Lecture 8Probabilities and distributions
nkkp )( ( ) 1 1 ( )n k kp k p k
n n
( ) ( ) 1p k p k
0 1p
Probability is the quotient of the number of desired events k through the total number of events n.
If it is impossible to count k and n we might apply the stochastic definition of probability. The probability of an event j is approximately the frequency of j during n observations.
What is the probability to win in Duży Lotek?
139838161
)!649(!6!491
p
The number of desired events is 1. The number of possible events comes from the
number of combinations of 6 numbers out of 49.
)!649(!6!4949
6 C
We need the number of combinations of k events out of a total of N events
)!(!!kNk
NkN
C nk
1001!0
Bernoulli distribution
knn
kn
n1
0
What is the probability to win in Duży Lotek?1 1 1 1 0.0000649 49 49 496 5 4 3
p
Wrong!
Hypergeometric distribution
A B C D E F G H I1 N 49 49 49 492 K 6 =+KOMBINACJE(B1;B2) 6 =+KOMBINACJE(D1;D2) 6 =+KOMBINACJE(F1;F2) 6 =+KOMBINACJE(H1;H2)3 n 6 =+KOMBINACJE(B2;B4) 6 =+KOMBINACJE(D2;D4) 6 =+KOMBINACJE(F2;F4) 6 =+KOMBINACJE(H2;H4)4 k 3 =+KOMBINACJE(B1-B2;B3-B4) 4 =+KOMBINACJE(D1-D2;D3-D4) 5 =+KOMBINACJE(F1-F2;F3-F4) 6 =+KOMBINACJE(H1-H2;H3-H4)5Combinations =+C2/(C3*C4) =+E2/(E3*E4) =+G2/(G3*G4) =+I2/(I3*I4)6 Probability =1/C5 =1/E5 =1/G5 =1/I57 Sum =+SUMA(C6:I6)
P = 0.0186
knKN
nK
nN
C Knkn,,
nNknKN
nK
p Knkn,,
64966649
66
64956649
56
64946649
46
64936649
36
,,Knknp
N
K=n+kn
We need the probability that of a sample of K elements out of a sample universe of N exactly n have a desired probability and k not.
Assessing the number of infected personsAssessing total population size
Capture – recapture methods
The frequency of marked animals should equal the frequency wothin the total population Assumption:
Closed populationRandom catchesRandom dispersalMarked animals do not differ in behaviour
resample
resampletotaltotal
resample
resample
mnm
NNm
nm
42176 N Nreal = 38
We take a sample of animals/plants and mark them
We take a second sample and count the number of
marked individuals
The two sample case
common
common
nmmN
Nm
mn 211
2
You take two samples and count the number of infected persons in the first sample m1, in the second sample m2 and the number of infected persons noted in both samples k.
12143 N
How many persons have a certain infectuous desease?
m species l species k species
In ecology we often have the problem to compare the species composition of two habitats. The species overlap is measured by the Soerensen distance metric.
lmkS
2
We do not know whether S is large or small.
To assess the expectation we construct a null model.Both habitats contain species of a common species pool. If the pool size n is known we can estimate how many joint species k contain two random samples of size m and l out of n.
n species Common species pool
Habitat A Habitat B
K
n n k n mk m k l k
pn nm l
nmlk
lk
nm
The expected number of joint species.Mathematical expectation
The probability to get exactly k joint species.Probability distribution.
0
0.1
0.2
0.3
0 3 6 9 12 15Species in common
Pro
babi
lity A
0
0.1
0.2
0.3
0 3 6 9 12 15Species in common
Pro
babi
lity B
0
0.1
0.2
0.3
0 3 6 9 12 15Species in common
Pro
babi
lity C
0
0.1
0.2
0.3
0 3 6 9 12 15Species in common
Pro
babi
lity D
Ground beetle species of two poplar plantations and two adjacent wheet fields near Torun (Ulrich et al. 2004, Annales Zool. Fenn.)
