sampling when we want to study populations. we don’t need to count the whole population. we...
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SamplingSamplingWhen we want to study populations. When we want to study populations.
We don’t need to count the whole population.We don’t need to count the whole population.
We take a sample that will REPRESENT the We take a sample that will REPRESENT the whole population.whole population.
How do we know when our How do we know when our sample is representative?sample is representative?
We can use a RUNNING MEAN.We can use a RUNNING MEAN.
Work out the mean of your data as you Work out the mean of your data as you collect it.collect it.
When the mean doesn’t change then your When the mean doesn’t change then your sample is representative.sample is representative.
Sampling works best Sampling works best when…when…
You take lots of samplesYou take lots of samples
When the samples are When the samples are RANDOMRANDOM
When the sample sizes are largeWhen the sample sizes are large
Samples are unbiasedSamples are unbiased
Reliable data…Reliable data…
Has been repeated many timesHas been repeated many times
We can then see any anomaliesWe can then see any anomalies
We can see any variation in our resultsWe can see any variation in our results
Accurate data…Accurate data…
When the method has been followed very When the method has been followed very closelyclosely
No errors in the processNo errors in the process
All results should be similar (close to the All results should be similar (close to the mean)mean)
Precise dataPrecise data
Has been carried out using equipment that Has been carried out using equipment that has good precision i.e. many decimal has good precision i.e. many decimal places or measured to the smallest places or measured to the smallest increment possible.increment possible.
How to take random How to take random samples.samples.
LEARN THIS!LEARN THIS! Dividing the area into a Dividing the area into a gridgrid (e.g. place a (e.g. place a
grid underneath a Petri dish)grid underneath a Petri dish) Coordinates are chosen at random by using Coordinates are chosen at random by using
a a random number generatorrandom number generator e.g. using a e.g. using a calculator or random number (such as a calculator or random number (such as a phone book!)to select co-ordinates phone book!)to select co-ordinates
Sample this area using the relevant method.Sample this area using the relevant method.
Methods of samplingMethods of sampling
For immobile organisms - QUADRATSFor immobile organisms - QUADRATS
Look at
•% cover
•Density of species
•Frequency of species.
Types of QuadratTypes of Quadrat
Frame Quadrat Point Quadrat
•These allow RAPID collection of data
•We do not need to define individual plants when collecting data about % cover
Sampling mobile speciesSampling mobile species
Mark – release – recapture.Mark – release – recapture.
To estimate a population size use this To estimate a population size use this equation… equation…
Total of 1Total of 1stst capture x Total of 2 capture x Total of 2ndnd Capture Capture
Total Marked in 2Total Marked in 2ndnd Capture Capture
How to catch mobile How to catch mobile organisms…organisms…
Use a beating tray!Use a beating tray! TrappingTrapping
Take care with marking Take care with marking the organisms!the organisms!
Always “Mark” the organisms in an area Always “Mark” the organisms in an area that is not visible. This will reduce the that is not visible. This will reduce the chance of attracting predatorschance of attracting predators
Things to consider…Things to consider…
Always allow your 1Always allow your 1stst capture to re-integrate capture to re-integrate into the environment before you carry out your into the environment before you carry out your 22ndnd capture. This will give representative data. capture. This will give representative data.
This process does not consider migrationThis process does not consider migration
This process does not consider breeding This process does not consider breeding seasons.seasons.
To look at the To look at the distribution of species in distribution of species in
a habitata habitat We can use a TRANSECTWe can use a TRANSECT
Transects…Transects…
Allow us to take a line through an areaAllow us to take a line through an area
This can give us a guide to follow when This can give us a guide to follow when looking systematically at the distribution of looking systematically at the distribution of organisms.organisms.
Particularly good for looking at zones like Particularly good for looking at zones like sea shores.sea shores.
What do you do with your What do you do with your data?data?
Test a Null HypothesisTest a Null Hypothesis
Your Null Hypothesis will be that the Your Null Hypothesis will be that the Independent Variable will have NO EFFECT Independent Variable will have NO EFFECT
on the Dependent Variableon the Dependent Variable
We use statistical analysis to prove or We use statistical analysis to prove or disprove the Null Hypothesis.disprove the Null Hypothesis.
Which statistical tests do Which statistical tests do we carry out?we carry out?
