Z-Score Calculator

IQ Score Calculator

Wechsler scale (WAIS / WISC): μ = 100, σ = 15.

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Enter an IQ score; get the percentile rank, z-score, and Wechsler classification.

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Worked Examples

IQ → Percentile · Gifted Cutoff

What percentile is an IQ of 130?

IQ 130 is the conventional cutoff for gifted classification on the Wechsler scale.

  1. Convert to a z-score: z = (130 − 100)/15 = 2.0.
  2. Read Φ(2.0) ≈ 0.9772 from the standard normal CDF.
  3. Convert to percentile: 0.9772 × 100 ≈ 97.7th percentile.
  4. Classify: IQ ≥ 130 → Very Superior.

About 2 to 3 children out of every 100 score at or above 130 — the textbook gifted-program threshold.

Percentile → IQ · Top 1%

What IQ is at the 99th percentile?

The 99th percentile is often used as a Mensa-style cutoff in casual conversation.

  1. Convert percentile to z: z = Φ⁻¹(0.99) ≈ 2.3263.
  2. Rescale to IQ: IQ = 100 + 15 × 2.3263 ≈ 134.9.
  3. An IQ of about 135 sits at the 99th percentile on the Wechsler scale.
  4. Classification: Very Superior (IQ ≥ 130).

Mensa accepts the top 2% — that corresponds to a Wechsler IQ of about 130.8 (z ≈ 2.05). The top 1% is a stricter bar at IQ ≈ 135.

IQ → Percentile · Average Range

What percentile is an IQ of 100?

The mean of the Wechsler distribution by construction. Useful as a sanity check.

  1. Convert to a z-score: z = (100 − 100)/15 = 0.
  2. Read Φ(0) = 0.5.
  3. Convert to percentile: 0.5 × 100 = 50th percentile.
  4. Classify: 90 ≤ IQ < 110 → Average.

By definition, IQ 100 is the population median on the Wechsler scale. Half the population scores below 100, half above.

Z-Score → IQ · Two SDs Below

What IQ is z = -2?

Two standard deviations below the mean is the historical cutoff for diagnosing intellectual disability.

  1. Rescale to IQ: IQ = 100 + 15 × (−2) = 70.
  2. Read the percentile: Φ(−2) ≈ 0.0228 → 2.3rd percentile.
  3. Classify: IQ < 70 → Extremely Low.
  4. Roughly 1 in 44 people score at or below this level.

An IQ of 70 by itself is not a diagnosis — clinical intellectual disability requires deficits in adaptive functioning as well, evaluated by a licensed psychologist.

IQ ↔ Z-Score (Wechsler scale)

The Wechsler intelligence scales (WAIS for adults, WISC for children) standardize IQ to a normal distribution with mean 100 and standard deviation 15. Convert any IQ to a z-score by subtracting 100 and dividing by 15, then read the percentile rank from the standard normal CDF Φ(z). The same recipe runs in reverse to find the IQ at any percentile.

z = (IQ − 100) / 15, IQ = 100 + 15·z, Percentile = Φ(z) · 100

How It Works

An IQ score is a standardized score on a normal distribution. The most widely used scales — Wechsler (WAIS, WISC) and most modern revisions — set the mean at 100 and the standard deviation at 15, so an IQ of 115 is exactly one standard deviation above the mean and an IQ of 70 is two standard deviations below. To translate an IQ into a percentile rank, convert it to a z-score using z = (IQ − 100)/15 and look up the cumulative area Φ(z) under the standard normal curve. To go the other way, take the inverse normal of the percentile divided by 100 and rescale: IQ = 100 + 15·z. The classification labels (Average, Superior, etc.) come from the WAIS-IV / WISC-V banding and are reported alongside the numeric score.

Example Problem

A child scores an IQ of 130 on the WISC-V. What percentile does this represent, and how does it classify?

  1. Convert to a z-score on the Wechsler scale: z = (130 − 100)/15 = 2.0.
  2. Read the cumulative area: Φ(2.0) ≈ 0.9772.
  3. Convert to a percentile rank: 0.9772 × 100 ≈ 97.7th percentile.
  4. Classify per the WISC-V band: IQ ≥ 130 → Very Superior (the top ~2.3% of the population).
  5. Interpret: roughly 2 to 3 children out of every 100 score at or above 130.

An IQ of 130 is the conventional cutoff for gifted programs. Because the Wechsler scale uses σ = 15, this corresponds to exactly z = +2 — a bright cutoff to remember.