Pool size 90 to 110 species.
There are much more species in common than expected just by chance.The ecological interpretation is that ground beetles colonize fields and adjacent
seminatural habitats in a similar manner. Ground beetles do not colonize according to ecological requirements (niches) but
according to spatial neighborhood.
K
n n k n mk m k l k
pn nm l
First steps in statistics
Experimental orobservational study
TheoryEnvisioned
methodof analysis
Motivation
Data analysis
What is interesting? Why is it interesting? Cui bono?
Literature
Planning
Data
Analysis
Interpretation
Defining the problemIdentifying the state of art
Formulating specific hypothesis to be tested
Study design, power analysis, choosing the analytical methods,
design of the data base,
Observations, experimentsMeta analysis
Statistical analysis, modelling
Comparing with current theory
PublicationScientific writing,
expertise
How to perform a biological study
Theory
Preparing the experimental or data collecting phase
• Let’s look a bit closer to data collecting. Before you start any data collecting you have to have a clear vision of what you want to do with these data. Hence you have to answer some important questions
• For what purpose do I collect data?• Did I read the relevant literature?• Have similar data already been collected by others?• Is the experimental or observational design appropriate for the statistical data
analytical tests I want to apply?• Are the data representative?• How many data do I need for the statistical data analytical tests I want to apply? • Does the data structure fit into the hypothesis I want to test?• Can I compare my data and results with other work?• How large are the errors in measuring? Do theses errors prevent clear final results?• How large might the errors be for the data being still meaningful?
How to lie with statistics
PO33%
PIS19%
LiD10%
Samoobrona10%
Unknown28%
Single sample notrepresentative
PS
Single sample too small
P
S
Multiple samples notrepresentative
PS
SS
S
S
S S
S
Multiple represenativesamples
PSS
S
S
S
S
S
S
Representative sampling
1
10
100
1000
10000
0.2 0.8 3.2 12.8 51.2
Body length class [mm]
Num
ber o
f species
0
500
1000
1500
2000
2500
3000
3500
4000
4500
0.2 0.8 3.2 12.8 51.2
Body length class [mm]
Num
ber o
f species
1
10
100
1000
10000
0.2 0.8 3.2 12.8 51.2
Body length class [mm]Num
ber o
f species
0
500
1000
1500
2000
2500
3000
3500
4000
4500
0.2 0.8 3.2 12.8 51.2
Body length class [mm]
Num
ber o
f species
0
10
20
30
40
50
1 2 3 4 5 6 7 8 9 10 11 12
Classes
Eve
nts
0
5
10
15
20
25
1 3 5 7 9 11Classes
Eve
nts
0
2
4
6
8
10
12
14
1 4 7 10 13 16 19 22Classes
Eve
nts
0102030405060708090
100
1 2 3Classes
Eve
nts
0.001
0.01
0.1
1
10
100
0.001 0.01 0.1 1 10 100Body weight [mg]
Mea
n de
nsity z
0.001
0.01
0.1
1
10
0.001 0.01 0.1 1 10 100Body weight class [mg]
Mea
n de
nsity z
0.001
0.01
0.1
1
10
100
0.001 0.01 0.1 1 10 100Body weight [mg]
Mea
n de
nsity z
0.001
0.01
0.1
1
10
0.001 0.01 0.1 1 10 100Body weight class [mg]
Mea
n de
nsity z
0.00
1.00
2.00
3.00
4.00
5.00
0 20 40 60 80 100 120
Numbers of storks
Birt
hrat
eNumber of storks nests and birthrates in Switzerland
0.00%
20.00%
40.00%
60.00%
80.00%
100.00%
1.6 1.65 1.7 1.75 1.8
Mean body height
% c
atho
lics
Worse Better
0369
12
A B C D EVariable
Var
iabl
e
A B C D ES1
0369
12
Var
iabl
e
Variable
0
3
6
9
12
A B C D EVariable
Var
iabl
e
ABCDE
02468
10
A B C D E0
20
40
60
80
020406080
A B C D E
VariableV
aria
ble
02468
10
A B C D E
Variable
Var
iabl
e
02468
10
A B C D E
Influence of variable 1 on variable 2y = f(x)R2 = n.s.
