Question

What does it mean when the Z score of a value is larger than 2?

Answer 1

Let x be the value drawn from a normal distribution with mean $\mu$ and standard deviation $\sigma$

The z score of x is $z=\frac{x-\mu}{\sigma}$ .

The value of x in terms of z score, mean and standard deviation is

$\frac{x-\mu}{\sigma}=z\implies x=\mu+z\sigma$

This indicates that the value of x is mean plus z standard deviations. Other way of stating this is, x is z standard deviations away from the mean.

When the z score of a value is larger that 2, it means that the value is more than 2 standard deviations away from the average (or mean).

It terms of probability we can get the P(Z>2) = 1-P(Z<2) = 1-(0.5+0.4772)=0.0228 (using standard normal tables)

This means that the probability of observing a value (with z score more than 2) is only 0.0228. That is there is only a 2.3% chance of observing this value and and hence it is not a common value.