Transcript Z-scores & Probability

Z-scores
Z-score or standard score
A
statistical techniques that uses the
mean and the standard deviation to
transform each score (X) into a zscore
 Why z-scores are useful?
Z-scores and location in a
distribution

The sign of the z-score (+ or –)

The numerical value corresponds to the
number of standard deviations between X
and the mean
The relationship between z-scores
and locations in a distribution
Transforming back and forth
between X and z

The basic z-score definition is usually
sufficient to complete most z-score
transformations. However, the definition can
be written in mathematical notation to create
a formula for computing the z-score for any
value of X.
X– μ
Deviation score
z = ────
Standard deviation
σ
Example

If  = 95 and  = 16 then a score of 124 has
a z-score of
Example

For a population with  =100 and  = 8,
find the z-score for each of the following
1.
2.
X= 84
X=104
What if we want to find out what
someone’s raw score was,
when we know their z-score?
Example:
 Distribution of exam scores has a mean of
70, and a standard deviation of 12.
 If an individual has a z-score of +1.00, then
would score did they get on the exam?

Another example …
If an individual’s z-score is -1.75, then what
score did they get on the exam?
 Again using a mean of 70, and a standard
deviation of 12.

Transforming back and forth
between X and z (cont.)

So, the terms in the formula can be
regrouped to create an equation for
computing the value of X corresponding to
any specific z-score.
X = μ + zσ
X=+z
So if =80 and  = 12 what X value
corresponds to a z-score of -0.75?
X=+z
=80  = 12
Z-scores
+0.80
-2.34
+1.76
-0.03
X
Distribution of z-scores

shape will be the same as the original
distribution

z-score mean will always equal 0

standard deviation will always be 1
Using z-scores to make
comparisons
Example:
 You got a grade of 70 in Geography and 64
in Chemistry. In which class did you do
better?

Z-scores and Locations

The fact that z-scores identify exact locations
within a distribution means that z-scores can
be used as descriptive statistics and as
inferential statistics.
As descriptive statistics, z-scores describe
exactly where each individual is located.
 As inferential statistics, z-scores determine
whether a specific sample is representative of
its population, or is extreme and
unrepresentative.

z-Scores and Samples

It is also possible to calculate z-scores for
samples.
z-Scores and Samples
Thus, for a score from a sample,
X–M
z = ─────
s
 Using z-scores to standardize a sample also
has the same effect as standardizing a
population.