= 5.5 02468
10
A B C D EVariable 1
Var
iabl
e 2
0.01
0.1
1
10
0.01 1 100
Variable 1
Var
iabl
e 2
02468
10
0 2 4 6 8 10Variable 1
Var
iabl
e 2
A B C D ES1 0
2468
10
A B C D EVariable 1
Var
iabl
e 2
Scientific publications of any type are classically divided into 6 major parts
•Title, affiliations and abstractIn this part you give a short and meaningful title that may contain already an essential result. The abstract is a short text containing the major hypothesis and results. The abstract should make clear why a study has been undertaken•The introductionThe introduction should shortly discuss the state of art and the theories the study is based on , describe the motivation for the present study, and explain the hypotheses to be tested. Do not review the literature extensively but discuss all of the relevant literature necessary to put the present paper in a broader context. Explain who might be interested in the study and why this study is worth reading!•Materials and methods A short description of the study area (if necessary), the experimental or observational techniques used for data collection, and the techniques of data analysis used. Indicate the limits of the techniques used.•ResultsThis section should contain a description of the results of your study. Here the majority of tables and figures should be placed. Do not double data in tables and figures. •DiscussionThis part should be the longest part of the paper. Discuss your results in the light of current theories and scientific belief. Compare the results with the results of other comparable studies. Again discuss why your study has been undertaken and what is new. Discuss also possible problems with your data and misconceptions. Give hints for further work.•AcknowledgmentsShort acknowledgments, mentioning of people who contributed material but did not figure as co-authors. Mentioning of fund giving institutions•Literature
Country Island/ Mainland Area [km2] DeltaT [°C] Lat Long
Days below zero
Min Max Mean Variance Skewness Kurtosis Species Sources
Albania m 28748 17 41.33 19.92 34 -4.28959 2.60059 -1.31798 1.87086 0.0616831 -0.210158 132 Thibaud, 1992; Thibaud & Peja, 1994, 1996; Kontschan et al., 2003; Traser & Kontschan, 2004; Deharveng, 2007Andorra m 468 14.7 42.5 1.5 60 -0.867014 1.58438 -0.0465939 1.22393 1.79255 3.39576 4 Deharveng, 2007Austria m 83871 20 48.12 14.57 92 -4.84426 2.60059 -1.26057 1.61122 -0.0019179 -0.0696599 486 Pomorski, 2006; Querner, 2008Azores i 2200 7 37.73 -28.01 1 -4.13091 1.93892 -1.15658 1.62273 -0.045433 0.101475 94 Gama, 2005a,bBaleary Islands i 5014 15 39.55 2.65 18 -3.98247 0.651808 -1.74797 1.19506 -0.0467805 -0.179688 42 Jordana et al., 2005; Deharveng, 2007; Palacios-Vargas & Simón Benito, 2009Belarus m 207650 23 53.87 28 144 -4.13091 1.82671 -1.02222 1.98246 0.0370745 -0.386657 48 