Final answer:
Z-scores enable us to compare disparate data sets by standardizing values to indicate how far a data point is from the mean. This allows for meaningful comparisons across different scales and contexts, such as academic achievements or athletic performances. The z-score calculation normalizes different scales to a common metric for comparison.
Step-by-step explanation:
Z-scores provide a standardized means of comparing values from different data sets by expressing how many standard deviations a data point is from the mean. This comparison is particularly useful because it allows us to compare things that appear incomparable, such as academic achievements in different subjects or performances in sports with different scoring systems. By transforming different scales to a common scale, using z-scores, we can make meaningful comparisons between different data sets.
The calculation for a z-score (z) is as follows: z=(x-μ)/σ, where x is the data value, μ is the mean of the data set, and σ is the standard deviation. Therefore, a z-score tells us if a value is above or below the mean and by how much, allowing for these comparisons to be made in a standardized manner. Even if the actual values and scales are different, a z-score of 1 indicates a value one standard deviation above the mean, which has the same implication across different contexts or units.
Z-scores are particularly useful in real-life scenarios, such as comparing students' SAT scores to determine college readiness or comparing two athletes' performances, even if they're from different sports. In terms of data sets, we may have one group with a mean score of 5 and a standard deviation of 6, and another group with a mean of 2 and a standard deviation of 1. If a person from group one scores a 17 (which is two standard deviations above the mean) and another from group two scores a 4 (also two standard deviations above the mean), we can say their performance is comparable in terms of their deviation from their respective means.