Descriptive statistics

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Descriptive statistics
for one variable

• ?????

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What to describe?

• What is the location or center of the data?
  (measures of location)
• How do the data vary? (measures of
  variability).

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Types of statistics

• Descriptive Statistics
• Gives numerical and graphic procedures to
  summarize a collection of data in a clear and
  understandable way

• Inferential Statistics
• Provides procedures to draw inferences about
  a population from a sample

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Reasons for using statistics

• aid in summarization
• aid in getting at whats going on
• aid in extracting information from the data
• aid in communication

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Frequency distribution

• The frequency with which observations are
  assigned to each category or point on a
  measurement scale.
• Most basic form of descriptive statistic
• May be expressed as a percentage of the total
  sample found in each category

Source Reasoning with Statistics, by Frederick
Williams Peter Monge, fifth edition, Harcourt
College Publishers.
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Frequency distribution

• The distribution is read differently depending
  upon the measurement level
• Nominal scales are read as discrete measurements
  at each level (no ordering)
• Ordinal measures show tendencies, but categories
  should not be compared (ordering exists, but not
  distance)
• Interval (distance exists, but no ratios) and
  ratio scales (ratios exist) all for comparison
  among categories

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• Sex N Mean Median TrMean StDev SE
  Mean
• female 126 91.23 90.00 90.83 11.32
  1.01
• male 100 96.79 110.00 105.62 17.39
  1.74
• Minimum Maximum Q1 Q3
• female 65.00 120.00 85.00
  98.25
• male 75.00 162.00 95.00
  118.75

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Source Protecting Children from Harmful
Television TV Ratings and the V-chip Amy I.
Nathanson, PhD Lecturer, University of California
at Santa Barbara Joanne Cantor, PhD Professor,
Communication Arts, University of
Wisconsin-Madison
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Source http//www.elonka.com/kryptos/ Web page
on cryptography
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Ancestry of US residents
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Source UCLA International Institute
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Source Cornell University website
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Source www.cit.cornell.edu/computer/students/band
width/charts.html
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Source www.cit.cornell.edu/computer/students/band
width/charts.html
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Source Verisign
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Search engine use
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The percentage of online searches done by US home
and work web surfers in July 2006
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NY Times
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Source Verisign
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Old Faithful Geyser
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• Duration in seconds of 272 eruptions of the Old
  Faithful geyser.
• library(datasets)
• gt faithful110,
• eruptions waiting
• 1 3.600 79
• 2 1.800 54
• 3 3.333 74
• 4 2.283 62
• 5 4.533 85
• 6 2.883 55
• 7 4.700 88
• 8 3.600 85
• 9 1.950 51
• 10 4.350 85

gt summary(faithful) eruptions waiting
Min. 1.600 Min. 43.0 1st
Qu. 2.163 1st Qu. 58.0 Median 4.000
Median 76.0 Mean 3.488 Mean
70.9 3rd Qu. 4.454 3rd Qu. 82.0
Max. 5.100 Max. 96.0
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Normal distribution

• Many characteristics are distributed through the
  population in a normal manner
• Normal curves have well-defined statistical
  properties
• Parametric statistics are based on the assumption
  that the variables are distributed normally
• Most commonly used statistics
• This is the famous Bell curve where many cases
  fall near the middle of the distribution and few
  fall very high or very low
• I.Q.

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Statistical properties of the normal distribution
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I.Q. distribution
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Measures of central tendency

• Mode (Mo) the most frequent score in a
  distribution
• good for nominal data
• Median (Md) the midpoint or midscore in a
  distribution.
• (50 cases above/50 cases below)
• insensitive to extreme cases
• --Interval or ratio

Source Reasoning with Statistics, by Frederick
Williams Peter Monge, fifth edition, Harcourt
College Publishers.
35
Measures of central tendency

• Mean
• The average scoretotal score divided by the
  number of scores
• has a number of useful statistical properties
• however, can be sensitive to extreme scores
• many statistics based on mean
• Sensitive to outliers
• Extreme cases that just happened to end up in
  your sample by chance

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Index of central tendency
Source http//www.uwsp.edu/psych/stat/5/skewnone.
gif
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Source Scianta.com
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Source www.wilderdom.com/.../L2-1UnderstandingIQ.
html
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Source CSAPs Data Pathways
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Measures of dispersion

• Look at how widely scattered over the scale the
  scores are
• Groups with identical means can be more or less
  diverse
• To find out how the group is distributed, we need
  to know how far or close individual members are
  from the mean
• Like mean, only meaningful for interval or
  ratio-level measures

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Measures of dispersion

• Range
• Distance between the highest and lowest
  scores in a distribution
• sensitive to extreme scores
• compensate by calculating interquartile range
  (distance between the 25th and 75th percentile
  points) which represents the range of scores for
  the middle half of a distribution
• Usually used in combination with other measures
  of dispersion.

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Range
Source www.animatedsoftware.com/
statglos/sgrange.htm
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Source http//pse.cs.vt.edu/SoSci/converted/Dispe
rsion_I/box_n_hist.gif
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• Average Deviation (Mean Deviation)
• Merits      
• 1. Easy to calculate and understand.     
•  2. This can be calculated from any
  average.      
• 3. It is less affected by extreme
  observations.      
• Demerits      
• 1. This is mathematically incomplete
  because it ignores negative signs.      
• 2. As it can be calculated from any
  average, it does not have certainty (i.e., it is
  not a well defined measure).      
• 3. Its use is very limited in statistical
  work.

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Measures of dispersion

• Variance (S2)
• Average of squared distances of individual points
  from the mean
• High variance means that most scores are far away
  from the mean. Low variance indicates that most
  scores cluster tightly about the mean. 

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Standard Deviation (SD)

• A summary statistic of how much scores vary from
  the mean
• Square root of the Variance
• expressed in the original units of measurement
• Used in a number of inferential statistics

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Variance vs. Standard Deviation
Standard Deviation
Variance
Population
Sample
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Skewness of distributions

• Measures look at how lopsided distributions
  arehow far from the ideal of the normal curve
  they are
• When the median and the mean are different, the
  distribution is skewed. The greater the
  difference, the greater the skew.

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• Distributions that trail away to the left are
  negatively skewed and those that trail away to
  the right are positively skewed
• If the skewness is extreme, the researcher should
  either transform the data to make them better
  resemble a normal curve or else use a different
  set of statisticsnonparametric statisticsto
  carry out the analysis

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Different Shapes of Distributions
Source http//faculty.vassar.edu/lowry/f0204.gif
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Skewness of distributions
Source http//www.polity.org.za/html/govdocs/repo
rts/aids/images/image022.gif
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Distribution of posting frequency on Usenet
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Kurtosis

• Measures of kurtosis look at how sharply the
  distribution rises to a peak and then drops away

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