You may recall that our two concerns about our data set were: Number one - Where
is the center of the data? and our second concern was: How close together or spread
out the scores were? The concept of "close together" or "spread out" is called
"variability". We now need a way to discuss the average amount of variability in a
distribution. Beginning with the concept that our mean is a model, we can begin
making predictions. So I am going to choose a player at random from our team
with an average height of 5 foot 10, and ask you to predict that players height. I
pick a player at random, you guess his height. I pick another player at random
you guess his height. It won't take too long for this game to get old, because
the mean is the best predictor for any individual in the sample, you should
always guess five-foot-ten. Of course, you are not going to be accurate every time,
but over the long run you will be right or close to right more often than you
will be wrong. When you use a mean to predict or estimate, your prediction or
estimate is likely to be off a little. This is because each measurement of the
subjects X in the sample varies from the mean, m. This is what is meant by the
scores have variability. We could also call it error. In fact, we could call it
measurement error. Measurement error is the discrepancy between the actual value
we measure, X, and the number we will use to represent that value, M. You use the
mean to predict that a player will be 5 foot 10, but in reality, he's six foot
zero. You now have two inches of measurement error. Still despite the
error the mean, m, is always your best guess, so we use
the mean as a model. But this tells us two things: one - each x-value will have a
deviation from the mean, even if that deviation is zero, so we can use
deviation scores as a measure of variability. And two: the more that the
scores deviate from the mean, the less well the mean represents reality. A
deviation score is the difference between an actual data point, X, and the
mean, M. Deviations are calculated by subtracting the mean from each score, X
minus the mean. The 6-foot zero player has a deviation score of positive two. A
five foot eight player has a deviation score of negative two. A five foot ten
player has a deviation score of zero. Always subtract the mean from the raw
score when you are computing deviation scores. Deviation scores convey two
things: one - whether the score is above or below the mean, and two: how far the score
is from the mean. The positive or negative value tells you whether the
score is above or below the mean. Positive deviation scores are always
above the mean, negative deviation scores are always below the mean the absolute
value of the deviation score tells us how far it is from the mean, so the
deviation score of positive two is two inches above the mean. If the mean is 5
foot 10, the person with a deviation score of positive two is six feet tall.
Here is a normal curve with a mean of six. Beneath the normal curve I have
created deviation scores above and below the
mean. A person with a deviation score of negative three is three points below the
mean, and has a raw score of three. A person with a deviation score of
positive one is one point above the mean, and so has a raw score of seven. The
deviation of negative one is a raw score of five. In fact, I could add all of these
raw scores based upon the deviation scores on a number line. A large
deviation score indicates a raw score that is far from the mean. The deviation
of negative three is farther from the mean than a positive one. Raw scores, far
from the mean have lower frequency than scores close to the mean. Frequency is
shown by how close the normal curve is to the x-axis. The closer the curve is to
the x-axis, the lower the frequency for that score. A score at the mean has a
deviation score of zero, and the highest frequency. The raw score of six has a
deviation of zero. A score far above the mean is a higher raw score. The deviation
of positive 3 is a raw score value of 9. A score far below the mean is a lower
raw score. The deviation of negative 3 is a raw score of 3. Now we can observe
something about the mean and variability. Obviously the closer all of the scores
in the sample are to the sample mean, the better the sample mean will predict any
particular score in the sample, because the sample variability is less. When the
scores are packed close to the mean, i.e. leptokurtic, the mean is a great
predictor. When the scores are spread out from the mean platykurtic, the mean is
a less robust predictor. Therefore deviation scores are a measure of the
variability in the distribution. Lower variability equals better prediction.