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Maths for Biology
July 6th 2015
Christian BokhoveCarys HughesHilary Otter
Rebecca D’SilvaNicky Miller
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Why this day
• Changes in the A level Biology curriculum• More maths• Maths used interdisciplinary• Principles
– Instruction but also hands-on tasks– Collaborative, doing it together, ask questions– Want to customise the course to where you
need support
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Introductions
• Christian Bokhove– Lecturer in Mathematics Education
• Carys Hughes• Hilary Otter• Rebecca D’Silva• Nicky Miller
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Objectives of the day
• Update and strengthen knowledge on some topics for the new Biology A level curriculum;– Exponential growth and decay, logarithms– Statistical tests
• Hear and discuss ideas for teaching them;• Want to hear your opinions for
improvements;• You leave with:
– Ideas, knowledge and some resources
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Schedule for the day
When What9:00 – 9:30 Welcome, introductions9:30 – 10:15 Exponential growth and decay10:15 – 12:30 Logarithms and log paper12:30 – 13:30 Lunch break13:30 – 14:00 Statistical tests14:00 – Mini workshops statistics
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MATHS CONFIDENCE SURVEYhttp://is.gd/m4bsurvey
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EXPONENTIAL GROWTH AND DECAY, LOGARITHMS
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Slide exam curriculum
From (i) A.0 - arithmetic and numerical computation
A.0.5 Use calculators to find and use power, exponential and logarithmic functions Candidates may be tested on their ability to: estimate the number of bacteria grown over a certain length of time
From (ii) A.2 – algebra A.2.5 Use logarithms in relation to quantities that range over several orders of magnitude Candidates may be tested on their ability to: use a logarithmic scale in the context of microbiology, e.g. growth rate of a microorganism such as yeast
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Standard form and powers
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Standard form and powers
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ACTIVITYLink to task
H
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VERY LARGE AND VERY SMALL• Negative and positive standard
form
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VERY LARGE AND VERY SMALL
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EXPONENTIAL GROWTH AND DECAY
(Before we can look at logarithms we need to deal with exponential growth and decay)
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ACTIVITY
Exponentials
Take an A4 sheet of paper.
How many times can you fold it in half?
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MYTHBUSTERS
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How many layers do you produce? HANDOUT
Number of folds
(x)
Number of layers
(y)
Mult Power Height
(cm)0 1 0.011 2 22 2*23 2*2*24 2*2*2*2 24
5
Table of results
H
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Plot your values on graph paper:
- 3 - 2 - 1 1 2 3 4 5 6 7 8
- 100
100
200
300
x
y
- 3 - 2 - 1 1 2 3 4 5 6 7 8
- 100
100
200
300
x
y
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Exponential growth
Imagine you contracted a virus (such as SARS) where you infected the first five people that you met, and they each infected the first five people that they met and so on…. There are 186,701 people living in Southampton. How many interactions would it take until everyone was infected?
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How many infections?
Number of interactions (x)
Number of infected people (y)
0 11 52 25345
Table of results
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General form:
y=bt
where b is the base and x is the power (or exponent)
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The exponential graph
- 2 2 4 6 8
- 200
200
400
x
y
ACTIVITY: Use Desmos or Geogebra online to graph
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How many layers do you produce?
Number of folds
(x)
Number of layers
(y)
Height
(cm)0 1 0.011 2 22 2*23 2*2*24 2*2*2*2 24
5
Table of results
y=2x
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Common features of y=bx
all curves pass through (0,1)exponential growth (and decay) takes place very rapidlyb > 0b 0b 1b > 1 has a positive gradient (PLOT THIS!)0 < b < 1 has a negative gradient (PLOT THIS!)
https://www.desmos.com/calculator/auubsaje fh
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HANDOUT
Exponential growth and decay worksheet
Exponential growth and decay practice sheet(and answers)
We are not doing all of these during the session.
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Logarithms
Logarithm is another name for a power
532log
232
2
5
So let’s say you know there are 32 layers in the folding task. How many times has someone
made a fold?
You could say taking ‘logarithms’ is the opposite of exponential growth or decay.
Exponential form
Logarithmic form
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Log examples: positive numbers
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ACTIVITY
In pairs decide whether or not each statement (selection)
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Further practice
There is further practice of conversion between logs and powers on:
www.bbc.co.uk/education/asguru
and theLogarithms practice sheet
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Why do we even need log paper?The exponential graph
- 2 2 4 6 8
- 200
200
400
x
y
ACTIVITY: Now with log paper. Demonstrate with Geogebra
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Log paper
• Log paper and powers of 2 5 10• Step by step example• Use of calculator• Most tools are rather poor at log paper!
