Question

You took a quiz in Mathematics but your teacher did not tell you your exact answer was. Instead,he informed you that 20% of your classmates higher than you. What was your score?

If your IQ is 85, what percentage of people have a higher IQ than you? IQ mean is 100 and the standard deviation is 15.

Answer 1

Let $X=$ IQ score: $X\sim N(\mu, \sigma^2)$

Then $Z=\dfrac{X-\mu}{\sigma}\sim N(0, 1)$

Given $\mu=100, \sigma=15$

If 20% of your classmates higher than you , what is your IQ?

$$P(X> x)=1-P(X\leq x)=$$ $$=1-P(Z\leq\dfrac{x-100}{15})=0.2$$

$$P(Z\leq\dfrac{x-100}{15})=0.8$$

$$\dfrac{x-100}{15}\approx0.841621$$

$$x\approx112.6$$

My score was 112.6.

If my IQ is 85, what percentage of people have a higher IQ than I?

$$P(X> 85)=1-P(X\leq 85)=$$ $$=1-P(Z\leq\dfrac{85-100}{15})=1-P(Z\leq-1)\approx$$ $$\approx1-0.158655=0.841345$$

If my IQ is 85, approximately 84% of people have a higher IQ than I.