PUAF 610 TA

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PUAF 610 TA

• Session 2

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Today

• Class Review- summary statistics
• STATA Introduction
• Reminder HW this week

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Review Two types of Statistics

• Descriptive statistics summarize numerical
  information.
• Inferential statistics uses a sample to infer the
  population.

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Summary statistic

• In descriptive statistics, summary statistics are
  used to summarize a set of observations.
• Typically,
• What is the central value?
• How widely are values spread from the center?
• Are there data that are very atypical?
• .

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Summary statistic

• a measure of location, or central tendency
• a measure of statistical dispersion
• a measure of the shape of the distribution

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Central tendency

• Central tendency relates to the way in which
  quantitative data tend to cluster around some
  value.
• A measure of central tendency is any of a number
  of ways of specifying the central value.

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Basic measures of central tendency

• Mean
• Median
• Mode

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Mean

• the sum of all measurements divided by the number
  of observations in the data set
• population mean (?) v. sample mean (x-bar)

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Example

• Assume 4 people take PUAF 610, and their final
  exam scores are 95, 87, 93, 83. Whats the mean
  for exam score?

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Example

• Mean (95879383)/489.5

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Median

• the middle observation, when data are ordered
  from smallest to largest
• the point of a distribution that divides the
  bottom 50 from the top 50 of the data. The
  median is the 50th percentile.

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Median

• If there is an odd number of observations, the
  median is the middle observation
• If there is an even number of observations, the
  median is the average of the two middle
  observations
• If the dataset is arranged in increasing order
  the median is located at position (n1)/2

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Example

• Calculate the sample median for the following
  observations 1, 5, 2, 8, 7.
• Start by sorting the values 1, 2, 5, 7, 8.
• The median is located at position (n1)/23,
  thus it is 5.
• An odd number of values.

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Example

• Calculate the sample median for the following
  observations 1, 5, 2, 8, 7, 2.
• Start by sorting the values 1, 2, 2, 5, 7, 8.
• The median is located at position (n1)/23.5,
  Thus, it is the average of the two middlemost
  terms (2 5)/2 3.5.
• An even number of values

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Mode

• the most frequent value in the data set
• It is possible for a distribution to have more
  than one mode or not to have a mode at all.

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Example

• The mode for the following data set
• (1) 1, 2, 2, 3, 4, 7, 9
• (2) 12, 26, 26, 53, 84, 71, 71, 79
• (3) 32, 46, 53, 94, 37, 29

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Comparing of Mode, Median and Mean

• Pros and Cons
• For descriptive purposes we might use the measure
  that suits the data.
• If we would like to infer from samples to
  populations, the mean is a measure of choice
  because it can be manipulated mathematically.

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Summary statistic

• a measure of location, or central tendency
• a measure of statistical dispersion, or variation
  
• a measure of the shape of the distribution

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Measures of Variation

• Variation is variability or spread in a variable
• Measures of variation are lengths of intervals on
  the measurement scale that indicate the spread of
  values in a distribution.

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Measures of Variation

• Range
• Quartiles
• Interquartile range
• Variance
• Standard Deviation

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Range

• the length of the smallest interval which
  contains all the data
• (highest value lowest value) 1

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Quartiles

• any of the three values which divide the sorted
  data set into four equal parts, so that each part
  represents one fourth of the sampled population.

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Quartiles

• first quartile (Q1) lower quartile cuts off
  lowest 25 of data 25th percentile
• second quartile (Q2) median cuts data set in
  half 50th percentile
• third quartile (Q3) upper quartile cuts off
  highest 25 of data, or lowest 75 75th
  percentile
• The difference between the upper and lower
  quartiles is called the interquartile range.

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Variance

• Describes how far values lie from the mean.
• Use the absolute values or to square the
  deviation scores to get rid of the minus signs.
• Averaging absolute values cannot be used in more
  advanced analyses.
• By averaging the sum of squared deviations (sum
  of squares) we can get a measure that is
  susceptible to further algebraic manipulations
  that are difficult or impossible with absolute
  values.

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Variance

• Less intuitive and more difficult to interpret,
  because it is measured in squared units rather
  than original units
• Do not use variance much
• (in population) and (in sample)

 
where µ is the mean and N is the number of
population.
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Standard deviation

• A widely used measure of the variability or
  dispersion.
• It shows how much variation there is from the
  "average.
• Standard deviation is obtained by taking a square
  root of the variance, i.e.
• (population) (sample)

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Standard deviation

• A low standard deviation indicates that the data
  points tend to be very close to the mean.
• A high standard deviation indicates that the data
  is spread out over a large range of values.

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Summary statistic

• a measure of location, or central tendency
• a measure of statistical dispersion, or variation
  
• a measure of the shape of the distribution

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Shape of the distribution

• Skewness
• Kurtosis

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Skewness

• a measure of the asymmetry of the distribution
• The skewness value can be positive or negative,
  or even undefined.

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Skewness

• negative skew The left tail is longer the mass
  of the distribution is concentrated on the right
  of the figure. It has relatively few low values.

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Skewness

• positive skew The right tail is longer the mass
  of the distribution is concentrated on the left
  of the figure. It has relatively few high values.

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Skewness

• A zero value indicates that the values are
  relatively evenly distributed on both sides of
  the mean.

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Kurtosis

• a measure of the "peakedness" of the distribution
  
• Higher kurtosis means more of the variance is the
  result of infrequent extreme deviations, as
  opposed to frequent modestly sized deviations

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Thats all for class review. So far so
good? Lets go to STATA!