Key Concepts

IQ scores on different scales are not directly comparable. The Wechsler family uses σ = 15. The Stanford-Binet 5 uses σ = 16, so a Stanford-Binet IQ of 132 is the same standing (z = +2) as a Wechsler IQ of 130. The Cattell scale uses σ = 24. Always check which scale a score is on before comparing. The deviation IQ (used by all modern tests) is fundamentally a z-score in disguise — it is just rescaled so that the mean is a familiar number (100) and one standard deviation is a familiar interval (15 in Wechsler).

Applications

  • Educational placement: gifted programs (typically IQ ≥ 130) and special education eligibility (typically IQ ≤ 70)
  • Clinical assessment: cognitive impairment screening, traumatic-brain-injury baseline comparison
  • Workplace and military testing: ASVAB-derived AFQT scores share the normal-distribution pedigree
  • Research: cognitive epidemiology, longitudinal IQ studies, intervention outcome analysis
  • Score comparison: converting between Wechsler, Stanford-Binet, and Cattell scales using z-scores
  • Percentile reporting: parents and clinicians often prefer percentile ranks over raw IQ for interpretation

Common Mistakes

  • Using σ = 16 (Stanford-Binet) when interpreting a Wechsler score, or vice versa — the calculator above assumes Wechsler
  • Treating IQ as a percentile directly — IQ 100 is the 50th percentile, not the 100th
  • Forgetting that IQ is normally distributed by construction — extreme IQs (above 145 or below 55) become rare quickly because of the bell curve, not because of a hard ceiling
  • Confusing classification bands across editions of the test — older WAIS/WISC editions used slightly different cutoffs (e.g., 'Mentally Retarded' was renamed 'Extremely Low')
  • Comparing modern scores to historical Flynn-effect-uncorrected norms
  • Using the calculator's classification as a clinical diagnosis — only a licensed psychologist can diagnose intellectual disability or giftedness, which require multiple measures and adaptive functioning data

Frequently Asked Questions

What IQ score is at the 95th percentile?

The 95th percentile corresponds to z ≈ 1.6449, so IQ = 100 + 15 × 1.6449 ≈ 124.7 on the Wechsler scale. An IQ of 125 is just above the 95th percentile.

What percentile is an IQ of 130?

IQ 130 is z = +2 on the Wechsler scale, corresponding to about the 97.7th percentile. This is also the conventional cutoff for gifted classification.

What is the difference between IQ scales?

The Wechsler scales (WAIS, WISC) use μ = 100 and σ = 15. The Stanford-Binet 5 uses σ = 16. The Cattell scale uses σ = 24. The same standing (e.g., +2 SD) gives different IQ numbers on each scale, so always check which scale a score is on. This calculator uses the Wechsler scale, which is the most common in clinical and educational use.

What is the IQ classification system?

Modern Wechsler classifications are: Extremely Low (< 70), Borderline (70–79), Low Average (80–89), Average (90–109), High Average (110–119), Superior (120–129), and Very Superior (≥ 130). Older editions used different labels for the same bands.

Is IQ really normally distributed?

Yes — by construction. Test publishers calibrate raw scores against a representative sample so the distribution of IQ closely matches a normal curve with the chosen μ and σ. This makes percentile lookup via the standard normal CDF mathematically valid.

What is the highest IQ ever recorded?

Reliable IQ tests measure poorly above about 4σ (IQ ≈ 160 on the Wechsler scale) because the calibration sample is too thin to characterize the tails. Any reported IQ above ~160 should be viewed skeptically — the test's standard error grows quickly out there.

How many standard deviations away is an IQ of 145?

IQ 145 is z = (145 − 100)/15 = +3 on the Wechsler scale — three standard deviations above the mean, corresponding to roughly the 99.87th percentile (about 1 in 740 people).

Can I use this calculator for the Stanford-Binet scale?

Not directly — this calculator assumes σ = 15 (Wechsler). For Stanford-Binet (σ = 16) you can convert: WAIS-equivalent = 100 + 15 × ((SB − 100)/16). Or use the Z-Score Calculator with custom μ and σ via the Normal Distribution Probability Calculator.

Reference: Wechsler scales (WAIS-IV, WAIS-5, WISC-V) standardize IQ to a normal distribution with μ = 100, σ = 15. The standard normal CDF Φ(z) used here follows Abramowitz & Stegun's rational approximation (|error| < 7.5 × 10⁻⁸).

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