A university program had been giving a departmental exit exam for years. This
was an exit exam that members of the department had created. They decided to
switch to a national test for their new exit exam. They came to me asking how
they could convert scores from the old exam to compare them with scores for the
new exam. The mean of the new exam was ten points higher than the mean for the
old exam. Should they add a constant, like ten points, to the lower scores? Now this
would raise the mean by 10 points, but by focusing only on the mean, we ignore
another important aspect of the distribution of scores - the standard
deviation. So I proposed a better solution. Convert the raw scores to
z-scores, and it worked! So Z scores are what we're going to learn about now. Let
me introduce you to the concept of z-scores. A z-score tells you how far a
raw score deviates from the mean when measured in standard deviation units. So
let's think about that. How far a raw score deviates from the mean. How would
you calculate the distance that one score falls from another? Subtraction. So
we will subtract the mean from the raw score. Now that gives us something we
call a deviation score. Next step, when measured in standard deviation units.
This is a little harder. How would you put that deviation score into standard
deviation units? Remember your choices are add, subtract, multiply, and divide.
Putting one thing in terms of another implies a ratio, division. So we will
subtract two numbers and then divide by a third number. We divide our deviation
score by the standard deviation, and that is a z-score. And here's what z-scores
look like. Notice what they all have in common they all begin with Z.
They all have a sign, positive or negative, and they all have three digits.
There are two parts to a z-score, the number and the sign. The sign tells you
whether the z score is above (positive) or below (negative) the mean. Now there are
very few always and nevers in statistics, but this is one. The positive z-score is
always above the mean. The negative z-score is always below the mean. Now in
the real world, sometimes the positive is assumed, so the sign is omitted. However
as we learn, I'd like you to include the positive or negative sign, just for
consistency. Now the number tells us the distance between the score and the mean
in terms of standard deviation. Regardless of the sign, the number
indicates exactly the same distance. A score of Z equals positive 1.23 is
exactly as far above the mean as a z-score of negative 1.23 is below the
mean. So here are some things to remember. Number one: a positive z-score indicates
that the raw score is larger than the mean. Of course a negative z-score
indicates that a raw score is smaller than the mean, and, number two, the mean is
z = 0.00. Number three: always include the sign, positive
or negative, and the number, round to two decimals, and include the leading zero if
the z-score is less than one. Remember the rule is that if a number cannot
exceed one, do not include a leading 0. For example, probability can only be
between 0 and 1, so no leading zero on probabilities. A correlation is always
between negative 1 and positive 1, so again, no leading 0, but a z-score can be
greater than 1, so you should include the leading zero.
So let's practice. Now what is wrong with this first z-score? They forgot to
include the sign. What's wrong with the second z score? Forgot to include the
leading zero. And what's wrong with that third z-score? We should round to two
decimals. So that is your introduction to z-scores, now let's figure out what to do
with them!
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