68-95-99.7 Rule question

Question

Consider a standardized Intelligence Quotient (IQ) test that results in IQ scores which are normally distributed with a mean of 100 and a standard deviation of 15. Use the 68-95-99.7 Rule to find the percentage of scores in each of the following scenarios:

Percentage of scores less than 70: ____________%

Percentage of scores greater than 85: ____________%

Percentage of scores between 100 and 130: ____________%

Percentage of scores between 85 and 130: ____________%

Answer 1

So it's important we contextualize what the 68-95-99.7 rule is to solve these questions.

 

68% of the data is within +/- 1 SD from the mean

95% of the data is within +/- 2 SDs from the mean

99.7% of the data is within +/- 3 SDs from the mean

 

So in this case:

 

68% of the data is within +/- 15 from 100, or 68% of the data is between 85-115.

95% is between 70 - 130

99.7% is between 55 - 145

 

So now use that information to answer the questions:

 

Percentage of scores less than 70:  ____________%

 

We know 70 is 2 SDs from the mean, so 95% of the data is between 70-130. That means 5% of the data is outside that range. Since it is a normal distribution, that means that 5% is split between the left and right tails of the distribution curve.  So 2.5% of the data is less than 70.

Percentage of scores greater than 85: ____________%

 

85 is -1 SD from the mean. We know there is 50% of the data on either side of the mean. We also know that half of the 68% of is between the mean and -1 SD, or 34%. So the amount of data greater than 85 is 34% + 50% = 84% (you can picture that as everything to the right of the mean and everything between the mean and -1 SD to the left).

Percentage of scores between 100 and 130: ____________%

 

Since 130 is 2 SDs from the mean, we know that 95%/2 = 47.5% of the data is between that point and the mean (100).

Percentage of scores between 85 and 130: ____________%

 

We can solve this by finding the area to the left of each point and then subtracting the difference.

 

In the second question, we established there is 84% of the data to the right of 85.  That means the other 16% of the data is less than 85.

 

In the third question, we established that there is 47.5% of the data between the mean (100) and 130.  We also know there is another 50% to the left of the mean. Therefore, there is 50 + 47.5 = 97.5% of the data to the left of 130.

 

So the percentage of data between 130 and 85 is the difference of the area to the left of these two points.  97.5% - 16% = 81.5%