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TASK LOG PAPER – SARS - PLOT
Number of interactions (x)
Number of people with the disease (y)
0 11 52 253 1254 6255 3,1256 15,6257 78,1258 390,625
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PEDAGOGY
How would you teach these topic?
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From sample exam question
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Discuss exam question
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Lunch break
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Statistical tests
These slides partly rely on the excellent resources from SteveJ64 from the TES
website
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Slide exam curriculumFrom (i) A.1 - handling data A.1.9 Select and use a statistical test Candidates may be tested on their ability to select and use: • the Chi squared test to test the significance of the
difference between observed and expected results • the Students t-test • the correlation coefficient A.1.11 Identify uncertainties in measurements and use simple techniques to determine uncertainty when data are combined Candidates may be tested on their ability to: • calculate percentage error where there are
uncertainties in measurement
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Normal distribution
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Suppose we have a crate of apples which are to be sorted by weight into small, medium and large. If we wanted 25% to be in the large category, we would need to know the lowest weight a “large” apple could be.
To solve a problem like this we can use a statistical model.
A model often used for continuous quantities such as weight, volume, length and time is the Normal Distribution.
The Normal Distribution is an example of a probability model.
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The Normal Distribution curve
2To fit the curve we use the mean, m, and variance, , of the data. These are the parameters of the model.
),(~ 2NXIf X is the random variable “ the weight of apples”, we write
Characteristics of the Normal Distribution
The Normal distribution model is a symmetric bell-shaped curve. We fit it as closely as possible to the data.
Reminder: s is the standard deviation.
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e.g.
)1,5(~ NX)1,4(~ NX
The axis of symmetry of the Normal distribution passes through the mean.
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e.g.
)50,4(~ 2NX
A smaller variance “squashes” the distribution closer to the mean.
)1,4(~ NX
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The percentages of the Normal Distribution lying within the given number of standard deviations either side of the mean are approximately:
SUMMARY
1 s.d. : 68%
2 s.d. : 95% 3 s.d. : 99·8%
68%
22
95%
33
99·8%
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Statistical tests
• Type of data collected– Measurements– Frequencies
• What are you looking for?– Associations– Differences
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MINI WORKSHOPS
• Break up in smaller groups to study a particular stats topic
• Slides and materials are available• Sample exam questions can also help
guide the work• At end we come back together and share
knowledge and experiences
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Chi-squared (χ2) test
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KARL PEARSON(1857-1936)
British mathematician, ‘father’ of modern statistics and a pioneer of eugenics!
(Pearson’s)
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Chi-squared (χ2) test
• This test compares measurements relating to the frequency of individuals in defined categories e.g. the numbers of white and purple flowers in a population of pea plants.
• Chi-squared is used to test if the observed frequency fits the frequency you expected or predicted.
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How do we calculate the expected frequency?
• You might expect the observed frequency of your data to match a specific ratio. e.g. a 3:1 ratio of phenotypes in a genetic cross.
• Or you may predict a homogenous distribution of individuals in an environment. e.g. numbers of daisies counted in quadrats on a field.
Note: In some cases you might expect the observed frequencies to match the expected, in others you might hope for a difference between them.
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Example 1: GENETICS
Comparing the observed frequency of different types of maize grains with the expected ratio calculated using a Punnett square.
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The photo shows four different phenotypes for maize grain, as follows:
Purple & Smooth (A), Purple & Shrunken (B), Yellow & Smooth (C) and Yellow & Shrunken (D)
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Gametes PS Ps pS ps
PS PPSS PPSs PpSS PpSs
Ps PPSs PPss PpSs Ppss
pS PpSS PpSs ppSS ppSs
ps PpSs Ppss ppSs ppss
The Punnett square below shows the expected ratio of phenotypes from crosses of four genotypes of maize.
A : B : C : D = 9 : 3 : 3 : 1
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H0 = there is no statistically significant difference between the observed frequency of maize grains and the expected frequency (the 9:3:3:1 ratio)
HA = there is a significant difference between the observed frequency of maize grains and the expected frequency
If the value for χ2 exceeds the critical value (P = 0.05), then you can reject the null hypothesis.
What is the null hypothesis (H0)?
