Z-Score Calculator

Calculate z-scores, probabilities, and percentiles for the standard normal distribution. Convert between raw scores and z-scores.

Z-Score Results

Z-Score

-1.0000

15.87th percentile

Raw Score85.0000
P(X < x)15.87%
P(X > x)84.13%

Calculate

Input Values

Results

Z-Score

-1.0000

Raw Score

85.0000

Percentile

15.87%

P(X < x)

15.8655%

Formula & Calculation

z = (X - μ) / σ

z = (85.00 - 100) / 15

z = -15.0000 / 15

z = -1.0000

Normal Distribution

μ (0)z = -1.00-3σ-2σ-1σ1σ2σ3σ
Red line shows z = -1.00, which is at the 15.9th percentile

Probability Summary

P(X < 85.00)15.8655%
P(X > 85.00)84.1345%
P(-z < Z < z) for |z| = 1.0068.2689%

Common Z-Score Reference

Z-ScoreP(Z < z)PercentileInterpretation
-30.13%0.1thVery low (rare)
-22.28%2.3thLow (unusual)
-115.87%15.9thBelow average
050.00%50.0thAverage (mean)
184.13%84.1thAbove average
1.64595.00%95.0th90% one-tail
1.9697.50%97.5th95% one-tail
297.72%97.7thHigh (unusual)
2.57699.50%99.5th99% one-tail
399.87%99.9thVery high (rare)

?How Do You Calculate Z-Scores?

A z-score measures how many standard deviations a value is from the mean. Formula: z = (x - mean) / standard deviation. A z-score of 0 means the value equals the mean. Positive z-scores are above the mean; negative are below. About 68% of data falls within z = +/-1, 95% within z = +/-2, 99.7% within z = +/-3.

What is a Z-Score?

A z-score (or standard score) indicates how many standard deviations an observation is from the mean of a distribution. Z-scores standardize data to a common scale with mean 0 and standard deviation 1, enabling comparison across different datasets. They are fundamental to statistical inference, hypothesis testing, and probability calculations.

Key Facts About Z-Scores

  • Z-score formula: z = (x - mean) / standard deviation
  • Z = 0 means the value equals the mean
  • Positive z-score: above average; negative z-score: below average
  • 68-95-99.7 rule: 68% within +/-1 SD, 95% within +/-2 SD, 99.7% within +/-3 SD
  • Z-score of 1.96 corresponds to 97.5th percentile (used in 95% confidence intervals)
  • Z-scores allow comparison of values from different distributions
  • In a normal distribution, z-scores directly give percentile rankings
  • Raw score from z: x = mean + (z * standard deviation)

Quick Answer

A z-score measures how many standard deviations a value is from the mean. Formula: z = (x - mean) / standard deviation. A z-score of 0 means the value equals the mean. Positive z-scores are above the mean; negative are below. About 68% of data falls within z = +/-1, 95% within z = +/-2, 99.7% within z = +/-3.

Frequently Asked Questions

A z-score (standard score) measures how many standard deviations a value is from the mean. Z = (X - μ) / σ. A z-score of 0 is the mean, +1 is one standard deviation above, -2 is two standard deviations below.
Z-scores indicate relative position: |z| < 1 is typical (68% of data), |z| < 2 is common (95%), |z| > 2 is unusual (5%), |z| > 3 is rare (0.3%). Positive z = above average, negative z = below average.
The standard normal distribution has mean = 0 and standard deviation = 1. Any normal distribution can be converted to standard normal using z-scores. The total area under the curve equals 1 (100%).
Use the standard normal table or calculator. P(Z < z) gives the area to the left. For P(Z > z), subtract from 1. For P(a < Z < b), calculate P(Z < b) - P(Z < a).
For 90% CI: z = 1.645. For 95% CI: z = 1.96. For 99% CI: z = 2.576. These represent the z-scores that capture the middle percentage of the distribution.

Last updated: 2025-01-15