Measures of Central Tendency - the Mean

Transcription

1 Measures of Cetral Tedecy - the Mea Dr Tom Ilveto Departmet of Food ad Resource Ecoomcs Overvew We wll beg lookg at varous measures of the ceter of the data - thk of t as a typcal value We wll start wth the mea I wll also talk about some math symbols we wll eed ad use to work wth data, especally cotuous data. Ad you wll start to see how we uses statstcs ad graphs to tell a story. I wll use two data sets to demostrate the mea, oe of whch s marrage rates for the 50 states ad Washgto D.C. I wo t drve wth studets Fastest Speed The fastest speed from past classes was 00. mph 3

2 Some Math Tools Sgma Notato The Greek symbol Stads for summato Alteratves to Math Formulas At tmes t s dffcult to use the proper math symbols Power Pot, Word, Quzzes, or e- mals As a alteratve, we wll use (ad you ca as well assgmets), the followg symbols Mea Sum(X)/ May of these follow the same usage Ecel or other spreadsheets 6 Alteratves to Math Formulas Multplcato * 5*3 5 Power 5^ Square Root SQRT or ^.5 ^ 5 SQRT(5) (5).5 (5)^.5 5 Summato Sum Sum() Alteratve Math Symbols Mea() µ mu " sgma Stadard Error SE 7 8

3 Cetral Tedecy The Mea or Arthmetc Average The cetral tedecy of a varable s the tedecy of the data to cluster or ceter about certa umercal values You mght also thk of ths as a typcal value The varablty s the spread of the data For cetral tedecy we wll focus o the mea, the mode, ad the meda The arthmetc mea or mea s the sum of the measuremets dvded by the umber of measuremets cotaed the data set For a sample, a statstc, we use wth a bar over t For a populato, a parameter, we use the Greek µ 9 0 There are two ways to epress the mea Fastest Speed eample ( / ) The sum of all the values, dvded by the umber of values The sum of each value weghted by the umber of values - a mathematcal epectato 57 Sum() 5,78.0 Mea 5,78.0/ O average, the fastest speed of dr lveto s studets s 00. mph Stem ad Leaf of Fastest Speed Stem Leaf represets 65 Suggesto for sgfcat dgts: for calculated statstcs, use oe more decmal place tha the orgal data. Cout

4 As a measure of cetral tedecy, the mea has several advatages: Sum of devatos about the mea equal zero The mea uses formato of all the values a varable We are addg all the values together, ad the dvdg by the sample sze Usg more formato s usually better The mea has two mportat mathematcal propertes:. The sum of the devatos about the mea equals zero. The sum of squared devatos about the mea s a mmum 3 #( " ) 0 ( " ) " " # " 0 Sum of squared devatos about the mea s a mmum Ths s called the Least Squares property ( " ) There s o other value or costat we could substtute the equato for the mea that would result a lower sum of squares. Other Propertes of the Mea We ca make fereces from a sample to a populato for the mea The mea forms the bass for a umber of other statstcs kow as Product Momet Statstcs But, the mea s sestve to outlers ad etremes the data. It s ot as resstat as other measures of cetral tedecy 5 6

5 The effect of a outler - Marrage Rate data Marrage rate data set 5 (50 states ad Washgto D.C.) sum().7 mea.7/ Removg the outlers o the Marrage Rate Data Revsed Marrage rate data set 9 sum() 358. mea 358./9 7.3 About a 5.% decrease from Let s look at the effect of outlers o the Studet Speed Data The mea oly chaged slghtly by removg the top fve scores, from 00. to 97.89, about a % decrease. 9 Closg thoughts o the mea ad Outlers Key pot: ad of themselves, outlers are ot wrog or bad. They should be eamed to determe they are ot part of the populato, Or f they are a mstake codg or measuremet. I wll preset you wth a strategy for accessg what s a outler ad the mpact of outlers o measures of cetral tedecy. Based o a probablty framework Ad the stadard devato 0