Transcript Descriptive Statistics

Statistics [0,I/2]
The Essential Mathematics
Two Forms of
Statistics
• Descriptive Statistics
• What is physically happening within
the data?
• Inferential Statistics
• What can I glean from a sample that
is pertinent to the population?
Descriptive Statistics
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Measures of Center
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mean, median, mode
Measures of Spread
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variance, standard deviation, range,
IQR, outliers
Measures of Shape
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kurtosis, skewness
Descriptive Statistics
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Measures of Center
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mean, median, mode
Measures of Spread
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variance, standard deviation, range,
IQR, outliers
Measures of Shape
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kurtosis, skewness
Measures of Center
• The expectation of what should happen
in a given situation at random
• Traditionally, we see that as the mean,
but that can also be the median or the
mode in certain contexts
Situation
• You are interested in the body mass of
full grown adults from one gender.
• If you were to find one person from that
gender at random, what would you
expect that person to weigh?
Mean
• Four types of means
• Arithmetic mean (typical
interpretation)
• Geometric mean
• Harmonic mean (most conservative)
• Quadratic mean (pooling operation)
Arithmetic Mean
• Unbiased estimator for the population
mean
• When should I be concerned with the
mean?
• Data should be symmetric
• equally likely to see something
relatively large as I am relatively
small
• Typically, the first thing to look at
Arithmetic Mean
• Add them up, divide by the number of
them
Symmetric without a
picture?
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Line the data up from worst to first
(maximum to minimum)
Find the one in the middle
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Subtract the minimum from the middle
and subtract the middle from the
maximum
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Are those two values equal?
Skewness (we’ll see that later)
Situation
• You are interested in the economic
conditions of a country (say the United
States).
• If you were to select a household at
random from the United States, how
much money do you expect that
household makes?
Median
• The exact middle observation of a set of
data
• This is the mean when a set is
symmetric
• When a set is asymmetric, these are
different
• Not responsive to questionable
influences
• The stoic of statistics
Median or Mean?
• Find the mean and the median
• How close are they?
• If they are “close”, use the mean
• If they are not close, typically use the
median (this indicates skew)
Situation
• You are an artificial intelligence
programmer and are interested in how
to assign algorithms for random
occurrences in a football game that
result in scores.
• What is the expected score that
happens on that play?
Weird Scenario...
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Football has a few ways of scoring, but we know what the
set is going to be composed of:
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Touchdown (typical): 7
Touchdown (2 pt. conversion): 8
Touchdown (failed conversion): 6
Field Goal: 3
Safety: 2
The “average score” on a play in football is probably
somewhere between 4.5 and 5
We should, however, expect the score to be either 3 or 7
Mode
• The mode is the most common
observation in a dataset
• Sparingly used, but can be important
• If observations recur, why is that
happening?
Questions:
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Which of the three makes sense based on
my understanding of what should happen?
Should this data be inherently symmetric?
Should this data be pulled one way or the
other?
Should this data be predisposed to particular
values?
Answer these questions before you see it!
Measures of Spread
• What is the variation found within my
data?
• Many different ways of looking at this
(based on your choice of mean or
median):
• Standard deviation/variance for mean
• Range/IQR for median
Variance
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Otherwise known as “residual error”
Find the mean
Take each observation and subtract the
mean from it
Square each value
Add them up
Divide by n-1
Variance
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If a set is “tight” to its mean, its variance will be low (we
will call this leptokurtic later)
If a set is “broad” to its mean, its variance will be higher
(we will call this platykurtic later)
Remember: the larger a residual, the higher the impact
of squaring it is
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52 = 25; 102 = 100, a factor of 4 when the residual
doubled
Why square it?
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If we didn’t, variance would always be 0,
rendering the statistic meaningless!
Why?
Variance allows us to see spread by making
negative values positive and then adding more
weight to something more distant (both effects
of squaring)
Why n-1?
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Degrees of freedom
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Makes us more conservative
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Dividing by larger numbers reduces
values; dividing by smaller numbers
assumes wider
We don’t have everything, so tend to
conservative
Standard Deviation
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Undoes the squaring procedure
Gives us the “average” distance between an observation
and the mean
If variance is high, standard deviation will be high; if low,
standard deviation will be low
Great metric for “how far” questions as it normalizes
observations
Range and IQR
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In the case of the median, percentile
observations are the focus
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Minimum, maximum
25%, 75%
Range = maximum - minimum
IQR = 75% - 25%
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IQR defines outliers
Skewness
• Is a distribution symmetric or biased?
• The signum of skewness is the
relationship between the mean and the
median
• Mean > median --> positive skew
• Mean < median --> negative skew
Reasons for left skew
• A test or task were too easy
• Ever taken an exam where nearly
everyone got a great grade, but
someone struggled? That’s left
skew...
Reasons for right
skew
• A variable naturally has a left bound
• Time-based data
• Economics
Right tail transform
• Right tail skews are typically
transformed using logarithms or square
roots
• Why?
Kurtosis
• Is data predisposed to a particular
central occurrence?
• Can’t be less than 1 (-2 excess)
• Positive values of kurtosis reflect high
peaks (predisposition)
• Negative values of kurtosis reflect flatter
peaks
Assignment
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You will be provided a dataset that comes from a
questionnaire about ecological values (New
Ecological Paradigm).
You will be shown all of the values mentioned in this
slide set and a bar graph of the responses
Determine the appropriate measure of central
tendency.
Determine whether or not you feel there are effects
such as biasing or predisposition occurring.
Remember: gut instincts...do not do any tests!
NEP
• For your reference:
• High values on odd questions favor
human endeavors
• High values on even questions favor
the environment