Lecture 1 Displaying data... 12 Statistics... 13 Statistical methd... 13 Variables... 13 Value... 15 Scre... 15 Type f Research... 15 Level f Measurement... 15 Numeric/Quantitative variables... 15 Ordinal/Rank-rder variables (in rder nly)... 15 Equal interval variables... 17 Categrical/Nminal variables... 17 Frequency table... 17 Making a frequency table... 17 Gruped frequency tables... 19 Histgrams... 19 Frequency plygns... 20 Shapes f distributins... 20 Number f peaks... 20 Is it rughly symmetrical?... 22 Kurtsis... 22 Discrete variable... 22 Cntinuus variable... 22 Flr effect... 22 Ceiling effect... 22 Nrmal curve... 22 Lecture 2 Central tendency and variability... 24 Central tendency... 25 Mean... 25
Imprtant cncepts... 25 X M = Calculating the mean N... 27 Mde... 27 Median... 27 Which central tendency measure... 28 Variability... 30 Measures f variability... 30 Range... 30 Interquartile range (IQR)... 30 Variance... 30 Calculating the variance... 30 Example: Number f therapy sessins... 31 Imprtant features f the variance... 31 Sum f Squares (SS) = å (X-M) 2... 32 The standard deviatin (Measures f variability)... 32 SD frmula... 32 Example: Number f therapy sessins... 32 Outlier... 32 Cmputatinal frmula... 32 Definitinal frmula... 33 Lecture 3 Standardised scres: Z scres... 34 Sme examples t cnsider... 35 Z scres... 35 Distributin f Z scres... 35 Calculating a Z scre frm a raw scre... 36 Example... 36 Interpreting Z scres... 36
Example... 36 Example... 37 Implicatins f Z scres... 37 Example: Cmparing scres frm different distributins... 37 The relative achievement f 3 friends.... 39 Calculating a raw scre frm a Z scre (frm Z scre t raw scre) X = SD x Z) + M (... 39 Example: IQ data... 39 Imprtant features f Z scres... 41 When the distributin is nrmal, Z scres tell us even mre... 41 The basis f percentages n a nrmal distributin... 41 Percentile... 41 Lecture 4 Crrelatin... 43 Types f Variables in Research... 44 Dependent Variable (DV)... 44 Independent Variable (IV)... 44 Examples... 44 Majr Types f Research Design... 44 Descriptin in an bservatinal study f tw cntinuus variables... 46 Graphing pairs f variables: Scatterplt... 46 Drawing a scatterplt... 46 Cnstructing a Scatterplt... 46 Patterns f linear relatinship... 47 Patterns f relatinship... 49 Quantifying the relatinship: Crrelatin... 51 Calculating the crrelatin cefficient r å Z xzy r =... 51 N Crssprducts (SZ X Z Y )... 52 Making sense f r: prprtinate reductin in errr r Cefficient f determinatin: r 2 tells us the prprtin f variability... 56 Lecture 5 Inferential statistics... 58 Intrductin t Inferential Statistics... 59
The nrmal curve... 59 Backgrund... 59 The nrmal distributin: areas under the nrmal curve... 59 1SD and the nrmal distributin... 59 2SD and the nrmal distributin... 60 Finding percentages using a nrmal curve table... 60 Tips fr using a nrmal curve table... 62 IQ scres example 1... 62 IQ scres example 2... 64 Finding raw scres frm percentages... 64 Prbability... 66 Calculating prbability... 66 Expected relative frequency... 67 Prbability and expectatin... 67 Z scres and prbability... 67 Samples and ppulatins... 67 Methds f sampling... 69 Ppulatin parameters and sample statistics... 69 Lecture 6 Hypthesis testing... 70 Errrs in hypthesis testing... 71 Example: brain affected by radiatin... 71 Tw pssibilities... 71 Statistical significance: The magical p <.05... 73 Interpretatin issues... 73 Hypthesis testing... 73 The prcess f hypthesis testing... 73 Step 1: Frmulating research and null hyptheses... 73 Step 2: Identifying the cmparisn distributin... 74 Step 3: Determining the cut-ff scre... 74
Step 4: Where des yur sample scre sit n the cmparisn distributin?... 76 Step 5: Decisin time: Shuld the null hypthesis be rejected?... 76 The implicatins f yur decisin... 76 One-tailed and tw-tailed hypthesis tests... 76 Directinal hyptheses... 76 Tw-tailed tests... 78 Cut-ff pints fr tw-tailed tests... 78 The nrmal curve: One- and tw-tailed tests... 78 Determining Cut-ff Pints with Tw-Tailed Tests... 78 Cmparisn f ne and tw-tailed tests... 80 Summary s far... 80 An example... 82 Errrs in hypthesis testing: Terminlgy... 82 Errrs when result is significant: Type 1 errr... 83 Errrs when result is nt significant: Type 2 errr... 84 Errrs in hypthesis testing... 84 Errrs in hypthesis testing: Table... 84 Crrect decisin... 85 Crrect decisin... 85 Type I errrs: when H 0 is actually true... 85 Crrect decisin... 85 Type II errrs: when H 1 is actually true and H0 is false... 85 Crrect decisin... 85 Relatinship between Type I and Type II errrs... 87 Pwer... 87 Jury Trial Example f Errrs... 88 Lecture 7 The distributin f means... 89 Distributin f means: The lgic... 90 Hypthesis testing with samples... 90