Standard deviation – Year 12 workStandard deviation – Year 12 work
Standard error and 95% confidence limitsStandard error and 95% confidence limits Chi SquareChi Square Spearman RankSpearman Rank
Standard Error and 95% Standard Error and 95% Confidence LimitsConfidence Limits
S.E. gives us the parameters that the S.E. gives us the parameters that the totaltotal population can fall into, no matter what sample population can fall into, no matter what sample you take, 95% of the time.you take, 95% of the time.
Standard error gives us our 95% confidence limits.Standard error gives us our 95% confidence limits.
This means that our results happen due to chance This means that our results happen due to chance less than 5% of the time. less than 5% of the time.
How to work out the How to work out the standard error.standard error.
S.E. = Standard deviationS.E. = Standard deviation
n n
We plot standard error as We plot standard error as bars on a graph.bars on a graph.
Sample 1 Sample 2
X
X
• Draw your axis
• Plot your means
• Plot TWO standard errors either side and draw a line.
Confidence limits
What do these bars tell What do these bars tell us?us?
X
X
Look to see whether the
bars overlap.
These do not overlap so we can say that the 2 sets
of data are significantly
different at the 95% confidence
limits
This means that we can REJECT the Null
Hypothesis
X
X
These 2 sets of data do overlap so we say that they
are not significantly different at the 95%
confidence limits
This means that we can ACCEPT the Null Hypothesis
CrabsCrabs Crabs were found on 2 different beaches; one Crabs were found on 2 different beaches; one
sandy and one rocky.sandy and one rocky.
On the sandy beach the crabs were - 5, 7, 8, 8, 7, On the sandy beach the crabs were - 5, 7, 8, 8, 7, 10, 14, 3, 6, 7, 11, 20, 21, 3, 17 cm10, 14, 3, 6, 7, 11, 20, 21, 3, 17 cm
On the rocky beach the crabs were - 10, 12, 15, 18, On the rocky beach the crabs were - 10, 12, 15, 18, 19, 22, 14, 23, 23, 29, 11, 12, 22, 18, 17 cm19, 22, 14, 23, 23, 29, 11, 12, 22, 18, 17 cm
What is your Null Hypothesis?What is your Null Hypothesis?
We can use standard error to test the Null Hypothesis.We can use standard error to test the Null Hypothesis.
Chi-SquareChi-Square 22
Chi-square (Chi-square (2) is used to decide if 2) is used to decide if differences between sets of data are differences between sets of data are significant. significant.
It compares your It compares your Observed data Observed data with the with the Expected data Expected data and tells you the probability and tells you the probability (P) of your Observed results being due to (P) of your Observed results being due to chance.chance.
Null HypothesisNull Hypothesis
Before we start an investigation we write Before we start an investigation we write a null hypothesis.a null hypothesis.
This tells us that we think there will NOT This tells us that we think there will NOT be any relationship in our results.be any relationship in our results.
We accept or reject this hypothesis at the We accept or reject this hypothesis at the end of the analysis.end of the analysis.
How to do Chi-square…How to do Chi-square…
Look at this exampleLook at this example Suppose you flip a coin 100 times. You know that Suppose you flip a coin 100 times. You know that
if the coin is fair or unbiased that there should be if the coin is fair or unbiased that there should be 50% of heads and tails.50% of heads and tails.
How do you know though that the coin really is fair How do you know though that the coin really is fair and not biased in some way?and not biased in some way?
We’re going to test this. What is your null We’re going to test this. What is your null hypothesis?hypothesis?
My results…My results…
Work out the Chi Square!
Outcome Observed Number, O
Expected Number, E
(O-E)² E
Heads 60
Tails 40
Total 100
Try this one…Try this one…
Observed Expected
red 34
pink 84
white 42
Total 160 160160
Work out Chi-Square!
40
80
40
Method…Method…
Actual numbers
Expected numbers
(O-E)2 (O-E)2/E
red flowers
34 4036 0.9
pink flowers
84 8016 0.2
white flowers
42 404 0.1
Total 160 160 1.2
What does this all mean?What does this all mean?
The Chi-Square value will help us to find the The Chi-Square value will help us to find the probabilityprobability of our results being due to of our results being due to chance, or whether something is significantly chance, or whether something is significantly influencing them.influencing them.
In Biology we say that if results occur due to In Biology we say that if results occur due to chance more than 5% of the time then we chance more than 5% of the time then we cannot say that they are significant.cannot say that they are significant.
Our Chi-Square value can help us find out Our Chi-Square value can help us find out what % of our results are due to chance.what % of our results are due to chance.
