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# What is a z score in regard to relative position?

Hi! A Z-score measures the "relative distance" of an observation in terms of standard deviations, from the mean. It gives an idea as to how far the observation is from its mean. A positive z-score indicates that the observation is above the mean and a negative value indicates that the observation is below the mean. The z-score does not have any units, that is why it is convenient to use in comparing observations relative to their groups, each having different units.

Suppose we have an observation, say, X, from a population mean µ and a standard deviation σ, then the Z score of the observation is given by the formula

That is, we deduct the value of the mean µ from the value of the observation X and divide it against the standard deviation.

The following is a rephrased example taken from the book Introduction to Statistics 3rd edition by Ronald E. Walpole.

Suppose a student had a grade of 82 in Chemistry and a grade of 89 in Economics, can we conclude that she is better in Economics than in Chemistry? Perhaps we should consider how this student performed relative to other students in each of her classes.

Assuming that the mean grade in Chemistry was 68 and the standard deviation was 8, whereas the distribution of economics grades had a mean of 80 and a standard deviation of 6, let us now compute the z scores corresponding to the Chemistry and Economics Grades

We see that the student had a grade in Chemistry that is 1.75 standard deviations above the mean of the Chemistry grade, whereas in Economics she was only 1.5 standard deviations above the mean of the Economics grades. Comparing these two z-scores, we can now say that the student's relative performance in Chemistry was higher than her performance in Economics.