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• How critical value• P-value• For the tests do not do all step-by-step recipes• Use the datasets from the separate
presentations to ask ‘what test would be appropriate here’.
• Central: chi squared, mini workshops– T-test– Spearman rank– SE and confidence intervals
• Mini workshop interpretation and reporting.
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Calculating χ2
χ2 = (O – E)2
E
O = the observed resultsE = the expected (or predicted) results
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Phenotype O E(9:3:3:1)
O-E (O-E)2 (O-E)2
E
A 271 244 27 729 2.99
B 73 81 -8 64 0.88
C 63 81 -18 324 4.00
D 26 27 -1 1 0.04
433 433 χ2= 7.91
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Compare your calculated value of χ2 with the critical value in your stats table
Our value of χ2 = 7.91Degrees of freedom = no. of categories - 1 = 3
D.F. Critical Value (P = 0.05)
1 3.842 5.993 7.824 9.495 11.07
Our value for χ2 exceeds the critical value, so we can reject the null hypothesis.
There is a significant difference between our expected and observed ratios. i.e. they are a poor fit.
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Example 2: ECOLOGY
• One section of a river was trawled and four species of fish counted and frequencies recorded.
• The expected frequency is equal numbers of the four fish species to be present in the sample.
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H0 = there is no statistically significant difference between the observed frequency of fish species and the expected frequency.
HA = there is a significant difference between the observed frequency of fish and the expected frequency
If the value for χ2 exceeds the critical value (P = 0.05), then you can reject the null hypothesis.
What is the null hypothesis (H0)?
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Calculating χ2
χ2 = (O – E)2
E
O = the observed resultsE = the expected (or predicted) results
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Species O E O-E (O-E)2 (O-E)2
E
Rudd 15 10 5 25 2.5
Roach 15 10 5 25 2.5
Dace 4 10 -6 36 3.6
Bream 6 10 -4 16 1.6
40 40 χ2= 10.2
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Compare your calculated value of χ2 with the critical value in your table of critical values.
Our value of χ2 = 10.2Degrees of freedom = no. of categories - 1 = 3
D.F. Critical Value (P = 0.05)
1 3.842 5.993 7.824 9.495 11.07
Our value for χ2 exceeds the critical value, so we can reject the null hypothesis.
There is a significant difference between our expected and observed frequencies of fish species.
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Example 3: CONTINGENCY TABLES
You can use contingency tables to calculate expected frequencies when the relationship between two quantities is being investigated.
In this example we will look at the incidence of colour blindness in both males and females.
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H0 = there is no statistically significant difference between the observed frequency of colour blindness in males and females.
HA = there is a significant difference between the between the observed frequency of colour blindness in males and females
If the value for χ2 exceeds the critical value (P = 0.05), then you can reject the null hypothesis.
What is the null hypothesis (H0)?
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Observed frequencies Males Females
Colour blind 56 14
Not colour blind 754 536
e.g.The expected frequency for colour blind males =
(56 + 14) x (56 + 754)
1360= 42
Expected Cell Frequency = (Row Total x Column Total)
n
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Observed: Males Females
•Colour blind 56 14•Not colour blind 754 536
Expected: Males Females
•Colour blind 42 28
•Not colour blind 768 522
Males Females
•Colour blind 4.7 14•Not colour blind 754 536
χ2 =… (O – E)2
E = 4.7 + 14 + 754 + 536 = 12.33
(O – E)2 / E
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Compare your calculated value of χ2 with the critical value in your table of critical values
Our value of χ2 = 12.33Deg of Freedom = (2 rows - 1) x (2 cols – 1) = 1
D.F. Critical Value (P = 0.05)
1 3.842 5.993 7.824 9.495 11.07
Our value for χ2 exceeds the critical value, so we can reject the null hypothesis.
There is a significant difference between our expected and observed frequencies.
The fraction of males with colour blindness is greater than that in females. The difference cannot be attributed to chance alone.
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Spearman rank
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Charles Spearman(1863-1945)British Psychologist and pioneer of IQ theory
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What can this test tell you?
Whether there is a statisticallysignificant correlation between two measurements from the same sample when you have 5-30 pairs of data.
If the correlation is negative or positive
Spearman’s Rank Correlation Coefficient
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What is the correlation coefficient r ?
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Does being good at maths make you better at biology?
Student Maths exam score
Biology exam score
Anand 57 83
Bernard 45 37
Charlotte 72 41
Demi 78 85
Eustace 53 56
Ferdinand 63 85
Gemma 86 77
Hector 98 87
Ivor 59 70
Jasmine 71 59
Is there a statistically significant correlation between these two sets of results?
