Statistical tests:

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Statistical tests:

Comparing two distributions: iKolmogorov Smirnov (KS) test

 why? Establish if 2 distributions are significantly different or if they are drawn from the same

originai distribution

 No assumption on the distribution kind (no models) technically named “non parametric test”

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N

N N_tot

D_max

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The KS test can also be used to compare an observed distribution with an expected one.

Let's assume that we got the following 4 values [1.5, 3.3, 4.7, 6.8] which are a fair representation of the observed data set. This means that if we acquire more date, 25% are expected to have a value between 0 and 1.5, 50% between 0 and 3.3 etc,

Let's compare the oserved distribution with an exponential distribution of mean = 5

An exponential distribution of mean has this form

 y=e

−1



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On the right the corresponding cumulative functions.

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What is the fraction of expected values below 1.5, 3.3 etc in a exponential distribution of mean

=5 ?

1−e^−0.2⋅1.5=0.26 1−e^−0.2⋅3.3=0.48

1−e^−0.2⋅4.7=0.61

1−e^−0.2⋅6.8=0.75

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Dati 1.5 3.3 4.7 6.8 Empirical

Fraction lowest estimate

0 0.25 0.50 0.75

Empirical fraction largest estimate

0.25 0.50 0.75 1.0

Theoretical

fraction 0.26 0.48 0.61 0.75

Largest

deviation 0.26 0.23 0.14 0.25

one must check (on Tables) if this distance is

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Size = 4. The two samples are not significantly different

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Test too allows one to compare two

distributions to verify if they are compatibie, provided that one can divide each distribution into classes.

²

Let's suppose that a typical environment (i.e. galaxy groups) is characterized by the following

morphological content:

10 % Ellipticals, 20 % S0s, 30 % Early Spirals (i.e. Sa >

Sb) and 40 % Late Sprials (i.e. > Sbc)

Let suppose that I have 3 distinct galaxy samples For which I want to check if their morphological

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Sample 1 : 18 E, 35 S0, 83 S early 104 S late

Total size 240 ( E 8%, S0 15%, S early 34%, S late 43%)

Sample 2 : 27 E, 50 S0, 80 S early 120 Slate

Total size 277 ( E 10%, S0 18%, S early 29%, S late 43%)

Sample 3 : 12 E, 20 S0, 100 S early 130 Slate

Totale 262 ( E 4%, S0 8%, S early 38%, S late 50%)

²=∑

i=1

k O_i−E_i² E_i

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Sample 1 :

Observed 18 35 83 104 Expected 24 48 72 96

²=∑

i=1

k O_i−E_i² E_i

²=18−24²

24 35−48²

48 83−72²

72 104−96² 96

²=36

24  13²

48 11²

72  64 96

2

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Sample 2 :

Observed 27 50 80 120 Expected 28 55 83 111

²=27−28²

28 50−55²

55 80−82²

83 120−111² 111

²= 1

28  25

55 4

83  81 111

²=1.27

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Sample 3 :

Observed 12, 20, 100, 130 Expected 26 52 79 105

²=12−20²

20 20−52²

52 100−79²

79 130−105² 105

²=64

20  32²

52 21²

79  25² 105

²=34.42

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the fredom degrees are 3 k=4 (the

classes) m =0 (as no parameter has been derived from the data)

_0.05² =3=7.82

=k−m−1

Looking at the 3^rd row of the Table we see that Sample 1 can be considered different at 90% c.l.

Sample 2 at 5% c.l. And Sample 3 at more than 99.5 % c.l.

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IDL

if we want a graphic window which is not erased By superposition we must digit

Idl> window, 0, retain=2

we can change the size of the window adding to the previous command also

xsize=1000,ysize=1000

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Exercise 3

Make a plot of the function sin x and cos x (with x ranging between and ) and of their sum.

All graphs must be in the same plot and have different colors.

− 

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Exercise 4

Make on the same page 3 distinct plots representing the following functions

y=e^x y=log x y=e⁻^x cos x

Save the result in a postscript file

How to make a multiple plot?

Idl> !p.multi=[0,2,3] ---> 2 colums, 3 rows idl>!p.multi=[0,1,2] ---> 1 column 2 rows

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How to save the result in a postscript file

Idl> set_plot, 'PS'

Idl> device,filename='namechosen.ps' Idl> plot,x,y

idl>device, /close Idl> set_plot, 'X'