Transcript Descriptive Statistics - Oakton Community College

Descriptive Statistics and
Inferential Statistics
A Picture Can Be Worth a
Thousand Words
Frequency Distributions
Large amounts of information can be
neatly organized and summarized
Graphs can show entire patterns of
scores very clearly
The horizontal line is called the
abscissa
The vertical line is called the ordinate
Frequency Distribution
Histogram: Graph of a frequency
distribution in which the number of
scores falling in each class
represented by vertical bars
Frequency polygon: Graph of a
frequency distribution in which scores
falling in each class represented by
points on a line
Scores Graphed on a Histogram for
Obedience to order to give shocks
Frequency Histogram for accuracy
of taste: Which is abscissa?
Opiate Deaths by Age: Which is the
Ordinate?
How Number in a Group Influences
our Level of Conformity
Studies Want to Compare Groups:
Average Score on Memory Test
Use Measures of Central Tendency
Mean: average score of each group
Median: middle score of each group
Mode: most frequent score
Measures of Variability Crucial
How spread out are the scores in
each group
Range: difference between highest
and lowest score
Standard Deviation: how much does
the typical score differ from the mean
SD is the important statistic in most
research studies
Standard Deviation
When scores are widely spread, the
standard deviation number gets larger
When the scores are close together,
the standard deviation score gets
smaller
Scores on different tests are
converted by using standard
deviations
Standardizing Scores From
Different Tests
Subtract mean from score and divide
by the standard deviation for the
group—now a Z-score
Susan had a score of 110 in a class
with a mean of 100 and a standard
deviation of 10
Her z-score is what?
Turning Scores on Different Tests
into Standard Scores
Susan’s score on the original test was
110. In order to compare her
performance to scores on other tests,
her score was turned into a standard
or z-score
Her score was subtracted from the
mean of 100, resulting in 10
10 was divided by the standard
deviation of 10, resulting in?
Z-scores allow researchers to
compare scores from different tests
Susan’s standard or z-score is +1
Now John took a different memory
test and received a score of 118.
Why can’t compare John’s score to
Susan’s original score of 110?
Importance of Standard Scores
John’s score of 118 on one test can
be compared to Susan’s score of 110
on a different score IF both scores are
turned into standard or z-scores
John’s score came from a class
having a mean of 100 and a standard
deviation of 18. What is his z-score?
Both John and Susan have the
equivalent Standard Scores
John: mean of 100 subtracted from
his score of 118 is 18. Divided by the
standard deviation of 18 provides the
z-score of +1
Now we know that Susan and John
scores are equivalent. Compared to
other students, each was an equal
distance above average.
Normal Curve
When chance events are recorded,
some outcomes have a high
probability and occur very often
Other events have a low probability
and occur rarely
Distributions resembles a normal
curve
Normal Curve
Bell-shaped, with large number of
scores in the middle, with very few
extremely high and low scores
Psychological Traits Follow
the Bell Curve
Measurements of height, memory
span and intelligence are distributed
along a normal curve
Most people have “average” height,
memory ability and intelligence
Fewer people found at the extremes
Standard Deviation and the
Normal Curve
Standard deviation measures the
proportion of curve above and below
the mean
68 percent of all cases fall between
one standard deviation above and
below the mean
The Real Extremes!
9 percent of all cases fall between 2
standard deviations above and below
the mean
99 percent of all cases fall between 3
standard deviations of the mean
Show the Percentages relative
to Standard Deviations
Relationship between standard
deviations and the normal curve
Standard scores relationship to
the Normal Curve
Z-scores of either – or + 3 are very
extreme—99.9 or 00.1
Z-scores of – or + 2 are also extreme:
97.7 or 02.3
Z-scores of plus or minus 1 are 84 or
16
Why need statistics?
Results of psychological studies often
expressed as numbers
These numbers need to be
summarized and interpreted to have
any meaning
Summarizing numbers with graphs
makes it easier to see patterns
Graphical statistics
Descriptive statistics organize and
summarize numbers
Histograms and Polygons represent
numbers pictorially.
Two basic questions about groups
of numbers
What is the central tendency of the
group of numbers?