Kuznetsova, 2002; Deharveng, 2007Belgium m 30528 15 50.9 4 50 -4.64414 1.93892 -1.14785 1.77014 -0.0967065 -0.216933 209 Janssens, 2008Bosnia and Herzegovina m 51197 20 43.82 18 114 -4.84426 2.60059 -1.03804 1.38211 0.327301 1.01551 145 Bogojević 1968; Deharveng, 2007; Lučić et al., 2007a; Curcic et al., 2007Bulgaria m 110971 21 42.65 25 102 -4.84426 1.93892 -1.0666 1.74795 -0.143106 -0.0350317 209 Rusek, 1965; Tsonev & Kazandjicva, 1991; Thibaud, 1995b; Pomorski & Skarzynski, 1999; Smolis et al., 2004; Pomorski 2006; Deharveng, 2007Canary Islands i 7270 5 27.93 -15.4 1 -6.95173 0.651808 -2.06754 1.85064 -0.0919176 0.779021 115 Gama, 2005b; Deharveng, 2007Corsica i 8680 13 41.92 8.73 11 -3.87025 1.82671 -1.1579 1.17258 0.315027 0.669108 60 Deharveng, 2007Crete i 8259 13 35.33 24.83 1 -3.76326 1.58438 -1.39088 1.74242 0.192635 -0.766043 108 Ellis, 1976; Schultz & Lymberakis, 2006Croatia m 56594 21 45.82 15.5 114 -3.98247 2.60059 -0.85965 1.71272 0.321948 -0.213202 141 Bogojević, 1968; Ozimec, 2002; Deharveng, 2007Czech Republic m 78866 19 50.1 15.5 119 -4.92946 2.60059 -1.43186 1.78157 0.0042882 -0.0565949 361 Rusek 1977, 1979, 1996, 2001, 2003, 2004; Rusek & Rusek, 1999; Rusek & Subrt, 1999; Jilova & Rusek, 2005; Deharveng, 2007 Denmark m 43093 16 55.63 12.57 85 -4.84426 1.93892 -1.47086 2.00751 -0.062116 -0.483545 222 Fjellberg, 2007aDodecanese Is. i 2663 14 36.4 23.73 2 -3.46924 1.58438 -1.16453 1.22197 0.235818 0.207184 36 Deharveng, 2007 Estonia m 45227 21 59.35 26 143 -3.87025 1.82671 -1.19182 1.52952 0.43837 0.424804 40 Kanal, 2004; Deharveng, 2007 Faroe Is. i 1399 7 62 -7 30 -4.64414 1.93892 -1.34386 1.79648 -0.164351 -0.0688218 85 Fjellberg, 2007aFinland m 338145 23 60.32 25 169 -4.64414 1.93892 -1.39185 1.7577 0.0182062 -0.334913 225 Fjellberg, 2007aFrance m 543965 15 48.73 2.3 50 -5.06348 1.93892 -1.43424 1.46779 0.014345 0.0427404 641 Liste des Collemboles français awailabe at: http://www.insecte.org/forum/viewtopic.php?p=405014#p405014Franz Josef Land i 16134 27 79.85 57.42 310 -2.8658 -0.280761 -1.00011 0.50044 -1.70537 3.45228 15 Babenko & Fjellberg, 2006Germany m 357021 19 52.38 13.42 97 -5.06348 2.60059 -1.422 1.69377 -0.0851031 -0.0590463 420 Pallisa, 2000, Deharveng, 2007Greece m 131992 17 37.9 23.73 2 -3.98247 1.58438 -1.06325 1.45253 -0.120441 -0.209457 103 Schultz & Lymberakis 2006; Pomorski, 2006; Deharveng, 2007; Ramel et al., 2008Hungary m 93054 22 47.43 20 100 -4.84426 2.60059 -1.29947 1.74949 0.0120114 -0.205994 408 Traser & Dányi, 2008; Dányi & Traser, 2008
ln Body weight Body weight distribution
The source data base
Each row gets a single data record.Columns contain variables.Variables can be of text or metric type.
Never use the original data base for calculations.Use only a replicate.Take care of empty cells.In calculated cells take care of impossible values.
