Statistics Statistical method Variables Value Score Type of Research Level of Measurement...

Transcription

1 Lecture 1 Displaying data Statistics Statistical methd Variables Value Scre Type f Research Level f Measurement Numeric/Quantitative variables Ordinal/Rank-rder variables (in rder nly) Equal interval variables Categrical/Nminal variables Frequency table Making a frequency table Gruped frequency tables Histgrams Frequency plygns Shapes f distributins Number f peaks Is it rughly symmetrical? Kurtsis Discrete variable Cntinuus variable Flr effect Ceiling effect Nrmal curve Lecture 2 Central tendency and variability Central tendency Mean... 25

2 Imprtant cncepts X M = Calculating the mean N Mde Median Which central tendency measure Variability Measures f variability Range Interquartile range (IQR) Variance Calculating the variance Example: Number f therapy sessins Imprtant features f the variance Sum f Squares (SS) = å (X-M) The standard deviatin (Measures f variability) SD frmula Example: Number f therapy sessins Outlier Cmputatinal frmula Definitinal frmula Lecture 3 Standardised scres: Z scres Sme examples t cnsider Z scres Distributin f Z scres Calculating a Z scre frm a raw scre Example Interpreting Z scres... 36

3 Example Example Implicatins f Z scres Example: Cmparing scres frm different distributins The relative achievement f 3 friends Calculating a raw scre frm a Z scre (frm Z scre t raw scre) X = SD x Z) + M ( Example: IQ data Imprtant features f Z scres When the distributin is nrmal, Z scres tell us even mre The basis f percentages n a nrmal distributin Percentile Lecture 4 Crrelatin Types f Variables in Research Dependent Variable (DV) Independent Variable (IV) Examples Majr Types f Research Design Descriptin in an bservatinal study f tw cntinuus variables Graphing pairs f variables: Scatterplt Drawing a scatterplt Cnstructing a Scatterplt Patterns f linear relatinship Patterns f relatinship Quantifying the relatinship: Crrelatin Calculating the crrelatin cefficient r å Z xzy r = N Crssprducts (SZ X Z Y ) Making sense f r: prprtinate reductin in errr r Cefficient f determinatin: r 2 tells us the prprtin f variability Lecture 5 Inferential statistics Intrductin t Inferential Statistics... 59

4 The nrmal curve Backgrund The nrmal distributin: areas under the nrmal curve SD and the nrmal distributin SD and the nrmal distributin Finding percentages using a nrmal curve table Tips fr using a nrmal curve table IQ scres example IQ scres example Finding raw scres frm percentages Prbability Calculating prbability Expected relative frequency Prbability and expectatin Z scres and prbability Samples and ppulatins Methds f sampling Ppulatin parameters and sample statistics Lecture 6 Hypthesis testing Errrs in hypthesis testing Example: brain affected by radiatin Tw pssibilities Statistical significance: The magical p < Interpretatin issues Hypthesis testing The prcess f hypthesis testing Step 1: Frmulating research and null hyptheses Step 2: Identifying the cmparisn distributin Step 3: Determining the cut-ff scre... 74

5 Step 4: Where des yur sample scre sit n the cmparisn distributin? Step 5: Decisin time: Shuld the null hypthesis be rejected? The implicatins f yur decisin One-tailed and tw-tailed hypthesis tests Directinal hyptheses Tw-tailed tests Cut-ff pints fr tw-tailed tests The nrmal curve: One- and tw-tailed tests Determining Cut-ff Pints with Tw-Tailed Tests Cmparisn f ne and tw-tailed tests Summary s far An example Errrs in hypthesis testing: Terminlgy Errrs when result is significant: Type 1 errr Errrs when result is nt significant: Type 2 errr Errrs in hypthesis testing Errrs in hypthesis testing: Table Crrect decisin Crrect decisin Type I errrs: when H 0 is actually true Crrect decisin Type II errrs: when H 1 is actually true and H0 is false Crrect decisin Relatinship between Type I and Type II errrs Pwer Jury Trial Example f Errrs Lecture 7 The distributin f means Distributin f means: The lgic Hypthesis testing with samples... 90

6 Samples frm ppulatins Sampling variability Minimising errr S what distributin d we need? Distributins f means Why des this distributin nrmalise? Characteristics f the distributin f means: # Characteristics f the distributin f means: # Measuring variability in sample means Standard Errr f the Mean Increase N, decrease Errr Characteristics f the distributin f means: # Three types f distributins: Ppulatins Three types f distributins: Samples Three types f distributins: Distributins f means Three types f distributins Cmparisn f Three Types f Distributins Hypthesis testing with samples Hypthesis testing against a knwn ppulatin Example Back t ur Nuclear Pwer Plant Twn Step 4: Where des yur sample mean sit? (this screen will be in the exam) Estimatin and cnfidence intervals