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Spearman’s Rank Correlation Coefficient: rs
Where:
N = the number of individuals in the sample
D = difference in the rank of the two measurements made on an individual
rs will be a number between – 1 and +1
This number can be compared with those in a table of critical values
rs = 1 – [ 6 x∑D2
N3 – N) ]
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H0 = there is no statistically significant correlation between Maths scores and Biology scores
HA = there is a statistically significant correlation between Maths scores and Biology scores
A negative value for rs implies a negative correlation
A positive value for rs implies a positive correlation
If the value for rs exceeds the critical value, then you can reject the null hypothesis
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Step 1: Rank each set of data
Student Maths exam score
Maths rank
Biology exam score
Biology rank
Alex 57 83
Bernard 45 37
Charlotte 72 41
Demi 78 85
Eustace 53 56
Ferdinand 63 85
Gemma 86 77
Hector 98 87
Ivor 59 70
Jasmine 71 59
3
1
7
8
2
5
9
10
4
6
(lowest to highest)
Where two or more scores are tied each is assigned an average rank
7
1
2
8
3
9
6
10
5
4
8.5
8.5
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Student Maths exam score
Maths rank
Biology exam score
Biology rank D D2
Alex 57 3 83 7
Bernard 45 1 37 1
Charlotte 72 7 41 2
Demi 78 8 85 8.5
Eustace 53 2 56 3
Ferdinand 63 5 85 8.5
Gemma 86 9 77 6
Hector 98 10 87 10
Ivor 59 4 70 5
Jasmine 71 6 59 4
∑D2 =
Step 2: Work out the differences in ranks (maths – biology)
- 4
0
5
- 0.5
- 1
- 3.5
3
0
- 1
2
Step 3: Work out the square of the differences
Step 4: Work out the sum of the square of the differences
16
0
25
0.25
1
12.25
9
0
1
4
68.5
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Step 5: Work out the correlation coefficient, rs
N = 10
∑D2 = 68.5
rs = 1 - 6 x 68.5)103 – 10)
= 1 - 6 x 68.5)1000 - 10
= 1 - 411990
= 1 – 0.415 = 0.585
rs = 1 – [ 6 x∑D2
N3 – N) ]Where:
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Step 6: Compare your calculated value of rs with the relevant critical value in your stats table
For N = 10 and P = 0.05, the critical value of rs is 0.65
Our value of rs is 0.585
Because this is below the critical value, we must accept H0
There is no statistically significant correlation between Maths scores and Biology scores
rs critical values (P=0.05)
No. of pairs Critical value
8 0.74
9 0.68
10 0.65
12 0.59
14 0.54
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Mad Geoff’s Chaotic Firework Factory
Do peoples stress levels increase the closer they live to Mad Geoff’s Chaotic Firework
Factory?
Cortisol is a stress hormone
The more stressed an individual is, the higher their blood cortisol levels will be
Acacia Ave.
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Resident Address Blood cortisol level (μg/ml)
Karl (Caretaker) 2 (Factory) 13.4
Lillie 8 22.6
Melanie 10 23.4
Nigel 12 18.6
Olga 12 17.4
Peter 14 16.8
Quentin 16 15.2
Rajesh 16 10.2
Susan 18 9.8
Toni 18 12.6
Uri 18 8.8
Vanessa 20 7.5
H0 = there is no significantly significant correlation between proximity to the fireworks factory and blood cortisol levels
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Resident Address Address rank
Blood cortisol
level (μg/ml)
Cortisolrank D D2
Karl 2 13.4
Lillie 8 22.6
Melanie 10 23.4
Nigel 12 18.6
Olga 12 17.4
Peter 14 16.8
Quentin 16 15.2
Rajesh 16 10.2
Susan 18 9.8
Toni 18 12.6
Uri 18 8.8
Vanessa 20 7.5
∑D2
1
2
3
4.5
4.5
6
7.5
7.5
10
10
10
12
6
11
12
10
9
8
7
4
3
5
2
1
-5
-9
-9
-5.5
-4.5
-2
0.5
3.5
7
5
8
11
25
81
81
30.25
20.25
4
0.25
12.25
49
25
64
121
513
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rs = 1 - 6∑D2
N3 – N)
N = 12
∑D2 = 513
rs = 1 - 6(513)123 - 12
= 1 - 6(513)1728 - 12
= 1 - 30781716
= 1 – 1.794 = - 0.794
Where:
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When comparing rs to the critical value ignore the sign on rs
For N = 12 and P = 0.05, the critical value of rs is 0.587
Our value of rs is 0.794
Because this is above the critical value, we can reject H0
There is a statistically significant correlation between proximity to the fireworks factory and blood cortisol levels
rs is negative so there is a negative correlation...
