When calculating \( Z \) scores for a sample as opposed to a population the formula for \( Z \) needs to be adjusted to

When calculating \( Z \) scores for a sample as opposed to a population, the formula needs to be adjusted to account for the sample standard deviation instead of the population standard deviation. The general formula for a \( Z \) score is: \[ Z = \frac{X - \mu}{\sigma} \] where: - \( X \) is the value for which you are calculating the \( Z \) score, - \( \mu \) is the mean of the population, - \( \sigma \) is the standard deviation of the population. For a sample, the formula is adjusted to: \[ Z = \frac{X - \bar{X}}{s} \] where: - \( \bar{X} \) is the sample mean, - \( s \) is the sample standard deviation. This adjustment is necessary because the sample standard deviation \( s \) provides a better estimate of variability when working with a sample rather than the population standard deviation \( \sigma \).