Questions: A population with μ=85 and σ=12 is transformed into z-scores. After the transformation, what is the standard deviation for the population of z-scores? σ=12 σ=1.00 σ=0 Cannot be determined from the information given

A population with μ=85 and σ=12 is transformed into z-scores. After the transformation, what is the standard deviation for the population of z-scores?
σ=12
σ=1.00
σ=0
Cannot be determined from the information given

Transcript text: A population with $\mu=85$ and $\sigma=12$ is transformed into $z$-scores. After the transformation, what is the standard deviation for the population of $z$-scores? $\sigma=12$ $\sigma=1.00$ $\sigma=0$ Cannot be determined from the information given

Solution

Solution Steps

To find the standard deviation of a population of z-scores, we need to understand that z-scores are standardized scores. The transformation of a raw score into a z-score involves subtracting the mean and dividing by the standard deviation. This process standardizes the distribution, resulting in a new distribution with a mean of 0 and a standard deviation of 1. Therefore, the standard deviation of the population of z-scores is always 1.

Step 1: Understanding the Transformation to Z-Scores

When a population is transformed into z-scores, each data point $ x $ is converted using the formula: \[ z = \frac{x - \mu}{\sigma} \] where $ \mu $ is the mean and $ \sigma $ is the standard deviation of the original population.

Step 2: Properties of Z-Scores

The transformation results in a new distribution of z-scores with a mean of 0 and a standard deviation of 1. This is because the transformation standardizes the data, making the spread of the z-scores uniform across different datasets.

Step 3: Determining the Standard Deviation of Z-Scores

Given that the standard deviation of z-scores is always 1, regardless of the original population's standard deviation, we can conclude that the standard deviation of the population of z-scores is: \[ \sigma = 1 \]