Transcript Statistics for the Behavioral and Social Sciences: A Brief

Statistics for the Behavioral
and Social Sciences:
A Brief Course
Fifth Edition
Arthur Aron, Elaine N. Aron, Elliot Coups
Prepared by:
Genna Hymowitz
Stony Brook University
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Some Key Ingredients for Inferential
Statistics
Chapter 4
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Chapter Outline
•
•
•
•
The Normal Curve
Sample and Population
Probability
Normal Curves, Samples and Populations,
and Probabilities in Research Articles
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Inferential Statistics
• Allow us to draw conclusions about
theoretical principles that go beyond the
group of participants in a particular
study
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The Normal Curve
• Normal Distribution
– histogram or frequency distribution that is a
unimodal, symmetrical, and bell-shaped
– a mathematical distribution
– Researchers compare the distributions of their
variables to see if they approximately follow
the normal curve.
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Why the Normal Curve Is
Commonly Found in Nature
• A person’s ratings on a variable or performance on a task
is influenced by a number of random factors at each point
in time.
• These factors can make a person rate things like stress
levels or mood as higher or lower than they actually are,
or can make a person perform better or worse than they
usually would.
• Most of these positive and negative influences on
performance or ratings cancel each other out.
• Most scores will fall toward the middle, with few very low
scores and few very high scores.
– This results in an approximately normal distribution
(unimodal, symmetrical, and bell-shaped).
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The Normal Curve and the Percentage of
Scores Between the Mean and 1 and 2
Standard Deviations from the Mean
•
There is a known percentage of scores that fall below any given point
on a normal curve.
– 50% of scores fall above the mean and 50% of scores fall below the mean.
– 34% of scores fall between the mean and 1 standard deviation above the
mean.
– 34% of scores fall between the mean and 1 standard deviation below the
mean.
– 14% of scores fall between 1 standard deviation above the mean and 2
standard deviations above the mean.
– 14% of scores fall between 1 standard deviation below the mean and 2
standard deviations below the mean.
– 2% of scores fall between 2 and 3 standard deviations above the mean.
– 2% of scores fall between 2 and 3 standard deviations below the mean.
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The Normal Curve Table and
Z Scores
• A normal curve table shows the percentages of
scores associated with the normal curve.
– The first column of this table lists the Z score
– The second column is labeled “% Mean to Z” and gives
the percentage of scores between the mean and that Z
score.
– The third column is labeled “% in Tail.”
.
Z
% Mean to Z
% in Tail
.09
3.59
46.41
.10
3.98
46.02
.11
4.38
45.62
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Using the Normal Curve Table to
Figure a Percentage of Scores
Above or Below a Raw Score
•
If you are beginning with a raw score, first change it to a Z Score.
– Z = (X – M) / SD
•
•
•
Draw a picture of the normal curve, decide where the Z score falls on it,
and shade in the area for which you are finding the percentage.
Make a rough estimate of the shaded area’s percentage based on the
50%–34%–14% percentages.
Find the exact percentages using the normal curve table.
– Look up the Z score in the “Z” column of the table.
– Find the percentage in the “% Mean to Z” column or the “% in Tail” column.
• If the Z score is negative and you need to find the percentage of scores above this
score, or if the Z score is positive and you need to find the percentage of scores
below this score, you will need to add 50% to the percentage from the table.
•
Check that your exact percentage is within the range of your rough
estimate.
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Using the Normal Curve Table to
Figure Z Scores and Raw Scores
•
•
•
•
Draw a picture of the normal curve and shade in the approximate area
of your percentage using the 50%–34%–14% percentages.
Make a rough estimate of the Z score where the shaded area stops.
Find the exact Z score using the normal curve table.
Check that your Z score is within the range of your rough estimate.
– From your picture, estimate the percentage of scores in the tail or between
the mean and where the shading stops.
• To figure the percentage between the mean and where the shading stops, you will
sometimes need to subtract 50 from your percentage.
– Look up the closest percentage in the appropriate column of the normal
curve table.
– Find the Z score for that percentage.
•
If you want to find a raw score, change it from the Z score.
– X = (Z)(SD) + M
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Example of Using the Normal
Curve Table to Figure Z Scores
and Raw Scores: Step 1
• Draw a picture of the normal curve
and shade in the approximate area
of your percentage using the 50%–
34%–14% percentages.
– We want the top 5%.
– You would start shading slightly to the
left of the 2 SD mark.
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Example of Using the Normal
Curve Table to Figure Z Scores
and Raw Scores: Step 2
• Make a rough estimate of the Z
score where the shaded area
stops.
– The Z Score has to be between +1
and +2.
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Example of Using the Normal
Curve Table to Figure Z Scores
and Raw Scores: Step 3
• Find the exact Z score using the normal
curve table.
– We want the top 5% so we can use the
“% in Tail” column of the normal curve
table.
– The closest percentage to 5% is 5.05%,
which goes with a Z score of 1.64.
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Example of Using the Normal
Curve Table to Figure Z Scores
and Raw Scores: Step 4
• Check that your Z score is within
the range of your rough estimate.
– +1.64 is between +1 and +2.
