Question

consider a sample with a mean equal to and a standard deviation equal to . calculate the standardized values for the following values.

Answer 1

The standardized values for the given sample with a mean
`(\( \bar{X} \))` of μ and a standard deviation
`(\( \sigma \))` of σ are calculated using the formula
`\( Z = \frac{{X - \bar{X}}}{{\sigma}} \)`. In this case, since the mean
`(\( \bar{X} \))`and standard deviation
`(\( \sigma \))` are both equal to 0, the standardized values for any given value
`\(X\)` would also be 0.

Step-by-step explanation:

In statistics, the standardized value (Z-score) for a data point in a sample is a measure of how many standard deviations it is from the mean of the sample. The formula for calculating the Z-score is
`\( Z = \frac{{X - \bar{X}}}{{\sigma}} \), where \(X\)` is the individual data point,
`\( \bar{X} \)` is the mean of the sample, and
`\( \sigma \)` is the standard deviation.

Given that the mean
`(\( \bar{X} \))` and standard deviation
`(\( \sigma \))`are both 0, the formula becomes
`\( Z = \frac{{X - 0}}{{0}} \)`, and any number divided by 0 is undefined. Therefore, the standardized values for any given value
`\(X\)` in this sample are 0.

This result makes sense intuitively, as a mean and standard deviation of 0 imply that all data points in the sample are identical, and therefore, any individual value in the sample is exactly at the mean, resulting in a Z-score of 0.