 68.3% of values lie within 1 standard deviation of the mean, 95.4% within 2, and 99.7% within 3. So a z-score beyond ±2 is uncommon (about 1 value in 22), and beyond ±3 is rare (about 1 in 370).
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean. A z-score of −2 is two standard deviations below the mean, and about 2.3% of a normal population is lower still.
How do I find the value for a given z-score?
Rearrange the formula: x = μ + z × σ. With a mean of 100 and a standard deviation of 15, a z-score of −2 is the value 100 − 2 × 15 = 70. Enter the mean, standard deviation, and z-score, and leave the value empty.
Should I use the population or the sample standard deviation?
The formula uses the population mean and standard deviation. If you only have a sample, the sample mean and sample standard deviation are the usual estimates, and the result is an estimate of the z-score.