Chapter Eight - Probability and Why It Counts
THE NORMAL CURVE (NORMAL DISTRIBUTION)
 The normal curve comes from the study of probability; it provides the basis for
determining the probability or likelihood that a particular event or finding is not
just the result of chance.
 The normal curve results from a long history of measuring various traits,
characteristics and/or events in the population over and over again and finding
that every time the sample is large enough and representative enough of the
larger sample, a normal distribution of “scores” will result.
 Researchers use the tools of probability and the study of the normal distribution
to determine the mathematical likelihood that a difference found between two
research variables is not the result of CHANCE!
THREE CHARACTERISTICS OF THE NORMAL CURVE
 Mean, median and mode are exactly the same
 Perfectly symmetrical about the mean
 Asymptotic—tails of the curve come closer and closer to the horizontal axis but
never touch
THE NORMAL CURVE EXPLAINS THE PROBABILITY OF VARIOUS ASPECTS OF
A TRAIT OR CHARACTERISTIC OCCURRING
 For example, intelligence is normally distributed within the population, and if you
could measure everyone’s intelligence using the same intelligence test, you
would find that intelligence is normally distributed, based on the mean and the
standard deviation of the population.
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Chapter Eight - Probability and Why It Counts
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INTELLIGENCE EXAMPLE, CONTINUED
 If an intelligence test has a mean of 100 and standard deviation of 15, then you
can assume the following things about intelligence in the population, based on
the normal curve.
 The majority (approx. 68%) fall between the -1.0 and +1.0 standard
deviation. Since the sd = 15, the majority of people have an intelligence of
between 85 and 100.
 Another 14% of the population fall on each side of the -1.0 and +1.0 sd, so
between -1.0 and -2.0, 14% of the population will have an IQ of between
70 and 85 and between +1.0 and +2.0, 14% of the population will have an
IQ between 115 and 130
 Approximately what percent of the population will have IQs below 70 and
above 130 (i.e. between the 2nd and 3rd sds?
 Almost 100% of the population falls between -3.0 and +3.0 standard
deviations.
STANDARD SCORES (Z SCORES IN PARTICULAR)
 Standard scores are comparable because they are standardized in units of
standard deviations.
 By converting raw scores to z scores, you can compare scores from two different
distributions that have different means and standard deviations. This is because
they are standardized in units of standard deviations.
 Positive z scores always fall to the right of the mean and are in the upper half of
the distribution. Negative z scores always fall to the left of the mean and are in
the lower half of the distribution.
 Very simply, z scores are the number of standard deviations from the mean.
 Once raw scores are converted to z scores, they are comparable.
Chapter Eight - Probability and Why It Counts
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THE RULE OF THUMB FOR TESTING HYPOTHESES
 When trying to decide if the difference you find between two variables is
significant, most researchers want to guarantee that an event seems to occur
only 5 times out of 100 (or 5%) (more to come later about what that event
represents).
 If the event occurs less than 5% of the time, we deem it to be unlikely relative to
all other events that could potentially occur. For example, if you were comparing
two methods for teaching reading, and you obtain a mean score that indicates
that the new reading approach is more effective than the traditional approach,
you would want to be able to demonstrate that this difference would be observed
in 95% of the cases and that only 5% of cases might show that the scores really
aren’t different (when accounting for the possibility of error) (more in the next
chapter). The point here is that if that the differences in scores in not due to
chance, but due to actual differences in the reading methods.
CONVERTING RAW SCORES TO Z SCORES
 Use the formula on page 142
 Identify the mean and standard deviation
 Subtract the mean from the raw score
 Divide by the standard deviation
DETERMINING THE PROBABILITY OF SPECIFIC Z SCORES
 Use Table B.1 in the appendix
 Draw a picture to help you
 Look up the value (percentage) of your score in the table. Always remember that
the absolute value is given, so this value may be above or below the mean. If
above, it will be added to .50; if below, it will be subtracted from .50
 Example: Mean = 100, sd=15
 What is the likelihood of a score falling above 140?
 What is the likelihood of a score falling below 80?
 What is the likelihood of a score falling between 95 and 105?