Samples frm ppulatins... 90 Sampling variability... 92 Minimising errr... 92 S what distributin d we need?... 92 Distributins f means... 92 Why des this distributin nrmalise?... 94 Characteristics f the distributin f means: #1... 94 Characteristics f the distributin f means: #2... 96... 96 Measuring variability in sample means... 96 Standard Errr f the Mean... 98 Increase N, decrease Errr... 98 Characteristics f the distributin f means: #3... 98 Three types f distributins: Ppulatins... 100 Three types f distributins: Samples... 100 Three types f distributins: Distributins f means... 100 Three types f distributins... 100 Cmparisn f Three Types f Distributins... 101 Hypthesis testing with samples... 101 Hypthesis testing against a knwn ppulatin... 101 Example 1... 101 Back t ur Nuclear Pwer Plant Twn... 103 Step 4: Where des yur sample mean sit?... 103 (this screen will be in the exam)... 105 Estimatin and cnfidence intervals... 105
Our example... 107 95% cnfidence intervals f sample... 107 Using cnfidence intervals t test hyptheses... 107 Our Class Example... 109 Hw cnfident are we... 110 Did we make an Errr?... 110 Lecture 8 t tests: single sample and dependent means... 112 Example #1: Stpstress... 113 Z tests à t tests: a general intrductin... 113 Estimating the ppulatin standard deviatin frm the sample data... 115 Why N-1? The mystery f Degrees f Freedm... 115 Estimating the standard deviatin f the cmparisn distributin... 115 Z frmula à t frmula (ne-sample tests)... 115 Shrt Cut t get SM... 117 The ne sample t test... 117 The cmparisn distributin... 117 The t distributin vs. nrmal distributin... 119 The t distributin vs. nrmal distributin re cut-ff scres... 119 Tips fr using the t table (A-2, p. 675)... 119 Wrking thrugh Example #1: Stpstress... 122 1. Stating the hyptheses... 122 2. Determining the characteristics f the cmparisn distributin... 122 3. Determine the critical value t reject H 0... 123 4. Determine the t value i.e., determine yur sample s scre n the cmparisn distributin (the t distributin)... 123 5. Cmpare the scres t make a decisin... 123 Anther way t use ur new t distributin... 123 The t test fr dependent means (repeated measures)... 124 Difference scres... 124 Single sample t dependent measures t test... 125 1. Stating the hyptheses... 125
2. Determining the characteristics f the cmparisn distributin... 125 3. Determine the critical value t reject H 0... 127 4. Determine the t value i.e., determine yur sample s scre n the cmparisn distributin (the t distributin)... 127 5. Cmpare the scres t make a decisin... 128 Cnfidence Intervals arund the Mean... 128 Using Cnfidence Intervals t Test Hypthesis f Mean Difference... 129 APA Style Write-Up... 129 Full APA Write-Up... 129 Assumptins f the t test... 129 Situatins where we use a t test fr dependent means... 131 Example #3: Neighburhd attachment... 131 Step 1... 131 Step 2... 131 Step 3... 133 Step 4... 133 Step 5... 133 Example # 4: Neighburhd Attachment; Repeated Measures Design... 133 Lecture 9 t test fr independent means... 136 The t test fr independent means... 137 The lgic underlying the independent means t test... 137 Wrking ur way t S difference Distributin f sample means... 137 Distributin f differences between means... 138 Identifying the distributin... 140 z-tests vs t-test... 140 Variance f Cmparisn Distributin... 140 Identifying the distributin... 142 Key Distributins in Hypthesis Testing... 142 Cmparisn Distributins... 142 Steps in the prcess f calculating independent grups t test... 144 Example: Dyslexia and clur verlays... 144
