The 'Z' score is simply a statistic that is used to evaluate a
hypothesis regarding a value or estimate. In order to use this
statistic, we must first assume that the value we want to test has a
normal distribution (follows a bell curve type of pattern). The
actual Z score is computed by dividing the value or estimate by its
standard error. That is:
Value (such as a Beta coefficient)/s.e.(standard error)
In order to truly understand why we do this, you should have some
understanding of statistics theory. The easiest way to explain this
is to imagine two overlapping 'bell curves'. Imagine, for a moment,
that you want to know whether or not a certain value that you have
estimated is statistically different than '0'. Since we assume that
this estimate follows a bell curve (normal) pattern, the mean of that
value (the center of the 'bell') lies at 0. Now imagine for a moment
a second curve that overlaps part of it. The closer the overlap, the
more likely the estimate is zero. But the more it deviates from the
curve whose mean is zero, the more likely that there is a
statistically significant difference. So where does the 'Z' score
come in?
The Z score gives us an idea of how many standard deviations from that
zero point our estimate is; the larger the number, the more probable
we have a statistical difference.
Here's an example:
We have a Beta coefficient (which we estimated in a simple regression)
and we want to know if it is statistically different than 0. We should
proceed first by stating our hypotheses. In this case we have a null
hypothesis:
H(null):BetaCoefficient=0
and an alternative hypothesis:
H(alternative): Beta is not 0.
At this point we choose an 'alpha' level, which is the probability
that our statistic is really zero in this case. This is also known as
a 'Type 1' error. Most of the time we choose either a .05 or .01
alpha level (in most scientific disciplines the .05 is sufficient).
The .05 means that there is a 5% chance that our value is no different
than zero (our null hypothesis), which means we might still have a 5%
chance that we have an estimate that is zero even though we reject the
null hypothesis. (Hypothesis testing is really just about rejecting
the null hypothesis).
Now, plug in your values and get a z-score. Say we get a value of
1.7. What do we do with this? We need a table of something called the
'standard normal distribution'. Reading this table is not as
difficult as might appear. Match the z score (to the nearest decimal)
with a value in the table. For 1.7, the value we get is .045.
So far so good, but we must understand that when we are analyzing the
'deviation from zero' we're actually analyzing a curve that can
deviate to the left or the right. Imagine, again, the overlapping
curves where the shift can be lower or higher than the central '0'
value. Since that deviation can occur on both sides, the overall
probability that we make an error is .045x2, or .09. That is, there is
an overall 9% probability that our value is actually zero. Remember
that we usually want no more than a 5% error, so going by this z
score, we cannot reject the null hypothesis that our estimate is
really zero.
For a .05 alpha, we want an (absolute) Z score of over 1.96 (or,
roughly, 2). This means our estimate deviates by two standard
deviations from the zero mean. In the standard normal table, a value
of 1.96 corresponds to .025. If you double this value, you get .05,
our alpha level.
Generally speaking, look for Z scores above (in absolute terms) 1.96
to reject the null hypothesis at a .05 alpha and z scores of above (in
absolute terms) 2.576 to reject the null hypothesis at a .01 alpha.
Sometimes just a .10 alpha is enough (although few disciplines accept
this) and you simply need an absolute z score above 1.645.
I hope this helps. |
