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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

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

Steps in Conducting an Assessment using Inventory

and Monitoring

- Develop Problem Statementmay include goals

- Develop specific objectives

- Determine important data to collect

- 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

- Collect data

- Analyze data

- Assess data in context of objectives

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.

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)

- 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

- We are interested in describing this population
- its total population size
- mean density/quadrat
- variation among plots

How did we obtain the sample statistics?

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)

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?

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

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?

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

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 ?

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

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

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.

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.

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- 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

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

The effect of sampling variation a function of

precision

All estimates come from the same population

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?

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

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?

Consider this example

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?

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

The interpretation of this chart (p. 76) should

now ( or soon!) be clear

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