  # z-score

## What is a Z-score?

The Z-score, or standard score, is a fractional representation of standard deviations from the mean value. Accordingly, z-scores often have a distribution with no average and standard deviation of 1. Formally, the z-score is defined as:

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One cannot calculate the z-score without first knowing the mean and standard deviation of the complete population.

## How does a Z-score work?

As described above, the z-score works by taking a sample score and subtracting the mean score, before then dividing by the standard deviation of the total population. It can be easy to think of this using an example. Imagine having a score of 70 out of 100, where the mean score is 60, and the standard deviation is 15. Using the z-score function, one can calculate the z-score of .6667 from the data. By comparing the z-score the standard normal distribution table, one can determine that the probability of a score being greater than .67 is .2514 or 25.14%. In short, the score is better than roughly 75% of the rest of the scores.

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### Applications of Z-scores

The z-score is a useful metric for comparing test results between tests of different scales. For example, the comparison of scores in the SAT vs ACT would be best done using z-scores. As the z-score usefully translates scores in terms of their distribution, rather than rating scale, it is a useful metric for comparing tests of these sorts. Using the z-score technique, one can now compare two different test results based on relative performance, not individual grading scale.