PSTAT 100: Lecture 02

Data Structure, and Tidy Data

Ethan P. Marzban

Department of Statistics and Applied Probability; UCSB

Summer Session A, 2025

Review: Data Semantics

Definition

Definition: Data

We define an observation to be a collection of values measured on certain attributes (aka variables). From a semantic standpoint, we define data to be a collection of observations collected on observational units.

Review: Data Semantics

Another Illustration

Review: Course Enrollments Dataset

Some Sample Observations

The enrollment for PSTAT 5A, titled “Understanding Data,” was 167 in Spring 2025
The enrollment for PSTAT 5A, titled “Understanding Data,” was 222 in Winter 2025
The enrollment for PSTAT 5H, titled “Statistics,” was 11 in Spring 2025

Wide Layout
Long Layout

      Course               Title S25 W25 F24 M24 S24 W24 F23 M23 S23 W23 F22
    PSTAT 5A  UNDERSTANDING DATA 167 222 213  76 258 283 549 110 212 279 499
    PSTAT 5H          STATISTICS  11  14  30  NA  31  24  25  NA  31  22   4
   PSTAT 5LS  STAT LIFE SCIENCES 355 340 348  59 304 299  NA  39 299 294  NA
     PSTAT 8  TRANS DS PROB STAT  70 105 137  22  83 134 132  18  86 156  70
    PSTAT 10  DATA SCIENCE PRINC 235 241 286  93 234 260 285  83 240 282 257
    PSTAT 99 INDEPENDENT STUDIES   1   2  NA  NA  NA  NA  NA  NA  NA  NA  NA
   PSTAT 100        DS_CONC&ANLS 109 131 108  NA  95  NA 104  48 125 127  NA
   PSTAT 105  INTRO N-PARAM METH  NA  79  NA  NA  NA  NA  39  NA  NA  NA  79
   PSTAT 109 STATISTICS FOR ECON  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA
   PSTAT 115 BAYES DATA ANALYSIS  81  56  74  25  74  96  16  15  38  70  63
  PSTAT 120A   PROB & STATISTICS 310 383 345  75 367 372 419  86 237 246 419
  PSTAT 120B   PROB & STATISTICS 312 329 236 136 382 241 261 102 256 256 165
  PSTAT 120C   PROB & STATISTICS 163  NA  37 129 131  39  19  81  65  31  87
   PSTAT 122 DESIGN OF EXPERMNTS 240 210 201  NA 177 162 167  NA 194 187 182
   PSTAT 123  SAMPLING TECHNIQUE  NA  NA  NA  NA  NA 137  NA  NA  NA  NA  NA
   PSTAT 126 REGRESSION ANALYSIS 197 214 272  94 147 200 123  84 149 175 182
   PSTAT 127    ADV STATS MODELS  NA  NA  72  NA  NA  NA  57  NA  NA  73  NA
   PSTAT 130    SAS BASE PROGRAM 116 120 121 108 122  97 120  95 119  98  97
   PSTAT 131  STAT MACHINE LEARN  96 156  64  80  93 138 150  55 108 197 178
   PSTAT 132 DATABASES FOR STATS  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA
   PSTAT 134   STAT DATA SCIENCE  46  91  78  NA  70  78  80  NA  87  42  33
   PSTAT 135  BIG DATA ANALYTICS  96  NA  NA  NA  NA  99  NA  NA  NA  93  NA
