TAHA Engineering Toolsby TAHA Knowledge Lab

Mathematics & Statistics

Z-Score Calculator

z = (x − μ) ÷ σ and back, with percentile, upper-tail and two-tailed probabilities.

Inputs

Calculate from
Bell curve with axis marked in standard deviations from −3 to 3; a value x at z = 1.5 with the area to its left shaded as the percentile.−3−2−10123μxz σpercentilez = (x − μ) ÷ σ
The z-score counts standard deviations σ from the mean μ. The shaded area left of x is the percentile. Diagram not to scale.

Result

z-score
1.5
Percentile P(X < x)
93.32%
P(X < x) as a probability
0.9331928
P(X > x) (upper tail)
0.066807201
P(|Z| > |z|) (two-tailed)
0.1336144
Meaning
1.5 standard deviations above the mean.

85 has a z-score of 1.5. For a normal distribution, 93.32% of values are lower (upper tail 0.066807201).

Educational estimate — check the assumptions below.

About this calculator

A z-score (standard score) says how many standard deviations a value lies above or below the mean. It puts different measurements on one scale: a z-score of 2 is equally unusual whether it is a shaft diameter, a test score or a cycle time. If the data are normally distributed, the z-score also gives the percentile — the share of values below it.

To find probabilities between two values, or the value for a given probability, use the Normal Distribution calculator.

Formula

Value → z-score

z = (x − μ) ÷ σ

Percentile = Φ(z)    upper tail = 1 − Φ(z)    two-tailed = 2 × (1 − Φ(|z|))

Φ is the standard normal cumulative distribution function.

z-score → value

x = μ + z × σ

Percentile = Φ(z)

Worked example

Mean 70, standard deviation 10, value 85:

  • z = (85 − 70) ÷ 10 = 1.5
  • Φ(1.5) = 0.93319 → the value is at the 93.32nd percentile; upper tail 0.066807; two-tailed 0.13361.
  • Back again: x = 70 + 1.5 × 10 = 85.

Assumptions and limitations

  • The z-score itself needs only the mean and standard deviation. The percentile and tail probabilities assume a normal distribution.
  • Φ is computed with a series near the centre and a continued fraction in the tails; it agrees with SciPy within 2.3 parts in 10¹³ (worst case from z = −38 to 8), also for very small tail probabilities.
  • Use the population mean and standard deviation if known; with sample estimates from few data the probabilities are approximate.

References

  • NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.1 (normal distribution).
  • Abramowitz, M. & Stegun, I. A. Handbook of Mathematical Functions, chapter 26.