Why Do IQ Scales Differ?
An IQ score is not an absolute count of solved questions, but a standardized statistical rank relative to a bell-shaped Gaussian normal distribution. In modern psychometrics (such as WAIS-IV, WISC, and Mensa qualification tests), the population mean is set to 100. However, the standard deviation ($\sigma$) differs across psychological batteries:
- Wechsler Scale (WAIS / WISC) & Mensa Standard: Standard deviation $\sigma = 15$. Used in over 85% of clinical, academic, and psychological assessments globally. Mensa cutoff (top 2%) corresponds to an IQ of 130.
- Stanford-Binet Scale (SB5): Standard deviation $\sigma = 16$. Mensa threshold corresponds to 132.
- Cattell Culture Fair Scale (CFIT): Standard deviation $\sigma = 24$. Tailored for non-verbal, culture-free fluid reasoning. The top 2% cutoff is 148!
Mathematical Scale Conversion Formula
To accurately convert an IQ score from one scale to another, we compute the standardized normal deviate (Z-score):
$$Z = \frac{\text{IQ}_1 - 100}{\sigma_1}$$
$$\text{IQ}_2 = 100 + Z \times \sigma_2$$
Example: A score of 130 on the Wechsler scale ($SD = 15$) yields $Z = (130 - 100) / 15 = +2.0\sigma$. On the Cattell scale ($SD = 24$), this equates to $100 + 2.0 \times 24 = 148$.