How to calculate IQ

Basic Concept of IQ Calculation

An intelligence quotient (IQ) is a standardized total score derived from a set of cognitive assessments designed to measure human intelligence. Today, an IQ score is a norm-referenced index that expresses how an individual's performance compares to the general population of the exact same age.

By design, the population average is always anchored at a predetermined mean of 100, with a standard deviation (SD) of 15 on major scales like the Wechsler tests. Standardization is strictly necessary to ensure consistency and reliability; by establishing norms based on a representative sample of the population, clinicians can reliably determine exactly where an individual's cognitive abilities rank relative to their peers.

Early Method: Ratio IQ

In the early 20th century, the concept of "mental age" was introduced. This metric mapped a child's test performance to the average age at which that specific level of performance was typical. To quantify this into a single score, the Ratio IQ method was popularized by Lewis Terman in 1916.

The exact formula for Ratio IQ is:

IQ = (Mental Age ÷ Chronological Age) × 100

For example, if an 8-year-old child completes an intelligence test and achieves a mental age typical of a 10-year-old, their calculation would be: (10 ÷ 8) × 100 = 125.

However, this method possessed fundamental mathematical limitations. The primary limitation was that mental age does not increase at a constant, linear rate indefinitely; it plateaus in late adolescence while chronological age continues to increase. If applied to adults, this formula would cause their IQ scores to artificially plummet as they grew older. Furthermore, the standard deviation of scores fluctuated drastically across different age groups, making cross-age comparisons mathematically invalid.

Estimating the IQ of Retirees (Ages 65+)

While ratio IQ is obsolete for general testing, specialized mathematical models have been proposed to estimate IQ in older populations using mental and chronological age. For estimating the IQ of humans above 65 years old, one modeled formula is:

IQ = 317.1332268 × (MA / CA) − 111.2947030 / CA

This specific mathematical model accounts for the declining rate of IQ in retirees and adjusts the mental age/chronological age ratio with specific constants optimized to prevent the artificial score drops seen in the classic ratio formula.

Modern IQ Calculation Method

To resolve the flaws of the ratio formula, psychometrics shifted to the "Deviation IQ" model, introduced by David Wechsler in 1939. This statistical system treats IQ scores as standardized deviations from an age-specific mean, placing the individual on a normal distribution (bell curve) relative to their peers.

The exact formula for the modern Deviation IQ is:

IQ = 100 + 15z

In order to find "z" (the standard score), the exact calculation is:

z = (raw score − age-group mean) / age-group SD

This model ensures a fixed mean and standard deviation, meaning the vast majority of people cluster around the average, with fewer individuals falling at the extreme high or low ends. Approximately 68% of the population scores within one standard deviation of the mean (between 85 and 115), and 95% fall within two standard deviations (between 70 and 130).

Measurement of IQ: Step-by-Step

1. Taking a standardized IQ test: The individual completes an assessment consisting of various discrete subtests administered under controlled, uniform conditions.
2. Raw score calculation: The test evaluator tallies the raw scores, which represent the total points earned on the individual's test performance.
3. Converting raw scores into scaled scores: The raw score is converted to a "scaled score" using age-specific normative tables. On tests like the WISC or WAIS, these individual subtest scaled scores range from 1 to 19, with a mean of 10 and a standard deviation of 3.
4. Comparing results to norm groups: The examiner adds the scaled scores to get a "Sum of Scaled Scores" (S).
5. Calculating FSIQ from the Sum of Scaled Scores: The Sum of Scaled Scores (S) follows a normal distribution with a mean of 100 and a standard deviation of 20. Because the final Full-Scale IQ requires a standard deviation of 15, examiners convert the sum into the final FSIQ using the statistical conversion formula:
   FSIQ = 100 + ((15 / 20) × (S - 100))
   Simplified, this exact statistical formula is expressed as:
   FSIQ = 25 + 0.75 × S
6. Prorating incomplete IQ tests: If a clinician is unable to administer the full battery of core subtests, they must "prorate" the available scores to estimate the final index or IQ.
7. Estimating a 3-subtest index using only 2 subtests: The formula requires taking the sum of the 2 available scaled scores and multiplying it by 1.5 (or 3/2).
8. Estimating Full-Scale IQ using only 8 or 9 core subtests: The formula requires multiplying the sum of the valid 8 or 9 scaled scores by (10/8) or (10/9), respectively.

Measurement of Intelligence in IQ Tests

Modern IQ tests measure specific facets of abstract reasoning, logic, and cognitive processing rather than acquired general knowledge. Different sections contribute to the total IQ by assessing specific, independent cognitive domains, such as fluid reasoning, verbal comprehension, working memory, and processing speed. Performance on these individual subtests generates specific index scores. Through composite scoring, these index scores are mathematically aggregated to produce the Full-Scale IQ (FSIQ), which serves as the most comprehensive estimate of global intellectual functioning.

Standardization and Norming Process

Test benchmarks are created through a process called standardization, which involves administering the test to a large, representative sample of the population. The role of these large population samples is to identify the mean performance and standard deviation for every specific age cohort so that accurate reference points can be established. Test subjects are only ever compared to a normative sample of their exact-age peers.

Statistical Formulas for IQ Test Variance (The Flynn Effect)

Test benchmarks must be periodically recalibrated and re-standardized due to the Flynn Effect, which is the continuous historical rise in average raw intelligence scores across generations. To maintain an average score of exactly 100, tests must be updated with new norms. When researchers conduct meta-analyses to measure the Flynn Effect between an old test and a newly normed test, they use specific statistical formulas to calculate the variance and precision of the difference in scores.

 

 

 

Interpreting the Final IQ Score

 

What the IQ number actually means is a reflection of statistical rarity and relative standing. Because the normal distribution is mathematically fixed, a specific deviation score always corresponds to a specific percentile ranking.

 

 

 

The classifications for IQ score ranges are interpreted as follows:

 

 

 

• 130 and above: Very Superior / Gifted (ranking at the 98th percentile or higher, representing approximately 2.1% to 2.3% of the population).
• 115 to 129: High Average to Superior (ranking between the 84th and 98th percentiles).
• 85 to 114: Average to Low Average (representing approximately 68% of the population, centered exactly at the 50th percentile for a score of 100).
• 70 to 84: Borderline to Low Average.
• Below 70: Extremely Low (representing the bottom ~2.3% of the population).