Download PUAF 610 TA - Public Policy PhD

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 (“xbar”)
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Example
• Assume 4 people take PUAF 610, and
their final exam scores are 95, 87, 93, 83.
What’s the mean for exam score?
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Example
• Mean= (95+87+93+83)/4=89.5
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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
(n+1)/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
(n+1)/2=3, 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
(n+1)/2=3.5, 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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That’s all for class review. So
far so good?
Let’s go to STATA!
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