   PSTAT 140 STATISTICAL PROCESS  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA
  PSTAT 160A  STOCHASTIC PROCESS 132 217 126  NA 100 153 132  NA 108 161 184
  PSTAT 160B  STOCHASTIC PROCESS 116  77  86  NA 106  73  43   7  82  81  58
   PSTAT 170  INTRO MATH FINANCE 116  65  92  19 125  65  59  17  77  57  55
   PSTAT 171   MATH COMPOUND INT  73  77  96  47  84  68 102  32  64  69 108
  PSTAT 172A    ACTUARIAL STAT I  NA  49  NA  NA  NA  60  NA  NA  NA  43  NA
  PSTAT 172B   ACTUARIAL STAT II  14  NA  NA  NA  20  NA  NA  NA  23  NA  NA
   PSTAT 173         RISK THEORY  NA  NA  64  NA  NA  NA  35  NA  NA  NA  43
   PSTAT 174         TIME SERIES  NA  89  83  NA 104  63  NA  NA 170 118  69
   PSTAT 175   SURVIVAL ANALYSIS  71  NA  NA  NA  55  NA  18  NA  NA  NA  73
   PSTAT 176    ADV MATH FINANCE  17  NA  NA  NA  15  NA  NA  NA  13  NA  NA
   PSTAT 177      FINANCIAL RISK  31  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA
  PSTAT 182T  ACTUARIAL TUTORIAL  12  20  15  NA  17   6   4  NA   6   4   7
   PSTAT 183  ACTUARIAL CONCEPTS  28  NA  NA  NA  44  NA  NA  NA  46  NA  NA
   PSTAT 190        ULA TRAINING  NA  NA  14  NA  NA  NA   5  NA  NA  NA  20
   PSTAT 191       ULA PRACTICUM  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA
   PSTAT 193 INTERNSHIP IN STATS   4   3   4  28   6   4   8  20   6   3   2
   PSTAT 194  GRP STDY ADV STDNT  NA  NA  NA  11  29  NA  NA  NA  28  NA  NA
   PSTAT 195      SPECIAL TOPICS  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA
   PSTAT 196   RESEARCH IN ACTSC  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA
  PSTAT 197A    DS CAPSTONE PREP  NA  NA  59  NA  NA  NA  58  NA  NA  NA  59
  PSTAT 197B DS CAPSTONE PROJECT  NA  58  NA  NA  NA  56  NA  NA  NA  53  NA
  PSTAT 197C DS CAPSTONE PROJECT  52  NA  NA  NA  53  NA  NA  NA  46  NA  NA
   PSTAT 199 INDEPENDENT STUDIES  23  22  18   6  19  17  23   6  21  17  21
 PSTAT 199RA INDEP RESEARCH ASST  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA
 M22 S22 W22
 116 194 278
  NA  NA  NA
  63 294 284
  23  59  NA
  87 278 287
  NA  NA  NA
  NA  95  88
  NA  NA  93
  NA  NA  NA
  NA  NA  81
  65 244 229
 174 216 241
 109 119  NA
  NA 137 119
  NA  NA  NA
  67 250 123
  NA  78  91
 138 122  98
  NA  75  76
  NA  NA  NA
  22  NA  66
  NA  NA  NA
  NA  NA  NA
  51 130  77
  54 109  72
  41  74  69
  49  70  68
  NA  NA  60
  NA  26  NA
  NA  NA  50
  NA  61  74
  NA  NA  NA
  NA  15  NA
  NA  NA  NA
  NA  12   5
  NA  68  NA
  NA  NA  NA
  NA  NA  NA
  35   5   4
  NA  66  NA
  NA  NA  NA
  NA  NA  NA
  NA  NA  NA
  NA  NA  46
  NA  47  NA
  10  19   5
  NA  NA  NA

