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Page 1: Stas$cs….) - cunygk12.net Brief Course in...Descrip$ve)Stas$cs)! Themostfundamentalbranchofstatisticsis$ descriptive(statistics! Statisticsusedtosummarizeordescribeasetofobservations$
Page 2: Stas$cs….) - cunygk12.net Brief Course in...Descrip$ve)Stas$cs)! Themostfundamentalbranchofstatisticsis$ descriptive(statistics! Statisticsusedtosummarizeordescribeasetofobservations$

Sta$s$cs….  L  �  Its  easy  to  be  a  really  good  methodologist  without  being  a  statistical  

genius  �  But,  you  have  to  have  some  knowledge  of  statistics  

�  The  statistical  skills  it  takes  to  be  a  good  methodologist  are  mainly  conceptual  rather  than  computational  

�  If  you  can  understand  the  logic  of  inferential  statistics  and  familiarize  yourself  with  a  few  relatively  simply  statistical  tests,  you  will  have  enough  of  the  working  knowledge  you’ll  need  to  be  a  great  methodologist  

�  Statistics  are  a  set  of  mathematical  procedures  for  summarizing  and  interpreting  observations  �  These  observations  are  usually  numerical  or  categorical  facts  about  

specific  people  or  things,  and  they  are  usually  referred  to  as  data  

Page 3: Stas$cs….) - cunygk12.net Brief Course in...Descrip$ve)Stas$cs)! Themostfundamentalbranchofstatisticsis$ descriptive(statistics! Statisticsusedtosummarizeordescribeasetofobservations$

Descrip$ve  Sta$s$cs  �  The  most  fundamental  branch  of  statistics  is  descriptive  statistics  �  Statistics  used  to  summarize  or  describe  a  set  of  observations  �  The  easy  ones!  �  Ex:  means,  medians,  modes,  percentages  �  Very  useful  for  the  general  population  

�  Think  of  sports  �  Bradshaw  ran  for  17  yards…then  2  yards…then  5  yards…then  8  yards…then  15  yards…then  1  yard…  

�  Wouldn’t  it  be  easier  to  say  that  he  ran  for  an  average  of    8  yards/carry?  

�  A  simple  mean  gives  us  more  information  a  lot  faster  

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Inferen$al  Sta$s$cs  � The  branch  of  statistics  used  to  interpret  or  draw  inferences  about  a  set  of  observations  is  referred  to  as  inferential  statistics  �  The  harder  ones!  

� We  will  talk  more  about  these  later  

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Central  Tendency  &  Dispersion  �  Descriptive  statistics  used  by  laypeople  are  often  incomplete  in  one  important  aspect  

�  Laypeople  make  frequent  use  of  descriptive  statistics  that  summarize  the  central  tendency  (basically,  the  average)  of  a  set  of  observations  

�  Most  people  are  unaware  of  an  equally  useful  and  important  category  or  descriptive  statistics  �  Those  that  summarize  the  dispersion,  or  variability  of  a  set  of  

scores  

�  Measures  of  dispersion  are  particularly  important  in  inferential  statistics  

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Central  Tendency  &  Dispersion  � One  common  measure  of  dispersion  is  the  range  of  a  set  of  scores  �  The  difference  between  the  highest  and  the  lowest  value  in  the  entire  set  of  scores  

�  Ex:  6.6,  5.6,  7.8,  6.7,  7.9….my  range  is  7.9-­‐5.6  =  2.3  

� Another  common  measure  of  dispersion  is  the  standard  deviation  �  The  average  measure  of  how  much  each  score  in  the  sample  differs  from  the  sample  mean  

�  Ex:  the  mean  IQ  score  is  100,  with  a  SD  of  15  

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Central  Tendency  &  Dispersion  � Measures  of  central  tendency,  like  the  mean,  tell  you  what  the  typical  person  is  like  

� Measures  of  dispersion,  like  the  standard  deviation,  tell  you  how  much  you  can  expect  specific  people  to  differ  from  this  typical  person  

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The  Shape  of  Distribu$ons  �  A  third  statistical  property  of  a  set  of  observations  is  the  shape  of  a  distribution  of  scores  

�  Arrange  them  in  order  from  lowest  to  highest,  and  graph  them  so  that  taller  parts  of  the  graph  represent  more  frequently  occurring  scores  