How much do the group of numbers
vary—or what is the variability?
Measures of Central
Tendencies
Mean is the average score: add all
scores and divide by total # of scores
Median found by arranging scores
from highest to lowest and selecting
the middle score
Mode is score that occurs most often
Measures of Variability
Range is the difference between the
highest score and the lowest score
Standard deviation shows how, on
average, all the scores differ from the
mean
Standard Scores
To change an original score into a
standard score (or z-score), you
subtract the mean from the score and
divide it by the standard deviation
This allows for meaningful
comparisons between scores from
different groups
Standard Deviations and
the Normal Curve
Any one standard score can be
placed on a normal curve relative to
how it compares to other scores.
Some scores are close to the mean
while other scores might be way
above or below the mean—by 2 or
even 3 standard deviations.
Comprehension Check
Let’s say you ask 100 people how
many minutes they sleep each night
and record their answers. How could
you show these scores graphically?
Comprehension Check
To find the average amount of sleep
for your subjects, would you prefer to
know the most frequent score (mode),
the middle score (median), or the
arithmetic average (mean)?
Comprehension Check
How could you determine how much
sleep times vary? That is, would you
prefer to know the highest and lowest
scores (range) or the average amount
of variation (standard deviation)?
Comprehension Check
Do you think that the distribution of
minutes of sleep would form a normal
curve? Why or why not?
If the number of minutes of sleep that
Subject A reports is two standard
deviations above the mean, what
percentage of people would sleep as
much or more than s/he does?
Comprehension Check
If the numbers of minutes Subject B
reports that he sleeps one standard
deviation below the mean, how many
people in the study sleep the same or
less?
If instead Subject B reports that he
sleeps two standard deviations below
the mean, how many people in the
study sleep the same or less?
Comprehension Check
__________________ statistics
summarize numbers so they become
more meaningful or easier to
communicate.
The measure of central tendency that
provides the average is the ________
Comprehension Check
If scores are placed in order, from the
smallest to the largest, the ________
defined as the middle score.
As a measure of variability, the ______
Is defined as the difference between the
highest and the lowest score.
Comprehension Check
As a measure of variability, the
________________ provides the
average amount of variation.
A z-score of -1 tells us that the score
stands ______ standard deviation
below the mean.
Comprehension Check
A z-score of +2 tells us that the score
stands two standard deviations _____
the mean.
In a normal curve 99 percent of all
scores can be found between ____
and ____ standard deviations from
the mean
Inferential Statistics
Let’s say a researcher studies the
effects of a new therapy on a small
group of depressed people. The
researcher would like to know if the
results of her study holds true for all
depressed people. Inferential
statistics provide techniques that
allow researchers to determine if their
results are generalizable or not.
Samples and Populations
Scientific studies wish to make
conclusions about entire populations
of subjects—such as all cancer
patients or all married couples.
However, it is totally impractical to
study the entire population of cancer
patients or the entire population of
married couples. How is this problem
resolved?
Representative Samples
Samples—that is smaller cross
sections of a population—are studies
to draw conclusions about the entire
population.
For any sample to be meaningful, it
must be representative. That is, it
must truly reflect the characteristics of
the entire population.
How obtain representative
samples?
A very important aspect of choosing a
sample is through random selection—
chosen totally at random.
That would mean that each member
of the population must have an equal
chance of being included in the
sample.
Significant Differences
Let’s say that in a memory
experiment, it was found that the
average memory score was higher for
the group given the drug than for
those who did not take the drug.
How can researchers determine if this
difference wasn’t just by chance?
Tests of Significant Differences
Any experimental result that could have
occurred by chance 5 times or less our of
100 (probability of .05) is considered to be
a significant result.
In the memory experiment, they find that
the probability is .025 that the group means
would differ as much as they do. This
allows the conclusion that, with reasonable
certainty, the drug did improve the scores.
Correlations—Rating Relationships
Psychologists are very interested in
detecting relationships between
events
Are children from single-parent
families more likely to achieve in
school?
Is wealth related to happiness
Is the chance of having a heart attack
related to being a hostile person?