7 Our example % cnfidence intervals f sample Using cnfidence intervals t test hyptheses Our Class Example Hw cnfident are we Did we make an Errr? Lecture 8 t tests: single sample and dependent means Example #1: Stpstress Z tests à t tests: a general intrductin Estimating the ppulatin standard deviatin frm the sample data Why N-1? The mystery f Degrees f Freedm Estimating the standard deviatin f the cmparisn distributin Z frmula à t frmula (ne-sample tests) Shrt Cut t get SM The ne sample t test The cmparisn distributin The t distributin vs. nrmal distributin The t distributin vs. nrmal distributin re cut-ff scres Tips fr using the t table (A-2, p. 675) Wrking thrugh Example #1: Stpstress Stating the hyptheses Determining the characteristics f the cmparisn distributin Determine the critical value t reject H Determine the t value i.e., determine yur sample s scre n the cmparisn distributin (the t distributin) Cmpare the scres t make a decisin Anther way t use ur new t distributin The t test fr dependent means (repeated measures) Difference scres Single sample t dependent measures t test Stating the hyptheses

8 2. Determining the characteristics f the cmparisn distributin Determine the critical value t reject H Determine the t value i.e., determine yur sample s scre n the cmparisn distributin (the t distributin) Cmpare the scres t make a decisin Cnfidence Intervals arund the Mean Using Cnfidence Intervals t Test Hypthesis f Mean Difference APA Style Write-Up Full APA Write-Up Assumptins f the t test Situatins where we use a t test fr dependent means Example #3: Neighburhd attachment Step Step Step Step Step Example # 4: Neighburhd Attachment; Repeated Measures Design Lecture 9 t test fr independent means The t test fr independent means The lgic underlying the independent means t test Wrking ur way t S difference Distributin f sample means Distributin f differences between means Identifying the distributin z-tests vs t-test Variance f Cmparisn Distributin Identifying the distributin Key Distributins in Hypthesis Testing Cmparisn Distributins Steps in the prcess f calculating independent grups t test Example: Dyslexia and clur verlays

9 Mean f the distributin f differences between means Estimated ppulatin variance frm bth samples The pled estimate f the ppulatin variance Weighting variance estimates accrding t df Calculating the variances f the tw distributins f means The distributin f the differences between the means Equal sample size The shape f the distributin f the differences between means Calculating the t scre crrespnding t yur samples Steps fr a t Test fr Independent Means Dyslexia and clur verlays example Step 1: State hyptheses Step 2: Determine characteristics f the cmparisn distributin Step 3: Determine the cut-ff scre Step 4: Calculate the t scre (determine sample scre n cmparisn distributin) Step 5: Decisin regarding H APA style write-up Assumptins f the t test fr independent means Effect size in t tests Chen s d Eta Squared η Easy t Calculate Effect Size and Pwer Lecture 10 Chi-square tests Statistical ptins Example: Attachment styles # Observed and expected frequencies: What we have vs. what we expect Determining Expected Frequencies: When all categries are equal Chi-square (c 2 ) test fr gdness f fit Expected and bserved frequencies Calculating the c 2 statistics

10 Example: Attachment styles # Testing significance: c 2 distributins Example: Attachment styles # Review f steps fr calculating the chi-square statistic Example: Attachment styles # c 2 distributins Heavy metal pllutin and mental health example: Chi-square (c 2 ) test fr independence H 0 : independent (unrelated) Example Cntingency table Calculating the expected frequencies Calculating the c 2 statistics Decisin Gender and reprted child abuse example: Assumptins f c 2 tests Effect size in c 2 tests (strength f relatinship in c 2 tests f independence) Chi-Square Tests in Research Articles Lecture 11 Intrductin t Qualitative Research Relevance f Qualitative Research Features f Qualitative Research Paradigms in Scial Research Imprtant cncept Psitivist Paradigm Scial Cnstructinist Paradigm Paradigms in Scial Research Quantitative vs. Qualitative Research Deductive Reasning Inductive Reasning Beynd Paradigm Wars

11 Prcess f Qualitative Research Thery in Qualitative Research Mre abut Thery Principles f Research Ethics Ethics f Qualitative Research Hw t Act Ethically Checklist fr Taking Ethical Issues int Accunt Summary

12 Lecture 1 Displaying data Variables Frequency tables Gruped frequency tables Histgrams Frequency plygns Shapes f distributins