�  There  are  three  main  types  of  distributions  �  Rectangular  �  Bimodal  �  Normal  

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The  Shape  of  Distribu$ons  � A  rectangular  distribution  contains  scores  that  are  about  equally  frequent  or  probable  �  Ex:  the  theoretical  distribution  representing  the  two  possible  outcomes  that  can  be  obtained  by  tossing  a  coin  

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The  Shape  of  Distribu$ons  �  In  a  bimodal  distribution,  two  distinct  ranges  of  scores  are  more  common  

than  any  other  �  They  are  relatively  rare  �  Usually  reflects  a  sample  that  only  contains  two  meaningful  subsamples  �  Ex:  the  heights  of  athletes  attending  the  annual  sports  banquet  for  a  very  large  

high  school  that  only  has  two  sports  teams  �  Women’s  gymnastics  �  Men’s  basketball  

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The  Shape  of  Distribu$ons  �  The  most  important  and  common  type  of  distribution  is  the  normal  distribution  �  A  symmetrical,  bell-­‐shaped  distribution  in  which  most  scores  

cluster  near  the  mean  and  in  which  scores  become  increasingly  rare  as  they  become  increasingly  divergent  from  this  mean  �  Ex:  height,  weight,  extroversion,  self-­‐esteem,  age  at  which  infants  begin  

to  walk  

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The  Shape  of  Distribu$ons  �  The  nice  thing  about  the  normal  distribution  is  that  if  you  know  that  a  set  of  observations  is  normally  distributed,  it  further  improves  your  ability  to  describe  the  entire  set  of  scores  �  More  specifically,  you  can  make  very  good  guesses  about  the  exact  proportion  of  scores  that  fall  within  any  given  number  of  standard  deviations  from  the  mean  

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The  Shape  of  Distribu$ons  �  About  68%  of  scores  will  fall  within  one  standard  deviation  of  the  mean  

�  About  95%  will  fall  within  two  SD  of  the  mean  

�  About  99%  will  fall  within  three  SD  

�  Ex:  Wechsler  Adult  Intelligence  Scale  �  Mean  of  100,  SD  of  15  �  68%  of  people  have  an  IQ  between  85  and  115  �  99%  fall  between  55  and  145  

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The  Shape  of  Distribu$ons  �  One  important  purpose  of  this  is  that  this  kind  of  analysis  can  be  used  to  put  a  particular  score  in  perspective,  which  is  the  first  step  towards  making  inferences  

�  Ex:  a  set  of  400  scores  on  an  astronomy  midterm:  �  Approximates  a  normal  distribution  �  Has  a  mean  of  70  �  A  standard  deviation  of  6  

�  Your  friend  got  an  84  �  How  impressed  should  you  be?  �  You  can  figure  out  that  she  scored  over  2  standard  deviations  

above  the  mean,  meaning  she  scored  in  the  top  1%  of  the  class  

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Descrip$ve  Sta$s$cs  �  This  only  covers  a  little  bit  on  descriptive  statistics,  and  you  will  learn  much  more  in  your  statistics  class  

�  Descriptive  statistics  provide  researchers  with  an  enormously  powerful  tool  for  organizing  and  simplifying  data  

�  At  the  same  time,  they  are  only  half  the  picture  though  

�  We  need  to  do  more  than  just  simplify  and  organize  our  data  �  We  need  to  be  able  to  draw  conclusions  about  populations  from  

our  sample  data  �  To  do  this,  we  need  to  rely  on  inferential  statistics  

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Inferen$al  Sta$s$cs  �  The  basic  idea  behind  these  kinds  of  statistics  is  that  decisions  about  what  to  conclude  from  a  set  of  research  findings  need  to  be  made  in  a  logical,  unbiased  fashion  

�  The  logic  of  statistical  testing  is  largely  a  reflection  of  the  skepticism  and  empiricism  that  are  crucial  to  the  scientific  method  

� When  conducting  statistical  tests  to  aid  in  the  interpretation  of  a  set  of  findings,  researchers  begin  by  assuming  that  the  null  hypothesis  is  true  �  They  begin  by  assuming  that  their  own  predictions  are  wrong  

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Inferen$al  Sta$s$cs  �  In  a  simple,  two-­‐groups  experiment,  this  would  mean  assuming  that  the  experimental  group  and  the  control  group  are  not  really  different  from  each  other  after  the  manipulation,  and  that  any  apparent  difference  is  due  to  luck  (failure  of  random  assignment)  