Therefore: the further one lives from the firework factory, the lower one’s blood cortisol levels.
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Pearson
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KARL PEARSON(1857-1936)British mathematician, ‘father’ of statistics and a pioneer of eugenics!
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What can this test tell you?
If there is a statisticallysignificant correlation between two measured variables, X and Y, and….
If that correlation is negative or positive
Pearson’s Correlation Coefficient
Note: The data must show normal distribution
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What is the correlation coefficient r ?
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Is there a significant correlation between an animal’s nose-to-tail length and its body mass?
Animal Mass (arbitrary units)
Length (arbitrary units)
1 1 2
2 4 5
3 3 8
4 4 12
5 8 14
6 9 19
7 8 22
If yes, then is the correlation positive (does a long tail mean a larger mass)?
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Pearson’s Correlation Coefficient
Where:n = the number of values of X and Y
r will always be a number between –1 and +1
This number can be compared with those in a table of critical values using: n – 2 degrees of freedom.
∑XY – [(∑X)(∑Y)]/n
{∑X2-[(∑X)2/n]} {∑Y2-[(∑Y)2/n]}r =
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H0 = there is no statistically significant correlation between length and body mass
HA = there is a statistically significant correlation between length and body mass
A negative value for r implies a negative correlation
A positive value for r implies a positive correlation
If the value for r exceeds the critical value, then you can reject the null hypothesis.
What is the null hypothesis (H0)?
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Construct the following results table
Animal(n = 7)
Mass (X)
Length (Y)
X2 Y2 XY
1 1 2
2 4 5
3 3 8
4 4 12
5 8 14
6 9 19
7 8 22
Total
Mean
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Construct the following table:
Animal Mass
(X)Length
(Y)X2 Y2 XY
1 1 2 1
2 4 5 16
3 3 8 9
4 4 12 16
5 8 14 25
6 9 19 81
7 8 22 64
Total X = 37
Mean 5.29
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Calculate values for X:
Animal Mass
(X)Length
(Y)X2 Y2 XY
1 1 2 1
2 4 5 16
3 3 8 9
4 4 12 16
5 8 14 25
6 9 19 81
7 8 22 64
Total X = 37 X2 = 251
Mean 5.29
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Calculate values for Y:
Animal Mass
(X)Length
(Y)X2 Y2 XY
1 1 2 1 4
2 4 5 16 25
3 3 8 9 64
4 4 12 16 244
5 8 14 25 196
6 9 19 81 361
7 8 22 64 484
Total X = 37 Y = 82 X2 = 251 Y2 = 1278
Mean 5.29 11.71
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Calculate values for XY:
Animal Mass
(X)Length
(Y)X2 Y2 XY
1 1 2 1 4 2
2 4 5 16 25 20
3 3 8 9 64 24
4 4 12 16 244 48
5 8 14 25 196 112
6 9 19 81 361 152
7 8 22 64 484 176
Total X = 37 Y = 82 X2 = 251 Y2 = 1278 XY = 553
Mean 5.29 11.71
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Use values obtained to populate the equation:
X = 37 Y = 82 X2 = 251 Y2 = 1278 XY = 553
∑XY – [(∑X)(∑Y)]/n
{∑X2-[(∑X)2/n]} {∑Y2-[(∑Y)2/n]}r =
553 – (37 x 82)/7
{251-[372/7]} {1278-[822/7]}r =
n = 7
=
r = 0.901
119.57
132.60
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Compare your calculated value of r with the relevant critical value in your stats table
Our value of r = + 0.901Degrees of freedom = n - 2 = 5
D.F. Critical Value (P = 0.05)
3 0.88
4 0.81
5 0.75
6 0.71
7 0.67
Our value for r exceeds the critical value, so we can reject the null hypothesis.
The + sign shows that any correlation is positive.
We can conclude that there is a significant positive correlation between the length of an animal and its body mass
i.e. a long tail is associated with a large body mass!
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Two sample t-test
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William Gosset (aka ‘Student’)(1876-1937)Worked in quality control at the Guinness brewery and could not publish under his own name.Former student of Karl Pearson
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What can this test tell you?