      Course               Title Quarter Enrollment
    PSTAT 5A  UNDERSTANDING DATA     S25        167
    PSTAT 5H          STATISTICS     S25         11
   PSTAT 5LS  STAT LIFE SCIENCES     S25        355
     PSTAT 8  TRANS DS PROB STAT     S25         70
    PSTAT 10  DATA SCIENCE PRINC     S25        235
    PSTAT 99 INDEPENDENT STUDIES     S25          1
   PSTAT 100        DS_CONC&ANLS     S25        109
   PSTAT 105  INTRO N-PARAM METH     S25         NA
   PSTAT 109 STATISTICS FOR ECON     S25         NA
   PSTAT 115 BAYES DATA ANALYSIS     S25         81
  PSTAT 120A   PROB & STATISTICS     S25        310
  PSTAT 120B   PROB & STATISTICS     S25        312
  PSTAT 120C   PROB & STATISTICS     S25        163
   PSTAT 122 DESIGN OF EXPERMNTS     S25        240
   PSTAT 123  SAMPLING TECHNIQUE     S25         NA
   PSTAT 126 REGRESSION ANALYSIS     S25        197
   PSTAT 127    ADV STATS MODELS     S25         NA
   PSTAT 130    SAS BASE PROGRAM     S25        116
   PSTAT 131  STAT MACHINE LEARN     S25         96
   PSTAT 132 DATABASES FOR STATS     S25         NA
   PSTAT 134   STAT DATA SCIENCE     S25         46
   PSTAT 135  BIG DATA ANALYTICS     S25         96
   PSTAT 140 STATISTICAL PROCESS     S25         NA
  PSTAT 160A  STOCHASTIC PROCESS     S25        132
  PSTAT 160B  STOCHASTIC PROCESS     S25        116
   PSTAT 170  INTRO MATH FINANCE     S25        116
   PSTAT 171   MATH COMPOUND INT     S25         73
  PSTAT 172A    ACTUARIAL STAT I     S25         NA
  PSTAT 172B   ACTUARIAL STAT II     S25         14
   PSTAT 173         RISK THEORY     S25         NA
   PSTAT 174         TIME SERIES     S25         NA
   PSTAT 175   SURVIVAL ANALYSIS     S25         71
   PSTAT 176    ADV MATH FINANCE     S25         17
   PSTAT 177      FINANCIAL RISK     S25         31
  PSTAT 182T  ACTUARIAL TUTORIAL     S25         12
   PSTAT 183  ACTUARIAL CONCEPTS     S25         28
   PSTAT 190        ULA TRAINING     S25         NA
   PSTAT 191       ULA PRACTICUM     S25         NA
   PSTAT 193 INTERNSHIP IN STATS     S25          4
   PSTAT 194  GRP STDY ADV STDNT     S25         NA
   PSTAT 195      SPECIAL TOPICS     S25         NA
   PSTAT 196   RESEARCH IN ACTSC     S25         NA
  PSTAT 197A    DS CAPSTONE PREP     S25         NA
  PSTAT 197B DS CAPSTONE PROJECT     S25         NA
  PSTAT 197C DS CAPSTONE PROJECT     S25         52
   PSTAT 199 INDEPENDENT STUDIES     S25         23
 PSTAT 199RA INDEP RESEARCH ASST     S25         NA
    PSTAT 5A  UNDERSTANDING DATA     W25        222
    PSTAT 5H          STATISTICS     W25         14
   PSTAT 5LS  STAT LIFE SCIENCES     W25        340
     PSTAT 8  TRANS DS PROB STAT     W25        105
    PSTAT 10  DATA SCIENCE PRINC     W25        241
    PSTAT 99 INDEPENDENT STUDIES     W25          2
   PSTAT 100        DS_CONC&ANLS     W25        131
   PSTAT 105  INTRO N-PARAM METH     W25         79
   PSTAT 109 STATISTICS FOR ECON     W25         NA
   PSTAT 115 BAYES DATA ANALYSIS     W25         56
  PSTAT 120A   PROB & STATISTICS     W25        383
  PSTAT 120B   PROB & STATISTICS     W25        329