No Raw data Classes Class means CounterNumber of occassions
FrequenciesCummulative
frquencies1 0.154497 0-0.1 0.05 20 20 0.1 0.12 0.919498 0.1-0.2 0.15 48 28 0.14 0.243 0.517978 0.2-0.3 0.25 83 35 0.175 0.4154 0.742013 0.3-0.4 0.35 107 24 0.12 0.5355 0.295932 0.4-0.5 0.45 127 20 0.1 0.6356 0.819647 0.5-0.6 0.55 149 22 0.11 0.7457 0.693982 0.6-0.7 0.65 172 23 0.115 0.868 0.194982 0.7-0.8 0.75 185 13 0.065 0.9259 0.276991 0.8-0.9 0.85 198 13 0.065 0.99
10 0.054868 0.9-1 0.95 200 2 0.01 111 0.386411
12 0.00286 +D10+0.1+LICZ.JEŻELI(B$2:B$2
01;"<1")=E11-E12 =F11/E$11 =G11+H12
13 0.129657
05
10152025303540
0 0.2 0.4 0.6 0.8 1
N
X
0
0.05
0.1
0.15
0.2
0 0.2 0.4 0.6 0.8 1
f(X)
X
Frequency distribution
0
0.05
0.1
0.15
0.2
0 0.2 0.4 0.6 0.8 1
f(X)
X
Frequency distribution
No Raw data Classes Class means CounterNumber of occassions
FrequenciesCummulative
frquencies1 0.154497 0-0.1 0.05 20 20 0.1 0.12 0.919498 0.1-0.2 0.15 48 28 0.14 0.243 0.517978 0.2-0.3 0.25 83 35 0.175 0.4154 0.742013 0.3-0.4 0.35 107 24 0.12 0.5355 0.295932 0.4-0.5 0.45 127 20 0.1 0.6356 0.819647 0.5-0.6 0.55 149 22 0.11 0.7457 0.693982 0.6-0.7 0.65 172 23 0.115 0.868 0.194982 0.7-0.8 0.75 185 13 0.065 0.9259 0.276991 0.8-0.9 0.85 198 13 0.065 0.99
10 0.054868 0.9-1 0.95 200 2 0.01 111 0.386411
12 0.00286 +D10+0.1+LICZ.JEŻELI(B$2:B$2
01;"<1")=E11-E12 =F11/E$11 =G11+H12
13 0.129657
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
F(X)
X
Cumulative frequency distribution
Probability density function (pdf)
max
min
max( ) ( ) 1x
x
F x f x dx max
1
( ) ( ) 1x
i iF x f x
Statistical or probability distributions add up to one.
0
0.05
0.1
0.15
0.2
0 0.2 0.4 0.6 0.8 1
f(X)
X
0
0.05
0.1
0.15
0.2
0 0.2 0.4 0.6 0.8 1
f(X)
X
Discrete distribution Continuous distribution
Discrete and continuous distributions
Probability generating function (pgf)
00.020.040.060.08
0.10.120.140.160.18
0 5 10 15x
f(xi)
symmetric
0
0.05
0.1
0.15
0.2
0.25
0 5 10 15x
f(xi)
left skewed
00.020.040.060.08
0.10.120.140.160.18
0.2
0 5 10 15x
f(xi)
right skewed
00.020.040.060.08
0.10.120.140.16
0 5 10 15x
f(xi)
bimodal
0
0.05
0.1
0.15
0.2
0.25
0 5 10 15x
f(xi)
decreasing
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 5 10 15x
f(xi)
U-shaped
Shapes of frequency distributions
Many statistical methods rely on a comparison of observed frequency distributions with theoretical distributions.
Deviations from theory (from expectation) (so called residuals) are measures of statistical significance.
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8 1
f(X)
X
Df(x)
Df(x)
If the Df(x) are too large we accept the hypothesis that our observations differ from the theoretical expectation.
The problem in statistical inference is to find the appropriate theoretical distribution that can be applied to our data.
Home work and literature
Refresh:
• Arithmetic, geometric, harmonic mean• Variance, standard deviation standard error• Central moments• Third and fourth central moment• Mean and variance of power and
exponental function statistical distributions• Pseudocorrelation• Sample bias• Coefficient of variation• Representative sample
Prepare to the next lecture:
• Bernoulli distribution• Pascal distribution• Hypergeometric distribution• Linear random number
Literature:
Mathe-onlineŁomnicki: Statystyka dla biologów.
