An IQ score has a mean of 100 and a standard deviation of 15. Genius is sometimes characterized as a person with an IQ score of 140 or above. What fraction of the population scores at the genius level? What would be the IQ score corresponding to the 90th percentile
The Correct Answer and Explanation is:
Correct Answers:
- Fraction of population with IQ ≥ 140:
Convert to z-score:
z=140−10015=4015≈2.67z = \frac{140 – 100}{15} = \frac{40}{15} \approx 2.67z=15140−100=1540≈2.67
From z-tables, the area to the left of z = 2.67 is approximately 0.9963.
Therefore, the fraction scoring 140 or above is:
1−0.9963=0.00371 – 0.9963 = 0.00371−0.9963=0.0037, or 0.37% of the population. - IQ score at the 90th percentile:
From z-tables, the z-score corresponding to the 90th percentile is approximately 1.28.
Convert back to IQ:
IQ=100+(1.28)(15)=100+19.2=119.2IQ = 100 + (1.28)(15) = 100 + 19.2 = \textbf{119.2}IQ=100+(1.28)(15)=100+19.2=119.2
Explanation:
IQ scores are generally modeled using a normal distribution. This distribution is characterized by a bell-shaped curve, symmetric around the mean. In this case, the distribution has a mean (μ) of 100 and a standard deviation (σ) of 15. The standard deviation measures the average spread of scores around the mean.
To determine the proportion of individuals who qualify as geniuses (IQ ≥ 140), the first step involves converting the raw score to a standard normal score, called a z-score. This standardizes the value and allows the use of the standard normal table. A z-score of approximately 2.67 indicates that a person’s IQ of 140 is 2.67 standard deviations above the mean. According to the standard normal distribution, about 99.63% of people score below this level. Thus, only about 0.37% of people score 140 or higher.
To find the IQ score corresponding to the 90th percentile, the z-score representing the 90th percentile is required. This value is about 1.28. Multiplying the z-score by the standard deviation and adding the mean yields the IQ that marks the top 10% of the population—approximately 119.2. This score defines the threshold below which 90% of the population falls.
