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Z-Score & Percentile Calculator

A z-score says how many standard deviations a value is from the mean; the percentile says what fraction of the population falls below it. Convert in either direction, compute the area under the standard normal curve between two z-scores, or look up critical z values for a 90 / 95 / 99% confidence interval.

i New to z-scores and percentiles? Read the 4-min primer ▾

What it is. A z-score says how many standard deviations a value sits above or below the mean. A z = +1.5 means “1.5 SDs above average.” A percentile says what fraction of the population is at or below a value - the 90th percentile means 90% scored at or below that point. For a normal (bell-shaped) distribution, z and percentile are the same statement in different units.

How to read it. z = 0 sits at the 50th percentile, z = 1 at the 84th, z = 1.96 at the 97.5th (the “magic number” for a 95% CI), and z = 3 at the 99.87th - only ~0.13% of the population is more than 3 SDs above the mean. Negative z-scores mirror the same percentages in the lower tail.

The recipe. Raw → z: z = (x − μ) / σ. z → percentile: look up Φ(z), the cumulative normal. Percentile → z: invert Φ. Tail area beyond z: 1 − Φ(z); two-tailed: 2(1 − Φ(|z|)). Sample mean → z: divide by SE = σ/√n.

Picking the mode. Have a raw score? Use raw→z. Have a z and want percentile or p-value? z→percentile or beyond. Need a critical value for a CI? Percentile→z (e.g., 0.975 → 1.96). Want a 68/95/99.7-style central area? Area-between two z. Skewed data? Trust percentiles, distrust z - this tool assumes the normal model.

5 modes · one tool · Wichura AS 241 inverse normal · Hart's algorithm · Runs in your browser

Try a real-world example to load.

📐 SAT score

A test score of 1280 in a distribution with mean 1050, SD 200. Convert to z, then percentile.

Output
R code RUNNABLE
R Reproduce in R 

        

      

      

        
Standard normal INTERACTIVE

        

          
          

          

          

        

      

    

  


  

    
Inference

    

  


  

    
      Read more
      Anatomy of the standard normal
    
    

      
Live recap - your inputs plugged in
Enter inputs above to see the derivation chain.

      

        
z = (x − μ) / σ
percentile = 100 · Φ(z)

        
Z-score from a raw value. How many standard deviations above (or below) the mean a raw score sits. Comparable across distributions with different units.
INx = your Raw score x, μ = your Mean μ, σ = your Standard deviation σ.OUTz-SCORE headline + percentile in the aux line. For sample means, set n > 1 to use SE = σ/√n.

      

      

        
P(Z ≤ z) = Φ(z)
percentile = 100 · Φ(z)
Right tail = 1 − Φ(z)
Two-tailed = 2 · (1 − Φ(|z|))

        
From z to percentile / tail area. The standard normal cdf Φ. We use Abramowitz–Stegun 26.2.17 for the cdf (max abs error 7.5e-8) and Wichura's AS 241 for the inverse.
INz = your z-score.OUTHeadline percentile; left/right/two-tailed in the recap.

      

      

        
z = Φ⁻¹(p)
For α-level two-sided crit:
z_crit = Φ⁻¹(1 − α/2)

        
Inverse: percentile to z. Used for confidence-interval critical values: 95% CI uses Φ⁻¹(0.975) ≈ 1.96; 99% uses Φ⁻¹(0.995) ≈ 2.576.
INp = your Percentile (auto-detects 0–1 fraction or 0–100%).OUTz-SCORE headline; if p > 0.5, also a two-tailed critical at α = 1 − p.

      

      

        
P(z₁ < Z < z₂) = Φ(z₂) − Φ(z₁)
P(outside) = 1 − P(z₁ < Z < z₂)

        
Area between two z's. Useful for “what fraction of the population is between μ ± k SDs” - the 68–95–99.7 rule comes from this.
INz₁ = your Lower z₁, z₂ = your Upper z₂ (auto-swapped if reversed).OUTHeadline area = Φ(z₂) − Φ(z₁); also the outside-interval entry in the recap.

      

      

        
Left tail:   P(Z ≤ z)  = Φ(z)
Right tail:  P(Z ≥ z)  = 1 − Φ(z)
Two-tailed:  P(|Z| ≥ |z|) = 2(1 − Φ(|z|))

        
Tail probabilities. Two-tailed p-values, “fraction of population beyond k SDs,” and the long-tail risk all use these formulas.
INz = your z-score, direction = your Tail selector.OUTHeadline tail probability; the recap shows all three for comparison.

      

    

  


  

    
      Caveats
      When this is the wrong tool
    
    

      

        
If you have…

        
Use instead

        
A t-distribution (small n) instead of z

        
Use the t critical values inside the CI calculator - standalone t-tool coming in Batch 6.

        
χ² or F for variance / ratio tests

        
The CI calculator uses both internally; standalone tools planned for Batch 6.

        
Heavily skewed data

        
Empirical percentiles only; z-scores are misleading. Run a normality check first - see the Normality Test Picker.

        
Need a CI, not just a critical value

        
Use the Confidence Interval Calculator - pass the critical value through automatically.

        
p-values smaller than ~10⁻¹⁵

        
Numerical noise dominates beyond that. Just report “p < 10⁻¹⁵.”

      

    

  


  

    
Further reading

    
    
Numerical accuracy: Φ(z) accurate to ~7.5 × 10⁻⁸ (Hart's algorithm); Φ⁻¹(p) to ~10⁻⁹ (Wichura AS 241).