Me a sure s o f Ce ntra l T e nde nc y L E CT URE 3 Ob je c - - PDF document

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Me a sure s o f Ce ntra l T e nde nc y

L E CT URE 3

Ob je c tive s

 De fine the te rms.  Ca lc ulate me a n, mo de , a nd me dia n.  Unde rsta nd simple sta tistic a l no ta tio n.

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Ce ntra l te nde nc y

 A la rg e pro po rtio n o f the va lue s fo r mo st va ria b le s a re fo und ne a r

the middle o f the ra ng e o f o b se rve d va lue s.

 De fine sta tistic a l ra ng e  De fine ma the ma tic a l ra ng e

 T

he c o nc e ntra tio n is typic a lly re fe re e d to a s the a ve ra g e o r the c e ntra l te nde nc y.

Ce ntra l te nde nc y

 Ca n b e use ful fo r c ha ra c te rizing a po pulatio n

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F ig ure 1. Numb e r o f e g g s la id b y re d- e a re d slide r pe r c lutc h

Figure 1. Lengths of pine needles (cm) from Larimer county, CO.

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Arithme tic me a n

 T

he c o mmo nly use d me a sure o f c e ntra l te nde nc y is the a rithme tic me a n.

 F

re q ue ntly re fe rre d to a s the me a n.

 Mo st c o mmo nly re fe rre d to (inc o rre c tly) a s the a ve ra g e .

Sta tistic a l no ta tio n

 Xi -- the i

th va lue o f the va ria b le in a da ta a rra y.

1 3 7 13 21 31 X1=1 X3=7 X6=31

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Sta tistic a l no ta tio n

 N = T

he to tal numb e r o f va lue s in the po pulatio n.

  = Me a n o f the po pula tio n (lo we r c a se mu)

Sta tistic a l no ta tio n

  = Summatio n  Summa tio n o f a ll Xi va lue s fro m X1 thro ug h XN

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Sta tistic a l no ta tio n

1 3 7 13 21 31

Arithme tic me a n

 Po pula tio n me a n ()

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Arithme tic me a n

1 3 7 13 21 31

Arithme tic me a n

 Sa mple me a n

n = sa mple size

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Me dia n

 T

he me dia n (M) is the middle me a sure me nt in a n o rde re d se t o f da ta .

 I

n a symme tric a l distrib utio n, the sample me dia n is unb ia se d a nd is a n e ffe c tive e stima te o f the po pulatio n me a n.

 Ho we ve r, mo st distrib utio ns a re no t pe rfe c tly symme tric a l a nd the n

me dia n b e c o me s a po o r me a sure o f the po pula tio n me a n.

 T

he me dia n is ra re ly use d.

Me dia n

 Orde r the da ta in a sc e nding o rde r.

 I

f the sa mple size (n) is o dd the n the sub sc ript in the e q ua tio n will b e a n inte g e r a nd will indic a te whic h da tum is the me dia n

 I

f the sa mple size (n) is e ve n the n the sub sc ript will b e midwa y b e twe e n two inte g e rs. T he da ta o n e a c h side o f midpo int a re the n use d a nd the me dia n is the midpo int b e twe e n the va lue s o f the two da ta .

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Me dia n

1 3 7 13 21 31 43

Me dia n

1 3 7 13 21 31

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Mo de

 T

he mo de is the mo st fre q ue ntly o c c urring va lue in a da ta se t o r a me a sure o f re la tive ly g re a t c o nc e ntra tio n.

 T

he mo de is e a sily de te rmine d fro m a fre q ue nc y ta b le , histo g ra m, o r po lyg o n.

 I

f two a dja c e nt va lue s ha ve e q ua l numb e r o f o c c urre nc e s the n the mo de is the midpo int b e twe e n the two va lue s.

Mo de

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Figure 1. Lengths of pine needles (cm) from Larimer county, CO. Mean = 9.7 cm Median = 9.7 cm “Mode” = 9.3 cm Mean = 9.8 cm Median = 6.8 cm “Mode” = 5.0 cm Figure 2. Fake data showing an exponential distribution.