Visualizing a Correlation
Construct scatter diagrams—one
variable being on the X axis and the
other being on the Y axis.
Variables with a positive correlation
means that when one goes up the
other also goes up
Also, when one variable goes down,
the other goes down
Positive Correlations
Amount a student studies for a test
and their test performance
The more a student studies, the
higher their test score will be.
The less a student studies, the lower
their test score will be.
Positive Correlation
How frequently parents read to their
child and the child’s reading ability.
The more parents read to their child,
the better their child’s reading ability.
The less parents read to their child,
the lower their child’s reading ability.
Strength of Correlation
The strength of the positive
correlation between studying and test
scores could be very strong--+.87 for
example.
But the strength of this positive
correlation for some types of tests
might be fairly low, more like +36.
Scatter Diagrams for
Variables with positive correlations
Increases in the X variable are match
by increases in the Y variable.
As German mark goes up, so does
the French mark
Positive Correlation between
Democracy and Capitalism
Negative Correlations
Two variables have a negative
association if the presence of one
predicts the absence of the other
Talking to your friends during class
will predict low comprehension of
class content.
Watching lots of TV predicts a low
GPA.
Diagrams for Variables with a
Negative Correlation
As values of one variable increases, the
values of other variable decreases.
Some variables have an
insignificant association
Correlation Coefficient
The coefficient is simply a number
falling somewhere between +1.00 and
–1.00.
If the number is close to zero, it
indicates a very weak or nonexistent
relationship.
Correlation Coefficient
If the correlation is +1.00, a perfect
positive relationship exists.
If the correlation is –1.00, a perfect
negative relationship exists.
Most correlations of significance have
a strength somewhere between .90
and .30.
Pearson r: Statistic used to
determine strength of correlations
The most commonly used correlation
coefficient is called the Pearson r.
Correlations often provide useful
information, like knowing there is a
correlation between smoking and lung
cancer
Correlations allow for Predictions
NOT Explanations
Colleges use students GPA’s and
scores on the ACT/SAT to help
predict which students will have the
best chance to succeed.
Such predictions are not perfect but
they are useful to a certain extent for
screening applicants
How to increase predictive
power of correlation coefficients
Square correlation coefficient—i.e., multiply
r by itself
Get a number telling the percent of
variance accounted for by correlation
Correlation between IQ and GPA is .6
Multiply .6 by .6 and learn 36 percent of
variation in college grads accounted for by
knowing IQ scores
Correlation and Causation
Finding correlation between two
variables does not automatically
mean one causes the other
Correlation does not demonstrate
causation
What does a correlation mean?
When a correlation exists, the best we can
say is that two variables are related
Not mean that it is impossible for two
correlated variables to have a cause-effect
relationship
Means cannot conclude causal link solely
on correlation
Reasons variables can be
correlated
Two correlated measures can be
related as a result of a third variable
The more hours students devote to
studying, the better their grades
Possible that grades and amount of
study both related to motivation or
interest a student has
Comprehension Check
In inferential statistics, observations of
a _________________ are used to
make inferences and conclusions
about an entire population.
A ________________ sample must
reflect characteristics of the entire
population.
Comprehension Check
A representative sample can be
obtained by selecting members of the
sample at _______________
The results of an experiment to be
considered statistical significant only if
the results could not have occurred by
______________
Comprehension Check
If the results of an experiment could
only have occurred by chance
______ times out of 100, then the
results would be considered
statistically significant.
If the results could have occurred 20
times out of 100, the result would not
be considered __________. Why?
Comprehension Check
A scatter diagram can be used to plot
and visualize a ________________
between two groups of scores.
If two variables have a negative
correlation, increases in one variable
will correspond to _________ in the
other variable.
Comprehension Check
In a positive correlation, increases in
X correspond to __________ in Y.
It is important to remember that a
correlation does not demonstrate
________________
Comprehension Check
This scatter diagram shows variables
that have a _________ correlation.
Correlation Check
This scatter diagram shows two
variables that have a _________
correlation.
Comprehension Check
This scatter diagram shows variables
that have close to zero association.