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

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. 1

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 2

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. 3

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 th va lue o f the va ria b le in a da ta a rra y.  X i -- the i 1 3 7 13 21 31 X 1 =1 X 3 =7 X 6 =31 4

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 X i va lue s fro m X 1 thro ug h X N 5

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 (  ) 6

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 7

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 . 8

Me dia n 1 3 7 13 21 31 43 Me dia n 1 3 7 13 21 31 9

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 10

Mean = 9.7 cm Median = 9.7 cm “Mode” = 9.3 cm Figure 1. Lengths of pine needles (cm) from Larimer county, CO. Mean = 9.8 cm Median = 6.8 cm “Mode” = 5.0 cm Figure 2. Fake data showing an exponential distribution. 11