13 Statistics Statistical methd Determining if true r nt. Descriptive - Infrmatin/data is summarised s as t be mre easily understd - describing data: e.g. what des the sample f 2000 represent Inferential - Inferring smething - used t draw cnclusins abut regularities in the data - Applying t the ppulatin. What peple in general may lk like frm the data cllected? - Prbability - Statistically significance Variables a characteristic that can have different values (e.g., age, religin, reactin time, anxiety level) smething which is able t vary r take different values is a variable - acrss peple: gender, height, weight - within peple: height, weight, jb satisfactin wrk with psychlgical materials - ften use scres n particular tests as variables - e.g., extrversin-intrversin scre Independent Variable (IV) - variable can change - nt dependent n ther variable, wrks independent - cause Dependent variable (DV) - affects by changes in the IV

14 DV depends n IV

15 Value A pssible number r categry that a scre can have (e.g., 1, 2, 3 r female) Just a number r categry. Number a variable can take, e.g Scre Particular persn s value n a variable (e.g., 3, 6 r Buddhist) Type f Research Observatinal / Naturalistic research - can t talk abut cause/effect - can talk abut relatinship - nt a cntrl envirnment Experimental / Cntrl research - cntrl envirnment - islate all ther variable - manipulate IV Level f Measurement Types f underlying numerical infrmatin prvided by a measure, such as equal-interval, rank-rder, and nminal (categrical) (Kinds f variables) Numeric/Quantitative variables - variables whse values are numbers (as ppsed t a nminal variable) - generally use numbers t dente different values f a variable, e.g. 68kg - 2 types f numeric variables Magnitude Equality f intervals: has magnitude and equal intervals Ordinal/Rank-rder variables (in rder nly) numeric variable in which the values are ranked, such as class standing r place finished in a race. Numeric variable in which values crrespnd t the relative psitin f things measured

16 difference in magnitude implied, N set magnitude between the 2 nt equal intervals between ranks grup has rder, e.g. race, 1 st 2 nd 3 rd,still a categry 1 st (10 secnds) 2 nd (11 secs) 3 rd (14 secs), magnitude ranks: e.g., place in class, rder in a hrse race e.g. GPA between being 2 nd and 3 rd in the class culd be different t 8 th and 9 th

17 Equal interval variables variable in which the numbers stand fr apprximately equal amunts f what is being measured Numeric variable in which differences between values crrespnd t differences in the underlying thing being measured has magnitude difference in magnitude implied equal intervals are assumed e.g., time elapsed, temperature, ages, GPA, weight, stress level e.g. GPA 2.5 and 2.8 means abut as much as the difference between a GPA f3 and 3.3 Categrical/Nminal - Variable with values that are names r categries (that is, they are names rather than numbers) variables Nminal cmes frm the idea that its values are names Variable in name nly. categry, number dn t necessary mean anything, just a categry, e.g. religin, gender (1=male, 2=female) Desn t dente anything abut the relative magnitude Frequency table - descriptive data - shws hw frequently each value f a variable ccurs - useful fr shwing verall tendencies - e.g., stress ratings f 30 students: 8,7,4,10,8,6,8,9,9,7,3,7,6,5,0,9,10,7,7,3,6,7,5,2,1,6,7,10,8,8 Making a frequency - make a list starting with the lwest scre ending with the highest table include values which didn t ccur - wrk thrugh yur scres and place a tick next t each value n yur list number f ticks = number f scres - make a neat table with values dwn left side and the number f ticks next t them

18

19 Gruped frequency tables when there are many values - table becmes awkward - use all values within an interval - use equal intervals - recrd frequency f all values in each interval Histgrams a type f bar graph a way f graphing the infrmatin in a frequency table the height f each bar is the frequency f each interval in the table can use the data frm frequency table r gruped frequency table

20 Frequency plygns a line graph f the infrmatin in a frequency table can use the data frm frequency table r gruped frequency table the height f each pint is the frequency f each value (r interval) Shapes f distributins frequency tables, histgrams, frequency plygns describe the distributin hw are scres distributed acrss a range f values? cmmn patterns and features: is there a single peak, tw, nne? is it rughly symmetrical? hw thick r heavy are the tails? Number f peaks Mdality: hw many peaks? Is there a single peak, tw, nne? 1 peak: unimdal 2 peaks: bimdal >2 peaks: multimdal withut any real peaks: rectangular Strictly speaking, a distributin is bimdal r multimdal nly if the peaks are exactly equal; hwever,

21 psychlgists use the terms mre infrmally t describe the general shape.

