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Question

1. IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. If a certain person has an IQ of 131,  what percentage of the population has a lower IQ?

 

 

 

2. IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. If a certain person has an IQ of 105, what percentage of the population has a lower IQ?

Answer 1

Nora,

 

IQ is a random variable which is approximately Normally distributed as  N(μ,σ) = N(100, 15).  Since IQ cannot be negative or very large positive (I think the highest ever recorded is somewhere around 280), it is not exactly Normally distributed since the Normal function has tails on both sides that go to infinity. However, within the practical "middle" range of us mere mortals it provides a very good approximation so we use it.  

 

The Normal distribution is is a symmetrical distribution about its mean.  Usually tables of values are only given for the right half so if you have a "less than" question, you must add a probability of 0.5 to the tabular data.  Moreover, the data are only supplied for a normalized or standardized version N(0,1) of it, so you first have to transform your random variable x to an "equivalent" standardized z-value as follows:

 

z = (x - μ)/σ .     In your case x = IQ.  With that preamble, let's answer your 2 questions.

 

(1) % of population with IQ < 131?

 

z₁ = (131 - 100)/15 = 31/15 = 2.067

Probability that z₁ < 2.067 = 0.50 (adj for z₁ <0) + Prob(0 ≤ z₁ < 2.067 [from N(0,1) area table]

                                        = 0.50  +  0.4806 (interpolated)

                                        = 0.98  (rounded off)

So 98 % of the general population have IQs below 131, and only about 2% exceed that number.

 

(2) % of population with IQ < 105?

z₂ = (105 - 100)/15 = 5/15 = 0.333
Probability that z₂ < 0.333   = 0.50 (adj for z₂ <0) + Prob(0 ≤ z₂ < 0.333 [from N(0,1) area table]
= 0.50 + 0.1306 (interpolated)
= 0.63 (rounded off)
So 63 % of the general population have IQs below 105, and only about 37% exceed that number.

 

Answer 2

 

1. 131 is roughly 2 standard deviations from the mean 100 ----> 2 times 15.5 plus 100. I expect you to know your statistical tables for the normal distribution; key numbers pop up in my memory, 95 % for 2 sigma, 68 % for 1 sigma or 1 standard deviations, 99 % for 3 sigma .

 

Therefore, with the mental image I have in my brain, the percentage of the population that has lower IQ is 95% + 5%/2 = 97.5 %

 

2. 105 is within one standard deviation, therefore, my mental estimation is around 50% + 34%/3 = 62%

 

PRECISION REQUIRES THE CORRECT USE OF STATISTICAL TABLES! I don't have them handy!