Z-scores are a means of answering the question ``how many
standard deviations away from the mean is this observation?'' If our
observation X is from a population with mean 
and standard
deviation 
, then
On the other hand, if the observation X is from a sample with mean
and standard deviation s, then
A positive (negative) Z-score indicates that the observation is greater than (less than) the mean.
EXAMPLE: In a certain city the mean price of a quart of milk is 63 cents and the standard deviation is 8 cents. The average price of a package of bacon is $1.80 and the standard deviation is 15 cents. If we pay $0.89 for a quart of milk and $2.19 for a package of bacon at a 24-hour convenience store, which is relatively more expensive? To answer this, we compute Z-scores for each:
Our Z-scores show us that we are overpaying quite a bit more for the milk than we are for the bacon.
Because of the Empirical rule (or the Chebychev's rule), the Z-score of a given observation also provides insight on how ``typical'' this observation is to the population. For example, by empirical rule, if data follow a bell-shaped curve, then approximately 95% of the data should have the Z-score between -2 and 2.