22 Is it rughly symmetrical? if nt symmetrical, skewed distributin - psitive skew: if tail pints t right - negative skew: if tail pints t left Kurtsis (width) Hw thick r heavy are the tails? Need t cmpare with the nrmal distributin, this quality is called kurtsis a) Nrmal b) Leptkurtic (Peaked) tails are thicker r heavier than nrmal curve mre easily recgnised by tp f curve being mre peaked than nrmal curve c) Platykurtic (Play sunds like flat) tails are thinner r lighter than nrmal curve Discrete variable Cntinuus Variable that has specific values and that cannt have values between these specific values Variable fr which, in thery, there are an infinite number f values between any tw values variable Flr effect Situatin in which many scres piles up at the lw end f a distributin (creating skewness t the right) because it is nt pssible t have lwer scre Ceiling effect Situatin in which many scres pile up at the high end f a distributin (creating skewness t the left) because it is nt pssible t have a higher scre Nrmal curve Specific, mathematically defined, bell-shaped frequency distributin that is symmetrical and unimdal;

23 distributins bserved in nature and in research cmmnly apprximate it.

24 Lecture 2 Central tendency and variability - Measures f central tendency mean mde median - Measures f variability range variance standard deviatin - Cautins and advice

25 Central Mst typical, cmmn scre, representative value f a grup f scres tendency Mean Sensitive t any scre = the average scre. = the sum f all the scres divided by the number f scres. = the typical r representative scre. best way f estimating what an individual unknwn scre might be. influenced by all scres in a distributin (s represents all scres but can be unduly influenced by extreme scres and, thus, can be biased). E.g. I ask 10 students hw much study they have dne in the last week and get the fllwing results: 10, 2, 4, 3, 4, 4, 6, 5, 5, 7 the ttal number f hurs studied = 50 the number f scres (bservatins) = 10 the mean number f hurs = 50/10 = 5 Imprtant cncepts it is like a balancing pint in a distributin the ttal distance frm the mean f all scres less than the mean = the ttal distance frm the mean f all scres greater than the mean belw mean ttal = -8, abve mean ttal = +8, sum f distances = 0 the mean can be a value r scre which des nt exist in the actual set f scres Scres Distance frm mean

26 Mean f the distributin f the number f dreams during a week fr 10 students.

27 Calculating the M = mean X N The mathematical frmula fr calculating the mean, M (smetimes µ r ) å: a Greek letter sigma means the sum f X X: a scre in the distributin f a variable X N: the number f scres in a distributin M M X = = N N 50 = = 5 10 Mde = the mst cmmn scre in a unimdal distributin = the peak f a histgram r a frequency plygn in a symmetrical unimdal distributin (nrmal distributin): the mde = the mean useful when nly a few values pssible as mde nly describes ne scre The mde as the high pint in a distributin s histgram, using the example f the number f dreams during a week fr 10 students. Median = the middle scre when all scres are ranked easy if there are an dd number f scres if even number, it falls halfway between the tw middle scres smetimes the median is a better measure f central tendency than the mean in skewed distributins, a few extreme scres can affect the mean. use when the data is heavily skewed, e.g. incme, huse prices

28 even versus dd number f cases: the middle scre when all scres are ranked if there is an even number f scres the median falls halfway between t tw middle scres scres median = 3.5 easy if there are an dd number f scres scres median = 4 Which central tendency measure Mde: nly few values Median: skewed Mean: nrmal in a symmetrical unimdal distributin, the mean = the mde = the median

29

30 Variability hw spread ut the scres are in a distributin Measures f variability tw distributins may have the same mean but ne may have a greater spread (r variability) in values in describing distributins numerically, need t be able t discuss the spread (r variability) f scres Range the simplest measure f spread is the range the range is simply the difference between the highest and lwest scres range = highest scre - lwest scre: 9-1 = 8 Interquartile range (IQR) IQR prvides the bundaries fr the middle 50% f scres Steps t find IQR find median find middle scre in tp and bttm halves X IQR = 8-3 = 5 usual t reprt IQR with median Variance Range nly describes tw, pssibly extreme, values IQR better but still nt representative f all scres Prefer a measure that cnsiders all values - like ur mean in central tendency The variance tells us hw spread ut a set f scres is arund their mean it is the average f each scre s squared deviatin arund the mean Variance: hw much individual scre differ frm the mean, square the value t cancel ut the minus Calculating the variance subtract the mean frm each scre (ne by ne) t get a deviatin scre (X-M) square (multiply by itself) each f these deviatin scres t get a squared deviatin scre