standard deviation

Question

IQ test scores are normally distributed with a mean of 100 and a standard deviation of 15. Find the x-score that corresponds to a z-score of 1.96.

Answer 1

If you are not interested in an exact answer, or if you would like to verify that your answer is in the ballpark, just notice that z=1.96 means that you are ALMOST 2 standard deviations above the mean.  (It's 1.96 standard deviations - but we are talking a rough estimate.)  Two standard deviations is 2 * 15 = 30.  So take the mean and add 100 + 30 = 130.  This is not an exact amount, but you can see that it is close to the answer that Varaia gave you of 129.4.

Answer 2

z=(x-μ)/σ    is the formula for calculating z-scores,  where x is the x-score, μ is the sample mean, and σ is the standard deviation.

If we use algebra to rewrite the formula we can solve for x.   

 

z=(x-μ)/σ      Step 1: Multiply both sides of the equation by σ

 

σz=x - μ         Step 2: add μ to both sides

 

σz + μ =x          Step 3: Calculate x with given information

 

(15)(1.96) + 100=x

 

x-score = 129.4