  PSTAT 120C   PROB & STATISTICS     W25         NA
   PSTAT 122 DESIGN OF EXPERMNTS     W25        210
   PSTAT 123  SAMPLING TECHNIQUE     W25         NA
   PSTAT 126 REGRESSION ANALYSIS     W25        214
   PSTAT 127    ADV STATS MODELS     W25         NA
   PSTAT 130    SAS BASE PROGRAM     W25        120
   PSTAT 131  STAT MACHINE LEARN     W25        156
   PSTAT 132 DATABASES FOR STATS     W25         NA
   PSTAT 134   STAT DATA SCIENCE     W25         91
   PSTAT 135  BIG DATA ANALYTICS     W25         NA
   PSTAT 140 STATISTICAL PROCESS     W25         NA
  PSTAT 160A  STOCHASTIC PROCESS     W25        217
  PSTAT 160B  STOCHASTIC PROCESS     W25         77
   PSTAT 170  INTRO MATH FINANCE     W25         65
   PSTAT 171   MATH COMPOUND INT     W25         77
  PSTAT 172A    ACTUARIAL STAT I     W25         49
  PSTAT 172B   ACTUARIAL STAT II     W25         NA
   PSTAT 173         RISK THEORY     W25         NA
   PSTAT 174         TIME SERIES     W25         89
   PSTAT 175   SURVIVAL ANALYSIS     W25         NA
   PSTAT 176    ADV MATH FINANCE     W25         NA
   PSTAT 177      FINANCIAL RISK     W25         NA
  PSTAT 182T  ACTUARIAL TUTORIAL     W25         20
   PSTAT 183  ACTUARIAL CONCEPTS     W25         NA
   PSTAT 190        ULA TRAINING     W25         NA
   PSTAT 191       ULA PRACTICUM     W25         NA
   PSTAT 193 INTERNSHIP IN STATS     W25          3
   PSTAT 194  GRP STDY ADV STDNT     W25         NA
   PSTAT 195      SPECIAL TOPICS     W25         NA
   PSTAT 196   RESEARCH IN ACTSC     W25         NA
  PSTAT 197A    DS CAPSTONE PREP     W25         NA
  PSTAT 197B DS CAPSTONE PROJECT     W25         58
  PSTAT 197C DS CAPSTONE PROJECT     W25         NA
   PSTAT 199 INDEPENDENT STUDIES     W25         22
 PSTAT 199RA INDEP RESEARCH ASST     W25         NA
    PSTAT 5A  UNDERSTANDING DATA     F24        213
    PSTAT 5H          STATISTICS     F24         30
   PSTAT 5LS  STAT LIFE SCIENCES     F24        348
     PSTAT 8  TRANS DS PROB STAT     F24        137
    PSTAT 10  DATA SCIENCE PRINC     F24        286
    PSTAT 99 INDEPENDENT STUDIES     F24         NA
   PSTAT 100        DS_CONC&ANLS     F24        108
   PSTAT 105  INTRO N-PARAM METH     F24         NA
   PSTAT 109 STATISTICS FOR ECON     F24         NA
   PSTAT 115 BAYES DATA ANALYSIS     F24         74
  PSTAT 120A   PROB & STATISTICS     F24        345
  PSTAT 120B   PROB & STATISTICS     F24        236
  PSTAT 120C   PROB & STATISTICS     F24         37
   PSTAT 122 DESIGN OF EXPERMNTS     F24        201
   PSTAT 123  SAMPLING TECHNIQUE     F24         NA
   PSTAT 126 REGRESSION ANALYSIS     F24        272
   PSTAT 127    ADV STATS MODELS     F24         72
   PSTAT 130    SAS BASE PROGRAM     F24        121
   PSTAT 131  STAT MACHINE LEARN     F24         64
   PSTAT 132 DATABASES FOR STATS     F24         NA
   PSTAT 134   STAT DATA SCIENCE     F24         78
   PSTAT 135  BIG DATA ANALYTICS     F24         NA
   PSTAT 140 STATISTICAL PROCESS     F24         NA
  PSTAT 160A  STOCHASTIC PROCESS     F24        126
  PSTAT 160B  STOCHASTIC PROCESS     F24         86
   PSTAT 170  INTRO MATH FINANCE     F24         92
