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Statistical Concepts

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Statistical Concepts Basic Principles An Overview of Today s Class What: Inductive inference on characterizing a population Why : How will doing this allow us to ... – PowerPoint PPT presentation

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Title: Statistical Concepts


1
Statistical Concepts Basic Principles
An Overview of Todays Class
What Inductive inference on characterizing a
population Why How will doing this allow us to
better inventory and monitor natural
resources Examples
Relevant Readings Elzinga pp. 77-85 , White et
al.
Key points to get out of todays lecture
Description of a population based on
sampling Understanding the concept of variation
and uncertainty
2
By the end of todays lecture/readings you should
understand and be able to define the following
terms
Accuracy/Bias Precision
Population parameters Sample statistics
Standard Error Confidence Interval
Mean Variance / Standard Deviation
3
Steps in Conducting an Assessment using Inventory
and Monitoring
  1. Develop Problem Statementmay include goals
  1. Develop specific objectives
  1. Determine important data to collect
  1. Determine how to collect and analyze data

principles of statistics allows us to better plan
how to collect the data AND analyze it - they
work in tandem
  1. Collect data
  1. Analyze data
  1. Assess data in context of objectives

4
The Relation between Sampling and Statistics
Can you make perfect generalizations from a
sample to the population?
There is uncertainty in inductive inference.
The field of statistics provides techniques for
making inductive inference AND for providing
means of assessing uncertainty.
5
Why sample?
Inductive inference process of generalizing
to the population from the sample.. Elzinga
p. 76
Target/Statistical Population
Sample Unit
Individual objects
(in this case, plants)
Elzinga et al. (200176)
6
  • We are interested in describing this population
  • its total population size
  • mean density/quadrat
  • variation among plots

At any point in time, these measures are fixed
and a true value exists. These descriptive
measures are called ?
Population Parameters
The estimates of these parameters obtained
through sampling are called ?
Sample Statistics
7
  • We are interested in describing this population
  • its total population size
  • mean density/quadrat
  • variation among plots

How did we obtain the sample statistics?
8
ALL sample statistics are calculated through an
estimator
An estimator is a mathematical expression that
indicates how to calculate an estimate of a
parameter from the sample data. White et al.
(1982)
9
You do this all the time!
The Mean (average) What is the formal estimator
you use?
(standard expression, but often denoted by a
some other character)
Which states to do what operations?

10
Estimating the amount of variability
Why?
Recall
There is uncertainty in inductive inference.
The field of statistics provides techniques for
making inductive inference AND for providing
means of assessing uncertainty.
  • Two key reasons for estimating variability
  • a key characteristic of a population
  • allows for the estimation of uncertainty of a
    sample

11
Think about this conceptually, before
mathematically
What characteristic of the population would
affect the level of similarity among each
groups samples?
How about sampling method?
12
Estimating the Amount of Variation within a
Population
The true population standard deviation is a
measure of how similar each individual
observation (e.g., number of plants in a quadrat)
is to the true mean
13
Populations with lots of variability will have a
large standard deviation, whereas those with
little variation will have a low value
High or low?
Counts of dock from wedn lab?
What would the standard deviation be if there
were absolutely no variability- that is, every
quadrat in the population had exactly the same
number ?
14
The Computation of the Standard Deviation
  • key is to get differences among observations,
    right?
  • then each difference is subtracted from the
    mean
  • consistent with definition

Does this make sense ?
For the pop Std Dev, we take the SQRT of the Var
15
The Computation of the Standard Deviation
The estimator of the variance that is what
produces the sample statistic, simply replaces N
with the actual samples (n), and the true
population mean with the sample mean
The estimator of the standard dev is simply the
SQRT of the var. Because of an expected small
sample bias, n-1 is usually used rather than n as
the divisor in both the var and stdev
16
Where Are We? We have computed a mean value of a
population and a sample We have computed the
variability of a population and a sample We now
can use the variability of the sample to tell us
something about uncertainty and the way we
sampled to tell us something accuracy.
17
Bias vs Precision
Bias (accuracy) Precision
Essentially, the closeness of a measured value
to its true value the average performance of an
estimator
The closeness of repeated measurements of the
same quantity the repeatability of a result.
18
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19
  • The level of bias is a function of your sampling
    scheme and
  • estimator used. Your are in control of this!
  • Precision is a function of the variance of the
    population, and
  • How you sample
  • Number of samples
  • Variability within samples (so quadrat SIZE and
    SHAPE
  • matters) compared to among samples
  • analytical techniques

20
Why does Bias and Precision matter in inventory
and monitoring of natural resources?
Lets imagine monitoring the density of dock in
Rons pasture through time
21
The effect of sampling variation a function of
precision
All estimates come from the same population
22
So how good are your parameter estimates?
Lets examine this with the estimation of the
population mean
What influences the reliability of the estimate
of the mean value?
23
Estimating the Reliability of a Sample Mean
Standard error the standard deviation of
independent sample means
Measures precision from a single sample (e.g.,
from a collection of quadrats)
Quantified the certainty with which the mean
computed from a random sample estimates the true
population mean
24
Estimating the Reliability of a Sample Mean
Formally, the SE is a function of the standard
deviation of the sample and the number of samples
SEs/SQRT(n)
Does this make sense?
25
Consider this example
26
Communicating the Reliability of a Sample Mean
Confidence Intervals
Provides an estimate of precision around a sample
mean or other estimated parameter Includes
two components confidence interval
width confidence level the probability that the
interval includes the true value
Whats the relation between the two?
27
Communicating the Reliability of a Sample Mean
Estimating the Confidence Interval
95 CI Mean /- 1.96(SE)
Intervals can be computed for any level of
confidence desired in a particular study
28
The interpretation of this chart (p. 76) should
now ( or soon!) be clear
29
Key points to get out of todays lecture
Description of a population based on
sampling Understanding the concept of variation
and uncertainty
Ability to define (and understand) the following
terms
Accuracy/Bias Precision
Population parameters Sample statistics
Standard Error Confidence Interval
Mean Variance / Standard Deviation
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