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 better inventory and monitor natural resources

Examples

Relevant Readings: Elzinga pp. 77-85 , White et al. Key points to get out of today’s lecture:

Description of a population based on samplingUnderstanding the concept of variation and uncertainty

By the end of today’s lecture/readings you should understand

and be able to define the following terms:

Population parameters

Sample statistics

Accuracy/Bias

Precision

Coefficient of variationMean

Variance / Standard Deviation

Inductive inference: “…process of generalizing to the population from the sample..”

Elzinga –p. 76

Why sample?

Elzinga et al. (2001:76)

Target/Statistical Population

Sample Unit

Individual objects(in this case, plants)

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

At any point in time, thesemeasures are fixed and a true value exists.

These descriptive measuresare called ?

Population Parameters

The estimates of these parametersobtained 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 indicateshow to calculate an estimate of a parameter from the sampledata.”

White et al. (1982)

You do this all the time!

The Mean (average):

What is the formal estimator you use?

y n y ii

n

1

1/ ( )

Which states to do what operations?

y is a sample statistic that estimates the population mean

Is y A sample statistic or population parameter ?

= population mean if all n units in the population are sampledy

(standard expression, but often denoted by a some other character)

=_

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:

Recall lab:

Each group collected data from 4m2 plots

Did each group get identical results?

What characteristic of the population would affect the level of similarityamong each groups’ samples?

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 plantsin a quadrat—the sample unit) is to the true mean

Can we develop a mathematical expression for this?

Populations with lots of variability will have a large standarddeviation, whereas those with little variation will have a low value

High or low?

What would the standard deviationbe if there were absolutely no variability-that is, every quadrat in the populationhad exactly the same number ?

The Computation of the Population Variance and Standard Deviation

• key is to get differences among observations, right?• then each difference is subtracted from the mean–

consistent with definition

First, we calculate the population variance

11

2/ ( )N X ii

N

Does this make sense ?

For the pop Std Dev, we take the SQRT of the Variance,

std= =SQRT(var)

Var= = 2

The Computation of the Sample Variance and 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 estimated variance.

Because of an expected small sample bias, n-1 is usually usedrather than n as the divisor in both the var and stdev

s n X Xii

n2

1

21 ( / ) ( )

Estimating the Sample Standard Deviation

s SQRT X X nii

n

( ( ) / )

1

2 1

Worksheet: compare the sample variation of mass of deer mice to mass of bison; which is more variable?

Coefficient of Variation:

A measure of relative precision

“The coefficient of variation is useful because,as a measure of variability, it does not depend upon the magnitude and units of measurements of the data.”

Elzinga et al: 142

CV= s/X * 100_

Usually expressed as a percent,

Using the coefficient of variation, what is more variable,mass of deer mice or bison?

Estimating the Reliability of a Sample Mean

Standard error: the standard deviation of independent sample means

Measures precision from a sample (e.g., density of plants from a collection of quadrats)

Quantified the certainty with which the mean computedfrom a random sample estimates the true population mean

Key points to get out of today’s lecture: Description of a population based on samplingUnderstanding the concept of variation and uncertainty

Ability to define (and understand) the following terms:

Population parameters

Sample statistics

Accuracy/Bias

Precision

Coefficient of variationMean

Variance / Standard Deviation

Friday’s class: from sampling variability to confidence intervals