Transcript Statistics for Behavioral and Social Sciences

Chapter 4
Some Key
Ingredients for
Inferential Statistics
The Normal Curve, Probability,
and Population Versus Sample
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Inferential Statistics
• Methods used by social and behavioral
scientists to go from results of research
studies to conclusions about theories or
applied procedures
• What most of statistics entails
• Beyond mere descriptive statistics
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
The Normal Curve
• Bell-shaped
• Unimodal
• Symmetrical
– Exactly half of the scores above
the mean
– Exactly half of the scores below
the mean
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
The Normal Curve
• There are known percentages of
scores above or below any given
point on a normal curve
– 34% of scores between the mean and
1 SD above or below the mean
– An additional 14% of scores between
1 and 2 SDs above or below the mean
– Thus, about 96% of all scores are
within 2 SDs of the mean (34% + 34%
+ 14% + 14% = 96%)
• Note: 34% and 14% figures can be
useful to remember
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Normal Curve Table
• Normal curve table gives the precise percentage of
scores between the mean (Z score of 0) and any
other Z score.
• Can be used to determine
– Proportion of scores above or below a particular Z
score
– Proportion of scores between the mean and a particular
Z score
– Proportion of scores between two Z scores
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Normal Curve Table Continued
• By converting raw scores to Z scores, can be used
in the same way for raw scores.
• Can also use it in the opposite way
– Determine a Z score for a particular proportion of
scores under the normal curve
• Table lists positive Z scores
– Can work for negatives too
– Why? Because curve is symmetrical
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Steps for Figuring Percentage
Above or Below a Z Score
• Convert raw score to Z
score, if necessary
• Draw a normal curve
– Indicate where Z score falls
– Shade area you’re trying to
find
• Make rough estimate of
shaded area’s percentage
• Find exact percentage with
normal curve table
• Check to verify that it’s
close to your estimate
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Steps for Figuring a Z Score or
Raw Score From a Percentage
• Draw normal curve, shading
approximate area for the
percentage desired
• Make a rough estimate of the Z
score where the shaded area
starts
• Find the exact Z score using
normal curve table
• Check to verify that it’s close to
your estimate
• Convert Z score to raw score, if
desired
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Probability
• Abbreviated as p, as in “p < .05”
• Expected relative frequency of a particular
outcome
• Probability = possible successful outcomes
divided by all possible outcomes
• Represented as
– Proportion (number between 0 and 1)
– Percentage (between 0% and 100%)
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Probability and Frequency
Distributions
• For any frequency distribution
the percentage of scores in a
particular region corresponds to
the probability of selecting a
score from that region.
• For example, the normal curve
• Histogram to the right
– 10 out of 50 people scored 7 or
higher.
– Thus, the probability of randomly
selecting a person with a score of 7
or higher is 10/50, or .20
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Sample vs. Population
• Sample
– Relatively small number of
instances that are studied in
order to make inferences
about a larger group from
which they were drawn
• Population
– The larger group from
which a sample is drawn
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Sample vs. Population Examples
• Population
a. pot of beans
b. larger circle
c. histogram
• Sample
a. spoonful
b. smaller circle
c. shaded scores
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall
Why Study Samples?
• Often not practical to study an entire population
• Instead, researchers attempt to make samples
representative of populations
– Random selection
• Each member of population has an equal chance of being sampled
• Good but difficult
– Haphazard selection
• Take steps to ensure samples do not differ from the population in
systematic ways
• Not as good but much more practical
Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall