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 sta nda rd de via tio n.  Ca lc ula te the dive rsity o f a sa mple o f no mina l o b se rva tio ns. 1

Va ria b ility  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. Va ria b ility  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 2

Me a sure s o f c e ntra lity a nd va ria b ility Me a sure s o f c e ntra lity a nd va ria b ility 3

Me a sure s o f c e ntra lity a nd va ria b ility Ra ng e  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  I n a da ta a rra y (sma lle st to la rg e st): Sa mple ra ng e = X n -X 1 4

Ra ng e  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 o n the e xtre me va lue s o f the da ta a rra y (la rg e st a nd sma lle st).  I 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 ). Bia se d a nd unb ia se d  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 o f a pa ra me te r.  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. 5

Sum o f sq ua re s  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. Sum o f sq ua re s  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. 6

Sum o f sq ua re s  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.  T 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). Sum o f sq ua re s  Po pula tio n sum o f sq ua re s  Sa mple sum o f sq ua re s 7

Va ria nc e  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 Va ria nc e  Sa mple va ria nc e No te diffe re nc e in the de no mina to r 8

Sta nda rd de via tio n  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 Sta nda rd de via tio n  Sa mple sta nda rd de via tio n 9

Co e ffic ie nt o f va ria tio n  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. Dive rsity  F o r no mina l sc a le da ta the re is no me a n to se rve a s a re fe re nc e fo r va ria b ility. I nste a d the c o nc e pt o f dive rsity (the distrib utio n o f o b se rva tio ns a mo ng c a te g o rie s) is use d.  T 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 ) 10

Dive rsity  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. Dive rsity  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 11

E ve nne ss  K = numb e r o f c a te g o rie s (spe c ie s) 12