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Summary Biometry
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Transcript of Summary Biometry
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BIOMETRY(SUMMARY)
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The usual course of events for conducting scientific work
The Scientific Method
Reformulate or
extend hypothesis
Develop a Working HypothesisObservationConduct an experiment
or a series of controlled
systematic observations
Appropriate statistical
tests
Confirm or
reject hypothesis
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The usual course of events for conducting scientific work
The Scientific Method
Reformulate or
extend hypothesis
Develop a Working HypothesisObservationConduct an experiment
or a series of controlled
systematic observations
Appropriate statistical
tests
Confirm or
reject hypothesis
In a group of crickets,
small ones seem to
avoid large ones
There will be movement away from
large cricket by small ones
Record the number of
times that small crickets
move away from smalland large crickets.
Chi square test
There is a significant
difference in the number
of times small crickets
move away from large
vs. small ones
Avoidance may depend on
previous experience
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Imagine that you are collecting samples (i.e. a number of
individuals) from a population of little ball creatures - Critterus
sphericales
Little ball creatures come in 3 sizes:
Small =
Medium =
Large =
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-sample 1
-sample 2
-sample 3
-sample 4
-sample 5
You end up with a total of five samples
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The real population
(all the little ball creatures that exist)
Your samples
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Each sample is a representation of the population
BUT
No single sample can be expected to accurately represent
the whole population
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To be statistically valid, each sample must be:
1) Random:
Thrown quadrat??Guppies netted from
an aquarium?
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1
2
34
5
6
7
8
9
10
11
12
13
14
15
Assign numbers from a random number table
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To be statistically valid, each sample must be:
2) Replicated:
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But not - Pseudoreplication
Not pseudoreplication Pseudoreplication
10
samples
from the
same
tree
10
samples
from 10different
trees
Sample size = 1Sample size = 10
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TYPES OF
DATA
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RATIO DATA
- constant size interval
- a zero point with some reality
e.g. Heights, rates,
time, volumes,
weights
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INTERVAL DATA
- constant size interval
- no true zero point
zero point depends on the scale used
e.g. Temperature
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Ordinal Scale
- ranked data
-grades, preference surveys
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Nominal Scale
Team numbers
Drosophila eye
colour
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The kind of data you are
dealing with is one
determining factor in thekind of statistical test you
will use.
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Statistics
and
Parameters
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Measures of:
Central tendency - mean, median,
mode
Dispersion - range, mean deviation,
variance, standard deviation,
coefficient of variation
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The real population
(all the little ball creatures that exist)
Central tendency - Mean
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The real population
(all the little ball creatures that exist)
Your samples
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The real population
(all the little ball creatures that exist)
m = SXiN
Central Tendency
1) Arithmetic mean
At Population level
Measuring the diameters ofall the little ball creatures that
exist
m - population mean
Xi - every measurement
in the population
N - population size
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Your samples
X = SXi
n
X = SXi
n
X = SXi
n
X = SXi
n
X = SXi
n
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X = SXin
Sample mean Sum of all measurements in
the sample
Sample size
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If you have sampled in an unbiased fashion
X = SXi
n
X = SXi
n
X = SXi
n
X = SXi
n
X = SXi
n
Each roughly equals m
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Central tendency - Median
Median - middle value of a population or sample
e.g. Lengths of Mayfly (Ephemeroptera) nymphs
5th value (middle of 9)
1 2 3 4 5 6 7 8 9
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Median valueMedian value
Odd number of values Even number of values
Median = middle value Median =+
2
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Odd number of values (i.e. n is odd)
Even number of values
Or - to put it more formally
Median = X(n+1)
2
Median = X(n/2) + X(n/2) + 1
2
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Frequency
(= number of times
each measurement
appears in the
population
Values (= measurements taken)
c. Mode - the most frequently occurring measurement
Mode
Central tendency - Mode
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Measures of Dispersion
Why worry about this??
-because not all populations are created equalDistribution of values in the
populations are clearly different
BUT
means and medians are the same
Mean & median
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Measures of Dispersion -
1. Range - difference between the highest and
lowest values
Remember little ball creatures and the five
samples
Range =
-
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Range - crude measure of dispersion
Note - three samples do not
include the highest value and - two samples do
not include the lowest
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Measures of Dispersion -
2. Mean Deviation
X is a measure of central tendency
Take difference between each measure and the mean
Xi - X
BUT
SXi - X = 0So this is not useful as it stands
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Measures of Dispersion -
2. Mean Deviation (contd)
But if you take the absolute value-get a measure of disperson
S |Xi - X|S |X
i- X|
and
n = mean deviation
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Measures of Dispersion -
3. Variance
-eliminate the sign from deviation from mean
Square the difference
(Xi - X)2
And if you add up the squared differences
- get the sum of squares
S(Xi - X)2 (hint: youllbe seeingthis a lot!)
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Measures of Dispersion -
3. Variance (contd)
Sum of squares can be considered at both thepopulation and sample level
ss = S(Xi - X)2SamplePopulation
SS = S(Xi - m)2
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s2 = S(Xi - X)2SamplePopulation
s2 = S(Xi - m)2
Measures of Dispersion -
3. Variance (contd)
If you divide by the population or sample size- get the mean squared deviation or VARIANCE
N n-1
Population variance Sample variance
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s2 = S(Xi - X)2Note something about the sample variance
n-1
Measures of Dispersion -
3. Variance (contd)
Degrees of freedom or df or n
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Measures of Dispersion -
4. Standard Deviation
- just the square root of the variance
s = S(Xi - m)2N
Population Sample
s= S(Xi - X)2n-1
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Standard Deviation - very useful
Most data in any population are within one
standard deviation of the mean
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NORMAL DISTRIBUTION
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Type of data
Discrete Continuous Other
distributions
2 categories &
Bernoulli process> 2 categories
Use a Binomial model
to calculate expected
frequencies
Use a Poisson distribution to
calculate expected
frequencies
From previous slide show
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Type of data
Discrete Continuous Other
distributions
2 categories &
Bernoulli process> 2 categories
Use a Binomial model
to calculate expected
frequencies
Use a Poisson distribution to
calculate expected
frequencies
Now were dealing with:
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Normal Distribution
- bell curve
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Central Limit Theorem
Any continuous variable influenced by numerous random
factors will show a normal distribution.
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Normal curve is used for:
2) Continuous random data
Weight, blood pressure weight, length, area, rates
Data points that would be affected by a large number of
random (=unpredictable) events
Blood pressure
age
physical activity
smoking diet
genes
stress
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Normal curves can come in different shapes
So, for comparison between them, we need tostandardize their presentation in some way
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Standarize by calculating a Z-Score
Z = value of a random variable - meanstandard deviation
Z = X -
s
or
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Example of a z-score calculation
The mean grade on the Biometrics midterm is 78.4
and the standard deviation is 6.8. You got a 59.7 on
the exam. What is your z-score?
Z = X - sZ = 59.7 - 78.4 = -2.75
6.8
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If you look at the formula for z-scores:
z = value of a random variable - meanstandard deviation
z is also the number of standard deviations a
value is from the mean
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Each standard deviation away from the mean defines
a certain area of the normal curve