Download Z- Scores

Z- Scores
The standard normal distribution is a normal distribution with a mean of 0 and a standard
deviation of 1. A Z-Score is a statistical measurement of a score's relationship to the
mean in a group of scores. A Z-score of 0 means the score is the same as the mean. A Zscore can also be positive or negative, indicating whether it is above or below the mean
and by how many standard deviations.
Z scores can be calculated using this formula.
z
X  Mean
SD
where X is a score from the original normal distribution, and SD is the standard
deviation of the original normal distribution.
The standard normal distribution is sometimes called the z distribution. A z score
always reflects the number of standard deviations above or below the mean a
particular score is. For instance, if a person scored a 70 on a test with a mean of 50 and
a standard deviation of 10, then they scored 2 standard deviations above the mean.
Converting the test scores to z scores, an X of 70 would be:
So, a z score of 2 means the original score was 2 standard deviations above the mean.
Using the diagram provided we can determine the proportion (or percentage) of people
who scored higher or lower than the any score in the distribution.
Z-scores also allow analysts to convert
scores from different data sets into scores
that can be accurately compared to each
other.
Suppose that you receive a score of 70 on a
class test for which the mean grade was 50
and the standard deviation was 10. Your
score is two standard deviations above the
mean. What percentage of the class scored
lower than you on the test?
Steps for solving
1. Calculate z score.
2. Draw a normal curve and shade in the area that you are looking for. (In this case all
scores below 2 standard deviations above the mean).
Use the curve on page 1 to determine the proportion of people that fall in the shaded
areas.
i.e.
.50 fall below 0 (i.e. by definition 50 % fall below the mean)
.3413 fall between 0 and +1.
.1359 fall between +1 and +2.
In total the proportion of scores that are lower than +2 z-cores is .8772.
Therefore, you scored higher than 87.72% of the class.
NOTE: Always begin from the mean, never from the end points. Remember a standard
normal curve is asymptotic, which means there are no end points.
The curve can be used when the z –score is a whole number (0, 1, 2, or 3). It cannot be
used when the z score is any number other than 0, 1, 2 or 3. In that case you need to use
a z Table. For example if the SD in the last example had been 40, the z-score would be
0.5.
You cannot just divide .3413 in half to determine the proportion of the
70  50
curve that falls between the mean and +0.5. You can however look it
z

up. From the table that proportion is .1915. Add .50 to that for all the
40
scores below the mean and you would find that your score is at the
69.15 percentile.
 0.5