The distribution of IQ scores is a nonstandard normal distribution with a mean of 100 and a standard deviation of 15. What are the value of the mean and standard deviation after all IQ score have been standardized by converting them to z scores?
What is a "nonstandard normal distribution"?
For Z scores:
Mean = 0
SD = 1
To find the value of the mean and standard deviation after standardizing the IQ scores, we need to convert the IQ scores to z-scores.
The z-score formula is given by:
z = (X - μ) / σ
Where:
- X is the value we want to convert to a z-score (the IQ score in this case)
- μ is the mean of the distribution (100 in this case)
- σ is the standard deviation of the distribution (15 in this case)
By substituting the values into the formula, we can determine the z-score for any given IQ score.
Let's consider an example:
Suppose we have an IQ score of 130.
To find the z-score for this IQ score, we use the formula:
z = (X - μ) / σ
z = (130 - 100) / 15
z = 30 / 15
z = 2
So, an IQ score of 130 would have a z-score of 2.
Now, regarding the values of the mean and standard deviation after all IQ scores have been standardized, we need to understand that standardizing the data does not change the mean and standard deviation values.
Therefore, even after converting all IQ scores to z-scores, the mean will remain 100, and the standard deviation will remain 15.