Hi Barb,

In probability and statistics the letter Z is used for the random variable which has the standard normal distribution, that is the normal distribution with mean 0 and standard deviation 1. Many times we have a random variable X from the normal distribution with some mean μ and standard deviation σ, and we convert it to the standard normal random variable Z using

the so called Z-score.

Lets look at a specific example. Suppose you have a normal distribution with mean μ = 5 and standard deviation σ = 3. If the random variable that satisfies this distribution is X then

Suppose you now take a measurement from this distribution and obtain a value X = 11. This gives Z = 2 or said using symbols

Multiplying both sides by σ gives

11 - μ = 2 σ

which says that 11 is 2 standard deviations more than the mean. Suppose we take another measurement X and this time get Z = -1.5. Then

X - μ = -1.5 σ

and hence X is one and a half standard deviations less than the mean.

Hence the Z-score tell us how far X is from the mean using the units of standard deviation.

I hope this helps,
Harley