Me a sure s o f Va ria b ility L E CT URE 4 Ob je c tive s De - - PDF document
Me a sure s o f Va ria b ility L E CT URE 4 Ob je c tive s De fine te rms. Dia g ra m re la tive diffe re nc e s in c e ntra l te nde nc y a nd va ria b ility. Ca lc ula te the ra ng e , sum o f sq ua re s, va ria nc e , a nd
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L E CT URE 4
De fine te rms. Dia g ra m re la tive diffe re nc e s in c e ntra l te nde nc y
a nd va ria b ility.
Ca lc ula te the ra ng e , sum o f sq ua re s, va ria nc e , a nd
sta nda rd de via tio n.
Ca lc ula te the dive rsity o f a sa mple o f no mina l
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A me a sure o f va ria b ility is a n indic a tio n o f the spre a d o f
me a sure me nts a ro und the c e nte r o f the distrib utio n.
Me a sure s o f va ria b ility a re pa ra me te rs o f the po pula tio n. Sa mple me a sure s tha t e stima te the va ria b ility o f the po pula tio n a re
sta tistic s.
Ra ng e Va ria nc e Sta nda rd de via tio n
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T
he diffe re nc e b e twe e n the hig he st a nd lo we st me a sure me nts in a g ro up o f da ta .
Sta tistic a l ra ng e vs. ma the ma tic a l ra ng e
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n a da ta a rra y (sma lle st to la rg e st): Sa mple ra ng e = Xn-X1
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Give s so me indic a tio n o f the va ria b ility o f da ta , b ut it o nly de pe nds
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t is unlike ly tha t the sa mple will c o nta in the e xtre me va lue s o f the po pula tio n so the sa mple ra ng e will c o nsiste ntly unde re stima te the po pula tio n ra ng e (a b ia se d e stima te ).
A b ia se d e stima to r will c o nsiste ntly unde r- o r o ve r-e stima te the va lue
An unb ia se d e stima to r will no t a lwa ys (o r e ve n o fte n) g ive the
c o rre c t va lue o f a pa ra me te r, b ut it will o ve r-e stima te the pa ra me te r a s o fte n a s it unde re stima te s the pa ra me te r.
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Sinc e the me a n is a use ful me a sure o f c e ntra l te nde nc y, it is po ssib le
to e xpre ss va ria b ility in te rms o f de via tio n fro m the me a n.
T
he sum o f a ll de via tio ns fro m the me a n will a lwa ys e q ua l ze ro .
Po sitive de via tio ns a re c a nc e lle d b y ne g a tive de via tio ns.
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Sq ua ring the de via tio ns fro m the me a n is o ne wa y o f e limina ting the
sig ns fro m the de via tio ns.
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he sum o f the sq ua re s o f the de via tio ns fro m the me a n is c a lle d the sum o f sq ua re s (SS).
Po pula tio n sum o f sq ua re s Sa mple sum o f sq ua re s
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T
he me a n sum o f sq ua re s is c a lle d the va ria nc e .
Sq ua re d units
Po pula tio n va ria nc e
Sa mple va ria nc e
No te diffe re nc e in the de no mina to r
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T
he sta nda rd de via tio n is the po sitive sq ua re ro o t o f the va ria nc e .
Po pula tio n sta nda rd de via tio n
Sa mple sta nda rd de via tio n
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T
he c o e ffic ie nt o f va ria tio n e xpre sse s sa mple va ria b ility re la tive to the sa mple me a n.
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va ria b ility. I nste a d the c o nc e pt o f dive rsity (the distrib utio n o f
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he inde x is use d in a re la tive fa shio n (c o mpa riso n no t a b so lute me a sure )
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T
a ke s into a c c o unt b o th numb e rs o f spe c ie s a nd the e ve nne ss o f the distrib utio n a mo ng c a te g o rie s.
n = numb e r o f individua ls in the sa mple fi = numb e r o f o b se rva tio ns in c a te g o ry i
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K
= numb e r o f c a te g o rie s (spe c ie s)