Summary Biometry

8/3/2019 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


Central tendency - Mean


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The real population


Your samples


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The real population


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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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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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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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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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


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


3. Variance (contd)

Degrees of freedom or df or n


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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

Summary Biometry

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