Download Summary Statistics: Measures of Location and Spread Measures of

1/3/2017
Measures of Location
Summary Statistics: Measures of
Location and Spread
 Illustrate
where the majority of
locations are found:
e.g., means, medians, modes
Ecological Analyses
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Measures of Spread
 Illustrate
Statistics versus Parameters
how variable the data are:

Statistics describe
the sample

Parameters describe
the [unknown?]
population
e.g., standard deviation, variance, standard
error
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Measures of Location: Mean
Arithmetic Mean
 Unbaised
 Arithmetic
estimate of  if:
Observations from random individuals
Samples are independent of eachother
Observations drawn from a large population
that can be described by a normal random
variable
Mean
All observations weighted equally in
calculation
X~N(,)
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Bias and Sample Size
Other Means
 Geometric
Mean
Example from exponential population
growth: when numbers are multiplied on an
arithmetic scale then can be added on a
logarithmic scale ...
So it depends how you use the ‘mean’
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Other Means
 Harmonic
Median and Mode
Mean
Uses the reciprocal (1/Y) of the data
Sensitive to extreme values that are small
Often in conservation biology: effective
population size
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Median and Mode
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Which measure of location?
 Arithmetic
Median: the ‘middle’
observation (unless tied)
 Mode: the observations
that occurs most
frequently

mean most common
Supported by Central Limit Theorem
 Geometric
mean most appropriate for
multiplicative measures
 Median or Mode when distribution
doesn’t match a standard probability
distribution
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Which measure of location?
Measures of Spread
 Researchers
use different measure of
location to support different points of
view...
 Pay attention to what measure is
supplied and always be suspicious of any
measure of location that is not
accompanied by a measure of spread!
 Variance
 Sum
and Standard Deviation
of Squares (SS):
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Measures of Spread
Degrees of Freedom
 Variance
 The
number of independent observations
that we have for estimating statistical
parameters
 For now ... n-1
 Unbiased
estimate of 2
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Measures of Spread
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Standard Error of the Mean
 Variance
 Think
 Standard
of the standard error (or the
mean) as an estimate of the standard
deviation of the POPULATION MEAN
Deviation
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Standard Error of the Mean
Standard Error of the Mean
 If
inference is about the
sample: provide SD (s)
 If the inference is about
the means: provide the
SE
Ecological Analyses
Skewness, Kurtosis, and Central Moments
A
central moment is the average of the
deviations of all observations in a
dataset from the mean of the
observations, raised to a power r:
Ecological Analyses
Skewness, Kurtosis, and Central Moments
r
= 1 (1st moment) always 0
r
= 2 (2nd moment) is the variance
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Skewness
Skewness
 g1 =
r
= 3 (3rd moment) divided by s3 =
skewness
0  normal distribution
 g1 >
0  right-skewed (longer
tail of observations to the right of the
mean
 Skewness
describes how the sample
differs in shape from a symmetrical
distribution
Ecological Analyses
 g1 <
0  left-skewed (longer
tail of observations to the left of the
mean
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Skewness
Kurtosis
 Based
on 4th central moment (r=4)
 Measures
the extent to which the
distribution is distributed in the tails
versus the center of the distribution
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Kurtosis
Kurtosis
 Clumped,
or platykurtic distributions
have g2 < 0 (less probability in the tails)
 Leptokurtic distributions have g2 > 0
(less probability in the center)
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Skewness and Kurtosis
Quantiles
 Should
be tested, but both measures
are sensitive to outliers ...
Ecological Analyses

Box plots of
quantiles can portray
the distribution of
data more accurately
than plots of means
and standard
deviations
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Other Measures

Distribution of Points
 For
normally distributed random
variables:
Coefficient of Variation (CV)
 Variability ‘independent of the mean’

Coefficient of Dispersion
 For discrete variables ‘varianceto-mean’ ratio
 Measure of clumping, but dependent on ‘scale’
67% of observations occur within 1 SD of
the mean
96% of observations occur within 2 SD of
the mean
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Confidence Intervals
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Confidence Intervals
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Confidence Intervals
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Confidence Intervals
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Confidence Intervals
Confidence Intervals
 Interpretation:
95% of the time such an interval will contain
the true value of 
NOT: “there is a 95% chance that the true 
occurs within the interval”  it either does or
does not ...
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