We have to use a probability table to find this We have to use a probability table to find this out…out…
Before we look at the Before we look at the probablility table…probablility table…
The number of variables you have, minus The number of variables you have, minus 1 = N-1 gives you your 1 = N-1 gives you your DEGREES OF DEGREES OF FREEDOM (dF)FREEDOM (dF)
Follow the numbers across until you find the Follow the numbers across until you find the one that is closest to, but not higher than, your one that is closest to, but not higher than, your Chi-Square result.Chi-Square result.
Read upRead up
Look at the probability.Look at the probability.
If it is 0.05 (5%) or less then it means that 5% If it is 0.05 (5%) or less then it means that 5% (or less) of your results are due to chance – (or less) of your results are due to chance – these would be significant results.these would be significant results.
You would reject the Null hypothesisYou would reject the Null hypothesis
Our CRITICAL VALUES
1- I have 4 dF and a Chi-Square value of 16.45. What is my conclusion?
2- I have 3 dF and a Chi-Square value of 4.25. What is my conclusion?
3- I have 5 dF and a Chi-Square value of 3.27. What is my conclusion?
4- I have 6 dF and a Chi-Square value of 13.98. What is my conclusion?
Spearman Rank… Spearman Rank… rrss
This statistical test tells us whether there is a This statistical test tells us whether there is a significant association between two sets of data.significant association between two sets of data.
E.g you could carry out Spearman Rank to prove E.g you could carry out Spearman Rank to prove a significant association between temperature a significant association between temperature and Enzyme Activity.and Enzyme Activity.
You MUST have at least 7 measurementsYou MUST have at least 7 measurements
Is there a significant association Is there a significant association between wing length in seeds and between wing length in seeds and
the distance they fall from the the distance they fall from the parent tree?parent tree?
Seed Number Length of wing/mm Distance from tree/ m
1 34 21
2 28 19
3 40 17
4 33 15
5 42 30
6 35 22
7 23 17
8 27 20
9 20 15
First, rank each column First, rank each column from lowest to highestfrom lowest to highest
Seed Number
Length of wing/mm
Distance from tree/ m
Rank 1 – wing length
Rank 2 – Distance from tree
1 34 21 6 7
2 28 19 4 5
3 40 17 8 3.5
4 33 15 5 1.5
5 42 30 9 9
6 35 22 7 8
7 23 17 2 3.5
8 27 20 3 6
9 20 15 1 1.5
Notice that for 2 results that are the same we rank them between two levels e.g. 1.5 and 1.5 instead of 1 and 2.
Next, find the difference Next, find the difference between the ranks.between the ranks.
Seed Number
Length of wing/mm
Distance from tree/
m
Rank 1 – wing length
Rank 2 – Distance from tree
Difference between ranks
D
1 34 21 6 7 1
2 28 19 4 5 1
3 40 17 8 3.5 4.5
4 33 15 5 1.5 3.5
5 42 30 9 9 0
6 35 22 7 8 1
7 23 17 2 3.5 1.5
8 27 20 3 6 3
9 20 15 1 1.5 0.5
Next, Square the differenceNext, Square the differenceSeed
NumberLength
of wing /mm
Distance from
tree/ m
Rank 1 – wing
length
Rank 2 – Distance from tree
Difference between
ranksD
Difference Squared
D2
1 34 21 6 7 1 1
2 28 19 4 5 1 1
3 40 17 8 3.5 4.5 20.25
4 33 15 5 1.5 3.5 12.25
5 42 30 9 9 0 0
6 35 22 7 8 1 1
7 23 17 2 3.5 1.5 2.25
8 27 20 3 6 3 9
9 20 15 1 1.5 0.5 0.25
How to work it out…How to work it out…
r r s s = 1- = 1- 6 x 6 x ΣΣDD22
nn33 -n -n
n = the number of pairs of items in the sample.D2= the difference between ranks squared
So in our example…So in our example…
1 – 6 x 47 729-9
Difference Squared
D2
1
1
20.25
12.25
0
1
2.25
9
0.25
1 – 282 720
= 0.61
So, what do we do with the
0.61?
r r s s = 1- = 1- 6 x 6 x ΣΣDD22
nn33 -n -n
Use the Critical Value TableNumber of pairs of measurements
Critical Value
5 1.00
6 0.89
7 0.79
8 0.74
9 0.68
10 0.65
12 0.59
14 0.54
16 0.51
18 0.48
If your Spearman Rank Value is less than the Critical Value then you ACCEPT the Null Hypothesis
If your Spearman Rank Value is more than the Critical Value then you REJECT the Null Hypothesis