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Example of Using the Normal
Curve Table to Figure Z Scores
and Raw Scores
• Check that your Z score is within the range of your
rough estimate.
– From your picture, estimate the percentage of scores in the tail or between
the mean and where the shading stops.
• To figure the percentage between the mean and where the shading
stops, you will sometimes need to subtract 50 from your percentage.
– Look up the closest percentage in the appropriate column of the normal
curve table.
– Find the Z score for that percentage.
• If you want to find a raw score, change it from the Z
score.
– X = (Z)(SD) + M
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Example of Using the Normal
Curve Table to Figure Z Scores
and Raw Scores: Step 5
• If you want to find a raw score,
change it from the Z score.
– X = (Z)(SD) + M
– X = (1.64)(16) + 100 = 126.24
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How Are You Doing?
• Use the partial normal curve table found below to
answer the following question:
• If the data from your study were normally distributed,
what percentage of scores would fall between the
mean and a Z score of .10?
Z
% Mean to Z
% in Tail
.09
3.59
46.41
.10
3.98
46.02
.11
4.38
45.62
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Sample and Population
• Population
– entire set of things of interest
• e.g., the entire piggy bank of pennies
• e.g., the entire population of individuals in the US
• Sample
– the part of the population about which you actually
have information
• e.g., a handful of pennies
• e.g., 100 men and women who answered an online
questionnaire about health care usage
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Why Samples Instead of
Populations Are Studied
• It is usually more practical to obtain
information from a sample than from the
entire population.
• The goal of research is to make
generalizations or predictions about
populations or events in general.
• Much of social and behavioral research is
conducted by evaluating a sample of
individuals who are representative of a
population of interest.
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Methods of Sampling
• Random Selection
– method of choosing a sample in which each individual
in the population has an equal chance of being
selected
• e.g., using a random number table
• Haphazard Selection
– method of selecting a sample of individuals to study by
taking whoever is available or happens to be first on a
list
• This method of selection can result in a sample that is
not representative of the population.
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Statistical Terminology for
Sample and Populations
• Population Parameters
– mean, variance, and standard deviation of a
population
– are usually unknown and can be estimated from
information obtained from a sample of the
population
• Sample Statistics
– mean, variance, and standard deviation you figure
for the sample
– calculated from known information
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Probability
• Expected relative frequency of a particular
outcome
– outcome
• term used for discussing probability for the result of an
experiment
– expected relative frequency
• number of successful outcomes divided by the number of
total outcomes you would expect to get if you repeated
an experiment a large number of times
• long-run relative-frequency interpretation of probability
– understanding of probability as the proportion of a
particular outcome that you would get if the experiment
were repeated many times
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Steps for Figuring Probability
• Determine the number of possible
successful outcomes.
• Determine the number of all possible
outcomes.
• Divide the number of possible successful
outcomes by the number of all possible
outcomes.
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Figuring Probability
• You have a jar that contains 100 jelly
beans.
• 9 of the jelly beans are green.
• The probability of picking a green jelly
bean would be
9 (# of successful outcomes) or 9%
100 (# of possible outcomes)
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Range of Probabilities
• Probability cannot be less than 0 or
greater than 1.
– Something with a probability of 0 has no
chance of happening.
– Something with a probability of 1 has a
100% chance of happening.
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p
• p is a symbol for probability.
– Probability is usually written as a decimal,
but can also be written as a fraction or
percentage.
– p < .05
• the probability is less than .05
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Probability, Z Scores, and the
Normal Distribution
• The normal distribution can also be
thought of as a probability distribution.
– The percentage of scores between two Z
scores is the same as the probability of
selecting a score between those two Z
scores.
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Normal Curves, Samples and
Populations, and Probability in
Research Articles
• Normal curve is sometimes mentioned in the
context of describing a pattern of scores on a
particular variable.
• Probability is discussed in the context of
reporting statistical significance of study
results.
• Sample selection is usually mentioned in the
methods section of a research article.
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Key Points
•
•
•
•
•
•
•
In behavioral and social science research, scores on many variables approximately
follow a normal curve which is a bell-shaped, symmetrical, and unimodal distribution.
50% of the scores on a normal curve are above the mean, 34% of the scores are
between the mean and 1 standard deviation above the mean, and 14% of the scores
are between 1 standard deviation above the mean and 2 standard deviations above
the mean.
A normal curve table is used to determine the percentage of scores between the
mean and any particular Z score and the percentage of scores in the tail for any
particular Z score. This table can also be used to find the percentage of scores above
or below any Z score and to find the Z score for the point where a particular
percentage of scores begins or ends.
A population is a group of interest that cannot usually be studied in its entirety and a
sample is a subgroup that is studied as representative of this larger group.
Population parameters are the mean, variance, and standard deviation of a
population, and sample statistics are the mean, variance, and standard deviation of a
sample.
Probability (p) is figured as the proportion of successful outcomes to total possible
outcomes. It ranges from 0 (no chance of occurrence) to 1 (100% chance of
occurrence). The normal curve can be used to determine the probability of scores
falling within a particular range of values.
Sample selection is sometimes discussed in research articles.
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