Measurement, Quantification and Analysis

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Measurement, Quantification and Analysis

• Some Basic Principles

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Three Major Issues

• 1) Biological and especially ecological data show
  high variability in quantitative traits
• 2) We almost never measure everything in field
  research rather we sample from larger
  populations or data sets
• Sampling leads to uncertainty about conclusions,
  so we always must estimate our uncertainty

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Variability
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Continuous data
All natural processes are variable, Whether
continuous or discreet
Discreet data
Plus, better sampling effort better describes
distributions
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In many processes, we observe characteristic
distributions
Binomial Few interacting factors
Normal Many interacting factors
2 factors One way to get AA or aa, 2 ways to get
Aa
4 factors One way to have AAAA or aaaa, 4 ways
to get AAAa or aaaA, and 6 ways to get AAaa
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Sampling and Estimation
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A characteristic of field biology is the attempt
to estimate parameters from highly variable
populations of uncertain true value.
To calculate the average in a sample Mean Sum
of all observations/number of observation
To estimate the variability of the
observations Variance Sum of (individual
observation Mean of observations)2
__________________________________________
___
Number of Individual Observations - 1
-1
Or to express this in the same units as the Mean
Standard deviation Square Root of the Variance
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All natural processes are variable, Whether
continuous or discreet What happens when we
estimate means? Select 5 observations at random.
Then 10. Then 25.
Probability

1. Better sampled populations yield better
   distributions
2. Larger sample sizes yield better estimates
3. Means will also be variable, and will have a
   characteristic distribution

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To estimate the variability of the means Divide
the standard deviation (the square root of the
variance) by the square root of the sample size
(why? Variability of the means is dependent upon
sample size.) Recall, To estimate the
variability of the observations Variance Sum
of (individual observation Mean of
observations)2
_____________________________________________
Number of
Individual Observations 1 To estimate the
variability of the means Divide the square root
of the variance, the standard deviation, by the
square root of the sample size. The bigger the
sample size, the less variable the means
This is the Standard Error, which is used to
calculate a Confidence Interval
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Uncertainty
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Confidence intervals represent a level of
confidence about the true value of the mean. In
other words, if you sample repeated with a given
sample size, a 95 CI means that in 95 of the
samples you collect, you will have the value of
the true mean.

• No matter how well we sample, we will
  miss-estimate the population parameter a
  certain percentage.
• What level of error are we willing to accept?
• With a 95 limit, 5 of the time.
• In theory, the tails are limitless, so we must
  set a criterion.
• Decision rule 5 error.
• Minimize this with replication

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Importance of Replication?
One sample Wrong 5 or 1/20 of the times you
sample Two replicated samples Wrong 1/20 x 1/20
or 1/400 Three replicated samples Wrong 1/20 x
1/20 x 1/20 or 1/8,000
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What confidence do we want? What error will we
accept?
One things we do frequently in science is compare
things. For example, if one population bigger
than another, which population are we sampling
from?
A
B
What kinds of errors can we make?
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Fundamental Principles

• Have clearly defined hypotheses
• Measure carefully
• Sample intensively large sample sizes reduce
  Beta-Error
• Replicate Replication reduces Alpha-Error

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Samples of Data Sets from Previous Projects that
required Quantification and Statistical Analysis
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Sum of Squares df Mean Square F Sig.
Forearm (mm) Between Groups 4053.985 2 2026.993 569.784 .000
Forearm (mm) Within Groups 152.971 43 3.557
Forearm (mm) Total 4206.957 45
Foot (mm) Between Groups 254.274 2 127.137 55.693 .000
Foot (mm) Within Groups 98.161 43 2.283
Foot (mm) Total 352.435 45