� The  main  thing  that  statistical  testing  does  is  it  tells  us  exactly  how  possible  it  is  that  someone  would  get  results  as  impressive  as  those  actually  observed  in  an  experiment  if  chance  alone  were  at  work  

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Inferen$al  Sta$s$cs  �  The  logic  of  statistical  testing  is  almost  identical  to  the  logic  of  what  happens  in  an  ideal  courtroom…  

�  Researchers  begin  by  assuming  that  the  null  hypothesis  is  correct  �  That  the  researcher’s  findings  reflect  change  variation  and  are  not  

real  

�  The  opposite  of  the  null  hypothesis  is  the  alternative  hypothesis  �  Any  observed  differences  between  the  experimental  and  the  

control  group  are  real  

�  We  assume  “wrong  until  proven  right”  like  a  court  assumes  “innocent  until  proven  guilty”  

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Inferen$al  Sta$s$cs  �  Jurors  decide  that  a  person  is  guilty  only  if  the  evidence  suggests  beyond  a  reasonable  doubt  that  the  defendant  committed  the  crime  in  question  �  The  statistical  equivalent  of  “beyond  a  reasonable  doubt”  is  the  alpha  level  �  In  most  cases,  the  alpha  level  is  set  at  .05  

�  That  is,  researchers  may  reject  the  null  hypothesis  and  conclude  that  their  hypothesis  is  correct  only  when  findings  as  extreme  as  those  observed  in  the  study  would  have  occurred  by  chance  alone  less  than  5%  of  the  time  

�  Even  if  a  researcher  does  find  statistically  significant  results,  they  need  to  provide  a  reason  for  why  they  are  true,  just  like  we  need  to  have  a  motive  for  a  person  to  commit  a  crime  

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Probability  Theory  �  All  inferential  statistics  are  grounded  firmly  in  the  logic  of  probability  theory  �  Deals  with  the  mathematical  rules  and  procedures  used  to  predict  and  understand  chance  events  

�  The  probability  of  an  event  is:  �  The  number  of  specific  outcomes  that  qualify  as  the  event  in  question  divided  by  

�  The  total  number  of  possible  outcomes  �  Ex:  the  probability  of  rolling  a  3  on  a  single  roll  with  a  standard  die  is  1/6,  or  .167  because:  �  There  is  only  one  roll  that  qualifies  as  a  3  �  There  are  6  equally  likely  outcomes  

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Factors  That  Influence  the  Results  of  Significance  Tests  �  Alpha  levels  and  Type  I  and  II  errors  

�  It  is  important  to  remember  that  when  a  researcher  conducts  a  statistical  test  and  obtains  a  significant  result,  it  does  not  always  mean  that  their  hypothesis  is  correct  

�  Even  if  an  experiment  is  perfectly  executed  with  no  design  flaws,  it  is  always  possible  that  their  results  were  due  to  chance  �  In  fact,  the  p-­‐value  (the  exact  probability)  we  observe  in  an  experiment  tells  us  exactly  how  likely  it  is  that  we  would  have  obtained  results  like  ours  even  if  nothing  but  dumb  luck  was  operating  in  our  study  

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Factors  That  Influence  the  Results  of  Significance  Tests  �  Statisticians  refer  to  this  worrisome  possibility  of  incorrectly  rejecting  the  null  hypothesis  when  it  is  true  as  a  type  I  error  �  The  likelihood  of  making  a  type  I  error  is  a  direct  function  of  where  we  set  our  alpha  level  �  If  we  set  it  at  .001,  then  this  means  that  we  are  taking  one  chance  in  1,000  of  falsely  rejecting  the  null  hypothesis  

�  We  can’t  set  our  alpha  at  .001  all  the  time,  because  that  would  be  foolish  

�  We  would  also  increase  the  likelihood  of  committing  a  type  II  error  

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Factors  That  Influence  the  Results  of  Significance  Tests  � A  type  II  error  is  when  we  fail  to  reject  the  null  hypothesis  when  it  is  false  

� This  is  why  we  usually  set  our  alpha  at  .05  

� Effect  size  also  influences  the  results  of  our  tests  �  The  magnitude  of  the  effect  in  which  the  researcher  is  interested  in  

�  Ex:  finding  a  correlation  between  height  and  foot  size  in  a  sample  of  NBA  players  

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Meta-­‐Analyses  �  Researchers  have  developed  a  special  set  of  statistical  techniques  to  summarize  and  evaluate  entire  sets  of  research  findings  

� Meta-­‐analyses  refer  to  the  use  of  techniques  to  analyze  the  results  of  studies  rather  than  the  responses  of  individual  participants  

�  Literally,  it  means  that  you  are  analyzing  analyses!  