   PSTAT 171   MATH COMPOUND INT     F24         96
  PSTAT 172A    ACTUARIAL STAT I     F24         NA
  PSTAT 172B   ACTUARIAL STAT II     F24         NA
   PSTAT 173         RISK THEORY     F24         64
   PSTAT 174         TIME SERIES     F24         83
   PSTAT 175   SURVIVAL ANALYSIS     F24         NA
   PSTAT 176    ADV MATH FINANCE     F24         NA
   PSTAT 177      FINANCIAL RISK     F24         NA
  PSTAT 182T  ACTUARIAL TUTORIAL     F24         15
   PSTAT 183  ACTUARIAL CONCEPTS     F24         NA
   PSTAT 190        ULA TRAINING     F24         14
   PSTAT 191       ULA PRACTICUM     F24         NA
   PSTAT 193 INTERNSHIP IN STATS     F24          4
   PSTAT 194  GRP STDY ADV STDNT     F24         NA
   PSTAT 195      SPECIAL TOPICS     F24         NA
   PSTAT 196   RESEARCH IN ACTSC     F24         NA
  PSTAT 197A    DS CAPSTONE PREP     F24         59
  PSTAT 197B DS CAPSTONE PROJECT     F24         NA
  PSTAT 197C DS CAPSTONE PROJECT     F24         NA
   PSTAT 199 INDEPENDENT STUDIES     F24         18
 PSTAT 199RA INDEP RESEARCH ASST     F24         NA
    PSTAT 5A  UNDERSTANDING DATA     M24         76
    PSTAT 5H          STATISTICS     M24         NA
   PSTAT 5LS  STAT LIFE SCIENCES     M24         59
     PSTAT 8  TRANS DS PROB STAT     M24         22
    PSTAT 10  DATA SCIENCE PRINC     M24         93
    PSTAT 99 INDEPENDENT STUDIES     M24         NA
   PSTAT 100        DS_CONC&ANLS     M24         NA
   PSTAT 105  INTRO N-PARAM METH     M24         NA
   PSTAT 109 STATISTICS FOR ECON     M24         NA
   PSTAT 115 BAYES DATA ANALYSIS     M24         25
  PSTAT 120A   PROB & STATISTICS     M24         75
  PSTAT 120B   PROB & STATISTICS     M24        136
  PSTAT 120C   PROB & STATISTICS     M24        129
   PSTAT 122 DESIGN OF EXPERMNTS     M24         NA
   PSTAT 123  SAMPLING TECHNIQUE     M24         NA
   PSTAT 126 REGRESSION ANALYSIS     M24         94
   PSTAT 127    ADV STATS MODELS     M24         NA
   PSTAT 130    SAS BASE PROGRAM     M24        108
   PSTAT 131  STAT MACHINE LEARN     M24         80
   PSTAT 132 DATABASES FOR STATS     M24         NA
   PSTAT 134   STAT DATA SCIENCE     M24         NA
   PSTAT 135  BIG DATA ANALYTICS     M24         NA
   PSTAT 140 STATISTICAL PROCESS     M24         NA
  PSTAT 160A  STOCHASTIC PROCESS     M24         NA
  PSTAT 160B  STOCHASTIC PROCESS     M24         NA
   PSTAT 170  INTRO MATH FINANCE     M24         19
   PSTAT 171   MATH COMPOUND INT     M24         47
  PSTAT 172A    ACTUARIAL STAT I     M24         NA
  PSTAT 172B   ACTUARIAL STAT II     M24         NA
   PSTAT 173         RISK THEORY     M24         NA
   PSTAT 174         TIME SERIES     M24         NA
   PSTAT 175   SURVIVAL ANALYSIS     M24         NA
   PSTAT 176    ADV MATH FINANCE     M24         NA
   PSTAT 177      FINANCIAL RISK     M24         NA
  PSTAT 182T  ACTUARIAL TUTORIAL     M24         NA
   PSTAT 183  ACTUARIAL CONCEPTS     M24         NA
   PSTAT 190        ULA TRAINING     M24         NA
   PSTAT 191       ULA PRACTICUM     M24         NA
   PSTAT 193 INTERNSHIP IN STATS     M24         28