If there is a statistically significant difference between two means, when:
The sample size is less than 25.The data is normally distributed
The t-test
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t-test
x1 = mean of first sample x2 = mean of second samples1 = standard deviation of first samples2 = standard deviation of second samplen1 = number of measurements in first samplen2 = number of measurements in second sample
x1 – x2
(s12/n1) + (s2
2/n2)
t =
SD = (x – x)2
n – 1
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Worked exampleDoes the pH of soil affects seed
germination of a specific plant species? • Group 1: eight pots with soil at pH 5.5• Group 2: eight pots with soil at pH 7.0• 50 seeds planted in each pot and the
number that germinated in each pot was recorded.
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H0 = there is no statistically significant difference between the germination success of seeds in two soils of different pH
HA = there is a significant difference between the germination of seeds in two soils of different pH
If the value for t exceeds the critical value (P = 0.05), then you can reject the null hypothesis.
What is the null hypothesis (H0)?
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Pot Group 1 (pH5.5)
(x – x)2 Group 2 (pH7.0)
(x – x)2
1 38 1.27 39 20.25
2 41 3.52 45 2.25
3 43 15.02 41 6.25
4 39 0.02 46 6.25
5 37 4.52 48 20.25
6 38 1.27 39 20.25
7 41 3.52 46 6.25
8 36 9.77 44 0.25
Mean 39.1 1.27 43.5 20.25
38.88 82.0
Construct the following table…
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Calculate standard deviation for both groups
SD = (x – x)2
n – 1
SD = (x – x)2
n – 1
Group 1:
Group 2:
= 38.888 – 1
=82.0
8 – 1
= 2.36
= 3.42
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Using your means and SDs, calculate value for t
x1 – x2
(s12/n1) + (s2
2/n2)
t =
39.1 – 43.5
(2.362/8) + (3.422/8)
t =
t = 2.99
-4.4
0.696 + 1.462=
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Compare our calculated value of r with the relevant critical value in the stats table of critical values
Our value of t = 2.99Degrees of freedom = n1 + n2 – 2 = 14
D.F. Critical Value (P = 0.05)
14 2.15
15 2.13
16 2.12
17 2.11
18 2.10
Our value for t exceeds the critical value, so we can reject the null hypothesis.
We can conclude that there is a significant difference between the two means, so pH does affect the germination rate for this plant.
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Standard error and confidence limits
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What can this test tell you?
If there is a statistically significant difference between two means, when:
The sample size is at least 30.The data are normally distributed.
NB: You can use this test to assess up to 5 means on the same graph.
Standard Error with 95% Confidence Limits
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Worked example
A student investigated the variation in the length of mussel shells on two different locations on a rocky shore.
The student measured the shell length of 30 mussels at each location.
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H0 = there is no statistically significant difference between the means of the two samples of mussels
HA = there is a significant difference between the means of the two samples of mussels
If the 95% confidence limit around the means do not overlap, then you can reject the null hypothesis.
What is the null hypothesis (H0)?
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Step 1: Calculate SD for both groups
SD = (x – x)2
n – 1
SD = (x – x)2
n – 1
Group 1:
Group 2:
= 361630 – 1
=1416
30 – 1
= 11.2
= 7.0
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Step 2: Calculate the SE for both groups
SD
n
SE =
SD
n
SE =
Careful with rounding off roundig offsGroup 1:
Group 2:
7.0
30 =
11.2
30
== 2.04
= 1.28
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Step 3: Calculate the 95% confidence limitsMean ± 2 x SE
Group 1:Upper limit = 61 + (2 x 2.04) = Lower limit = 61 – (2 x 2.04) =
Group 2:Upper limit = 33 + (2 x 1.28) = Lower limit = 33 – (2 x 1.28) =
Note: in recent exam specs maybe 1.96 instead of 2 (more
precise)
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0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
0.5 1 1.5 2
With no overlap, we can conclude that there is a significant difference between the two means. Shell lengths differ significantly between the two locations.
Step 3: Plot means and confidence limits
Mean ± 2 x SE
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EXAM QUESTIONS
• Should guide you quite a lot:
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Conclusion
• Hopefully some skills, knowledge and confidence added
• Please fill in the evaluation form and http://is.gd/m4bsurvey2
• In a month’s time we would like to send you some follow-up questions.– Impact form– Maths confidence
• Thank you for your attention.