Final answer:
Multiplying each score in a data set by a constant will multiply the standard deviation by the absolute value of that constant. The standard deviation is a measure of how spread out data values are from the mean and is integral to the calculation and interpretation of z-scores for standardizing different data sets.
Step-by-step explanation:
When each score in a data set is multiplied by a constant, the standard deviation of the resulting data set is also multiplied by the absolute value of that constant. The standard deviation is a measure of variability, indicating how spread out the data values are from the mean. For instance, if we have a set of scores and the standard deviation is five, multiplying each score by three would result in the standard deviation being fifteen, which is three times the original standard deviation.
It's essential to understand that the standard deviation, whether denoted by s for a sample or σ for a population, is never negative and is zero only when all data values are equal. Standard deviation is particularly useful when combined with the concept of z-scores, as any score can be standardized by calculating its z-score, showing how many standard deviations it is from the mean.
When dealing with z-scores and transforming scores in a normal distribution to the standard normal distribution, a z-score helps to standardize different data sets for comparison by expressing how many standard deviations a value is above or below the mean.