Transcript Intro to Stats

Statistics Intro
Univariate Analysis
Central Tendency
Dispersion
Review of Descriptive Stats.
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Descriptive Statistics are used to
present quantitative descriptions in a
manageable form.
This method works by reducing lots of
data into a simpler summary.
Example:
– Batting average in baseball
– Cornell’s grade-point system
Univariate Analysis
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This is the examination across cases of one
variable at a time.
Frequency distributions are used to group
data.
One may set up margins that allow us to
group cases into categories.
Examples include
– Age categories
– Price categories
– Temperature categories.
Distributions
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Two ways to describe a univariate
distribution
A table
A graph (histogram, bar chart)
Distributions (con’t)
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Distributions may also be displayed
using percentages.
For example, one could use
percentages to describe the following:
– Percentage of people under the poverty
level
– Over a certain age
– Over a certain score on a standardized test
Distributions (cont.)
A Frequency Distribution Table
Category
Under 35
36-45
46-55
56-65
66+
Percent
9%
21
45
19
6
Distributions (cont.)
A Histogram
66+
56-65
46-55
36-45
Percent
Under
35
45
40
35
30
25
20
15
10
5
0
Central Tendency
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An estimate of the “center” of a
distribution
Three different types of estimates:
– Mean
– Median
– Mode
Mean
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The most commonly used method of
describing central tendency.
One basically totals all the results and
then divides by the number of units or
“n” of the sample.
Example: The HSS 292 Quiz 1 mean
was determined by the sum of all the
scores divided by the number of
students taking the exam.
Working Example (Mean)
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Lets take the set of scores:
15,20,21,20,36,15, 25,15
The Mean would be 167/8=20.875
Median
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The median is the score found at the
exact middle of the set.
One must list all scores in numerical
order and then locate the score in the
center of the sample.
Example: If there are 500 scores in the
list, score #250 would be the median.
This is useful in weeding out outliers.
Working Example (Median)
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Lets take the set of scores:
15,20,21,20,36,15, 25,15
First line up the scores.
15,15,15,20,20,21,25,36
The middle score falls at 20. There are
8 scores, and score #4 and #5
represent the halfway point.
Mode
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The mode is the most repeated score in
the set of results.
Lets take the set of scores:
15,20,21,20,36,15, 25,15
Again we first line up the scores
15,15,15,20,20,21,25,36
15 is the most repeated score and is
therefore labeled the mode.
Central Tendency
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If the distribution is normal (i.e., bellshaped), the mean, median and mode
are all equal.
In our analyses, we’ll use the mean.
Dispersion
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Two estimates types:
– Range
– Standard deviation
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Standard deviation is more
accurate/detailed because an outlier
can greatly extend the range.
Range
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The range is used to identify the highest
and lowest scores.
Lets take the set of
scores:15,20,21,20,36,15, 25,15.
The range would be 15-36. This
identifies the fact that 21 points
separates the highest to the lowest
score.
Standard Deviation
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The standard deviation is a value that
shows the relation that individual scores
have to the mean of the sample.
If scores are said to be standardized to
a normal curve, there are several
statistical manipulations that can be
performed to analyze the data set.
Standard Dev. (con’t)
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Assumptions may be made about the
percentage of scores as they deviate from
the mean.
If scores are normally distributed, one can
assume that approximately 69% of the scores
in the sample fall within one standard
deviation of the mean. Approximately 95% of
the scores would then fall within two standard
deviations of the mean.
Standard Dev. (con’t)
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The standard deviation calculates the
square root of the sum of the squared
deviations from the mean of all the
scores, divided by the number of
scores.
This process accounts for both positive
and negative deviations from the mean.
Working Example (stand. dev.)
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Lets take the set of scores 15,20,21,20,36,15,
25,15.
The mean of this sample was found to be
20.875. Round up to 21.
Again we first line up the scores.
15,15,15,20,20,21,25,36.
21-15=6, 21-15=6, 21-15=6,20-21=-1,20-21=1, 21-21=0, 21-25=-4, 36-21=15.
Working Ex. (Stan. dev. con’t)
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Square these values.
36,36,36,1,1,0,16,225.
Total these values. 351.
Divide 351 by 8. 43.8
Take the square root of 43.8. 6.62
6.62 is your standard deviation.