Mean f the distributin f differences between means... 144 Estimated ppulatin variance frm bth samples... 146 The pled estimate f the ppulatin variance... 146 Weighting variance estimates accrding t df... 146 Calculating the variances f the tw distributins f means... 148 The distributin f the differences between the means... 148 Equal sample size... 148 The shape f the distributin f the differences between means... 148 Calculating the t scre crrespnding t yur samples... 148 Steps fr a t Test fr Independent Means... 149 Dyslexia and clur verlays example Step 1: State hyptheses... 149 Step 2: Determine characteristics f the cmparisn distributin... 151 Step 3: Determine the cut-ff scre... 151 Step 4: Calculate the t scre (determine sample scre n cmparisn distributin)... 152 Step 5: Decisin regarding H 0... 152 APA style write-up... 152 Assumptins f the t test fr independent means... 153 Effect size in t tests... 154 Chen s d... 154 Eta Squared η 2... 154 Easy t Calculate... 154 Effect Size and Pwer... 156 Lecture 10 Chi-square tests... 158 Statistical ptins... 159 Example: Attachment styles #1... 159 Observed and expected frequencies: What we have vs. what we expect... 161 Determining Expected Frequencies: When all categries are equal... 161 Chi-square (c 2 ) test fr gdness f fit... 161 Expected and bserved frequencies... 161 Calculating the c 2 statistics... 163
Example: Attachment styles #1... 163 Testing significance: c 2 distributins... 164 Example: Attachment styles #1... 164 Review f steps fr calculating the chi-square statistic... 164 Example: Attachment styles #2... 164 c 2 distributins... 166 Heavy metal pllutin and mental health example:... 166 Chi-square (c 2 ) test fr independence... 168 H 0 : independent (unrelated)... 170 Example... 170 Cntingency table... 170 Calculating the expected frequencies... 172 Calculating the c 2 statistics... 172 Decisin... 174 Gender and reprted child abuse example:... 174 Assumptins f c 2 tests... 176 Effect size in c 2 tests (strength f relatinship in c 2 tests f independence)... 176 Chi-Square Tests in Research Articles... 176 Lecture 11 Intrductin t Qualitative Research... 177 Relevance f Qualitative Research... 178 Features f Qualitative Research... 178 Paradigms in Scial Research... 180 Imprtant cncept... 180 Psitivist Paradigm... 180 Scial Cnstructinist Paradigm... 180 Paradigms in Scial Research... 182 Quantitative vs. Qualitative Research... 182 Deductive Reasning... 182 Inductive Reasning... 183 Beynd Paradigm Wars... 183
Prcess f Qualitative Research... 185 Thery in Qualitative Research... 185 Mre abut Thery... 187 Principles f Research Ethics... 187 Ethics f Qualitative Research... 189 Hw t Act Ethically... 189 Checklist fr Taking Ethical Issues int Accunt... 189 Summary... 190
Lecture 1 Displaying data Variables Frequency tables Gruped frequency tables Histgrams Frequency plygns Shapes f distributins
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
DV depends n IV
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. 0-10 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
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
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
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
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,
psychlgists use the terms mre infrmally t describe the general shape.
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;
distributins bserved in nature and in research cmmnly apprximate it.
Lecture 2 Central tendency and variability - Measures f central tendency mean mde median - Measures f variability range variance standard deviatin - Cautins and advice
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 10 5 7 2 6 1 5 0 5 0 4-1 4-1
4-1 3-2 2-3 Mean f the distributin f the number f dreams during a week fr 10 students.
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 10 + 7 + 6 + 5 + 5 + 4 + 4 + 4 + 3+ 2 = = 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
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 2 3 4 5 median = 3.5 easy if there are an dd number f scres scres 2 3 4 5 5 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
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 1 3 3 6 7 7 8 8 8 9 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 1 3 3 6 7 X 7 8 8 8 9 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