   PSTAT 194  GRP STDY ADV STDNT     M24         11
   PSTAT 195      SPECIAL TOPICS     M24         NA
   PSTAT 196   RESEARCH IN ACTSC     M24         NA
  PSTAT 197A    DS CAPSTONE PREP     M24         NA
  PSTAT 197B DS CAPSTONE PROJECT     M24         NA
  PSTAT 197C DS CAPSTONE PROJECT     M24         NA
   PSTAT 199 INDEPENDENT STUDIES     M24          6
 PSTAT 199RA INDEP RESEARCH ASST     M24         NA
    PSTAT 5A  UNDERSTANDING DATA     S24        258
    PSTAT 5H          STATISTICS     S24         31
   PSTAT 5LS  STAT LIFE SCIENCES     S24        304
     PSTAT 8  TRANS DS PROB STAT     S24         83
    PSTAT 10  DATA SCIENCE PRINC     S24        234
    PSTAT 99 INDEPENDENT STUDIES     S24         NA
   PSTAT 100        DS_CONC&ANLS     S24         95
   PSTAT 105  INTRO N-PARAM METH     S24         NA
   PSTAT 109 STATISTICS FOR ECON     S24         NA
   PSTAT 115 BAYES DATA ANALYSIS     S24         74
  PSTAT 120A   PROB & STATISTICS     S24        367
  PSTAT 120B   PROB & STATISTICS     S24        382
  PSTAT 120C   PROB & STATISTICS     S24        131
   PSTAT 122 DESIGN OF EXPERMNTS     S24        177
   PSTAT 123  SAMPLING TECHNIQUE     S24         NA
   PSTAT 126 REGRESSION ANALYSIS     S24        147
   PSTAT 127    ADV STATS MODELS     S24         NA
   PSTAT 130    SAS BASE PROGRAM     S24        122
   PSTAT 131  STAT MACHINE LEARN     S24         93
   PSTAT 132 DATABASES FOR STATS     S24         NA
   PSTAT 134   STAT DATA SCIENCE     S24         70
   PSTAT 135  BIG DATA ANALYTICS     S24         NA
   PSTAT 140 STATISTICAL PROCESS     S24         NA
  PSTAT 160A  STOCHASTIC PROCESS     S24        100
  PSTAT 160B  STOCHASTIC PROCESS     S24        106
   PSTAT 170  INTRO MATH FINANCE     S24        125
   PSTAT 171   MATH COMPOUND INT     S24         84
  PSTAT 172A    ACTUARIAL STAT I     S24         NA
  PSTAT 172B   ACTUARIAL STAT II     S24         20
   PSTAT 173         RISK THEORY     S24         NA
   PSTAT 174         TIME SERIES     S24        104
   PSTAT 175   SURVIVAL ANALYSIS     S24         55
   PSTAT 176    ADV MATH FINANCE     S24         15
   PSTAT 177      FINANCIAL RISK     S24         NA
  PSTAT 182T  ACTUARIAL TUTORIAL     S24         17
   PSTAT 183  ACTUARIAL CONCEPTS     S24         44
   PSTAT 190        ULA TRAINING     S24         NA
   PSTAT 191       ULA PRACTICUM     S24         NA
   PSTAT 193 INTERNSHIP IN STATS     S24          6
   PSTAT 194  GRP STDY ADV STDNT     S24         29
   PSTAT 195      SPECIAL TOPICS     S24         NA
   PSTAT 196   RESEARCH IN ACTSC     S24         NA
  PSTAT 197A    DS CAPSTONE PREP     S24         NA
  PSTAT 197B DS CAPSTONE PROJECT     S24         NA
  PSTAT 197C DS CAPSTONE PROJECT     S24         53
   PSTAT 199 INDEPENDENT STUDIES     S24         19
 PSTAT 199RA INDEP RESEARCH ASST     S24         NA
    PSTAT 5A  UNDERSTANDING DATA     W24        283
    PSTAT 5H          STATISTICS     W24         24
   PSTAT 5LS  STAT LIFE SCIENCES     W24        299
     PSTAT 8  TRANS DS PROB STAT     W24        134
    PSTAT 10  DATA SCIENCE PRINC     W24        260
