Question

If IQ scores are normally distributed with a mean of 100 and a standard deviation of 15, what is the mean IQ score after the scores have been standardized by converting them to z-scores?

Answer 1

`\mu = 0 , \sigma=1`

Explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

Solution to the problem

Let X the random variable that represent the IQ scores of a population, and for this case we know the distribution for X is given by:

`X \sim N(100,15)`

Where
`\mu=100` and
`\sigma=15`

If we standardize the variable with the z score given by:

`Z= (x -\mu)/(\sigma)`

We got a normal standard distribution with parameters
`Z\sim N (0,1)`

`\mu = 0 , \sigma=1`