�  It  can  help  us  determine  in  what  contexts  a  specific  effect  holds  true  �  Ex:  gender  conformity  experiment,  pg.  291  

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T-­‐Tests  � One  of  the  most  important  statistical  tests  

� Used  to  compare  TWO  GROUPS  or  TWO  MEANS  

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Condi$ons  of  a  t-­‐test  

� The  groups  must  be  independent  �  Their  outcomes  don’t  affect  each  other  

� The  response  variable  that  you  are  measuring  is  quantitative  �  Its  values  have  numerical  meaning  and  represent  quantities  

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What  do  we  use  it  for?  �  Typically  used  to  study  the  mean  of  a  population  

�  Not  to  study  individuals  within  a  population  

�  It  is  especially  useful  if  your  data  set  is  small  or  if  you  don’t  know  the  standard  deviation  of  the  population  

�  What  your  actual  distribution  looks  like  depends  on  your  sample  size  �  Smaller  samples  will  yield  large  standard  deviations  and  therefore  a  

flatter  curve  

�  Also  based  on  the  degrees  of  freedom  �  How  big  is  your  sample?  

�  The  number  of  items  that  are  free  to  vary…so  the  formula  is…  

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When  are  t-­‐tests  not  enough?  

� But,  if  you  are  extensively  studying  a  population,  at  some  point  two  means  might  not  be  enough…  

� This  is  where  ANOVAs  come  in  

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ANOVA  �  One  of  the  most  commonly  used  statistical  tests  at  a  more  advanced  level  

�  Stands  for  analysis  of  variance  �  All  about  examining  the  variance  in  a  variable  and  trying  to  figure  out  where  that  variance  comes  from  

�  To  compare  several  populations  regarding  some  quantitative  variable  �  The  populations  you  are  comparing  constitute  different  groups,  denoted  by  another  variable  (ex:  age,  ethnicity,  etc.)  

�  Particularly  useful  when  you  are  comparing  groups  who  receive  different  treatments  

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ANOVA  �  Has  several  components  

�  Sum  of  squares  �  Pieces  of  variability  

�  F-­‐test  �  Compares  how  much  each  group  differs  compared  to  how  much  

variability  is  within  each  group  �  ANOVA  table  

�  Ex:  applied  to  a  one-­‐factor/one-­‐way  ANOVA  �  Comparing  the  responses  based  on  only  one  treatment  variable  

�  Two-­‐way  ANOVA  has  two  treatment  variables,  or  two  IVs  

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ANOVA  Condi$ons  �  The  groups  are  independent  

�  Their  outcomes  don’t  affect  each  other  

�  The  population  you  have  sampled  is  normally  distributed  �  You  can  guarantee  this  by  using  proper  random  sampling  and  assignment  

�  Variance  among  groups  is  equal  (to  start)  

�  After  you  have  assured  all  of  this,  decide  on  your  Ho  and  Ha  

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ANOVA  �  Verify  your  assumptions  

�  Come  up  with  your  hypotheses  

�  Determine  your  p-­‐value  

�  Collect  your  data  and  analyze  

�  If  you  find  significance,  you  can  run  multiple  comparisons  �  Tells  you  where  exactly  the  differences  are  (between  which  groups)  

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There’s  more…  �  You  can  do  two-­‐way,  three-­‐way,  four-­‐way  ANOVAs  or  Factorial  ANOVAs,  but  those  are  for  when  you  have  more  than  one  independent  variable  

�  If  you  use  these,  you  have  to  interpret  main  effects,  as  well  as  interactions  

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References  Pelham,  B.  W.  and  Blanton,  H.  (2012).  Conducting  research  in  psychology:  Measuring  the  weight  of  smoke,  4th  ed.  Belmont,  CA:  Wadsworth,  Cangage  Learning.  


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