Point scores have a mean of 100 and a standard deviation of 15. Nick has an IQ of 112. What is the difference between Nick’s IQ score and the mean? (0) Convert Nick’s IQ score to a z-score.
The Correct Answer and Explanation is:
Correct Answer:
Nick’s IQ score = 112
Mean = 100
Standard deviation = 15
Step 1: Difference from the mean:Difference=112−100=12\text{Difference} = 112 – 100 = 12Difference=112−100=12
Step 2: Convert to a z-score:z=X−μσ=112−10015=1215=0.8z = \frac{X – \mu}{\sigma} = \frac{112 – 100}{15} = \frac{12}{15} = 0.8z=σX−μ=15112−100=1512=0.8
Explanation
In statistics, a z-score is a standardized measure that indicates how many standard deviations a data point is from the mean of a distribution. It is commonly used in the context of a normal distribution, where most values cluster around the average. The formula for calculating the z-score is:z=X−μσz = \frac{X – \mu}{\sigma}z=σX−μ
In this expression, XXX is the observed value, μ\muμ is the mean of the distribution, and σ\sigmaσ is the standard deviation.
For IQ scores, the mean (μ\muμ) is given as 100, and the standard deviation (σ\sigmaσ) is 15. Nick’s IQ is 112, which is above the mean. To understand how far above the average this score is, the difference between the score and the mean is computed:112−100=12112 – 100 = 12112−100=12
This raw difference indicates Nick’s score is 12 points higher than the average. To convert this raw difference into a more interpretable scale, the z-score is calculated:z=1215=0.8z = \frac{12}{15} = 0.8z=1512=0.8
A z-score of 0.8 means Nick’s IQ is 0.8 standard deviations above the mean. In the context of a normal distribution, this places his IQ above approximately 78.8% of the population. This statistical transformation allows comparisons across different distributions and scales by placing scores on a common metric based on their relative position to the mean.
Using z-scores helps in assessing how unusual or typical a data point is within a distribution, aiding in interpretation and decision-making across various fields such as education, psychology, and research.
