What does a z score tell you?

The Z-Score tells you the position of an observation in relation to the rest of its distribution, measured in standard deviations, when the data have a normal distribution.

You usually see position as an X-Value, which gives the actual value of the observation. This is intuitive, but doesn't allow you to compare observations from different distributions. Also, you need to convert your X-Scores to Z-Scores so you can use the Standard Normal Distribution tables to look up values related to the Z-Score.

For example, you want to know if an eight year old's pitching speed is unusually good compared to his or her league. If the mean little league pitch speed is 30 mph with a standard deviation of 4 mph, is a 38 mph pitch unusual? 4 mph is an X-Score. You convert to a Z-Score with this formula:

#Z=(X-mu)/sigma#

So the Z-Score is

#Z=(38-30)/4=2#

The probability of a Z-Score of 2 is 0.022; this makes this little league pitcher unusually fast. Is he or she more unusual than a professional player who pitches 92 mph, if the mean professional pitch is 89 mph and the standard deviation is 3 mph? The Z-Score of the professional is:

#Z=(92-89)/3=1#

The little leaguer's Z-Score was 2, and the professional's was 1, so the little leaguer is more unusual than his or her professional counterpart. You could not tell this by comparing X-Scores.