    PSTAT 99 INDEPENDENT STUDIES     W24         NA
   PSTAT 100        DS_CONC&ANLS     W24         NA
   PSTAT 105  INTRO N-PARAM METH     W24         NA
   PSTAT 109 STATISTICS FOR ECON     W24         NA
   PSTAT 115 BAYES DATA ANALYSIS     W24         96
  PSTAT 120A   PROB & STATISTICS     W24        372
  PSTAT 120B   PROB & STATISTICS     W24        241
  PSTAT 120C   PROB & STATISTICS     W24         39
   PSTAT 122 DESIGN OF EXPERMNTS     W24        162
   PSTAT 123  SAMPLING TECHNIQUE     W24        137
   PSTAT 126 REGRESSION ANALYSIS     W24        200
   PSTAT 127    ADV STATS MODELS     W24         NA
   PSTAT 130    SAS BASE PROGRAM     W24         97
   PSTAT 131  STAT MACHINE LEARN     W24        138
   PSTAT 132 DATABASES FOR STATS     W24         NA
   PSTAT 134   STAT DATA SCIENCE     W24         78
   PSTAT 135  BIG DATA ANALYTICS     W24         99
   PSTAT 140 STATISTICAL PROCESS     W24         NA
  PSTAT 160A  STOCHASTIC PROCESS     W24        153
  PSTAT 160B  STOCHASTIC PROCESS     W24         73
   PSTAT 170  INTRO MATH FINANCE     W24         65
   PSTAT 171   MATH COMPOUND INT     W24         68
  PSTAT 172A    ACTUARIAL STAT I     W24         60
  PSTAT 172B   ACTUARIAL STAT II     W24         NA
   PSTAT 173         RISK THEORY     W24         NA
   PSTAT 174         TIME SERIES     W24         63
   PSTAT 175   SURVIVAL ANALYSIS     W24         NA
   PSTAT 176    ADV MATH FINANCE     W24         NA
   PSTAT 177      FINANCIAL RISK     W24         NA
  PSTAT 182T  ACTUARIAL TUTORIAL     W24          6
   PSTAT 183  ACTUARIAL CONCEPTS     W24         NA
   PSTAT 190        ULA TRAINING     W24         NA
   PSTAT 191       ULA PRACTICUM     W24         NA
   PSTAT 193 INTERNSHIP IN STATS     W24          4
   PSTAT 194  GRP STDY ADV STDNT     W24         NA
   PSTAT 195      SPECIAL TOPICS     W24         NA
   PSTAT 196   RESEARCH IN ACTSC     W24         NA
  PSTAT 197A    DS CAPSTONE PREP     W24         NA
  PSTAT 197B DS CAPSTONE PROJECT     W24         56
  PSTAT 197C DS CAPSTONE PROJECT     W24         NA
   PSTAT 199 INDEPENDENT STUDIES     W24         17
 PSTAT 199RA INDEP RESEARCH ASST     W24         NA
    PSTAT 5A  UNDERSTANDING DATA     F23        549
    PSTAT 5H          STATISTICS     F23         25
   PSTAT 5LS  STAT LIFE SCIENCES     F23         NA
     PSTAT 8  TRANS DS PROB STAT     F23        132
    PSTAT 10  DATA SCIENCE PRINC     F23        285
    PSTAT 99 INDEPENDENT STUDIES     F23         NA
   PSTAT 100        DS_CONC&ANLS     F23        104
   PSTAT 105  INTRO N-PARAM METH     F23         39
   PSTAT 109 STATISTICS FOR ECON     F23         NA
   PSTAT 115 BAYES DATA ANALYSIS     F23         16
  PSTAT 120A   PROB & STATISTICS     F23        419
  PSTAT 120B   PROB & STATISTICS     F23        261
  PSTAT 120C   PROB & STATISTICS     F23         19
   PSTAT 122 DESIGN OF EXPERMNTS     F23        167
   PSTAT 123  SAMPLING TECHNIQUE     F23         NA
   PSTAT 126 REGRESSION ANALYSIS     F23        123
   PSTAT 127    ADV STATS MODELS     F23         57
   PSTAT 130    SAS BASE PROGRAM     F23        120
   PSTAT 131  STAT MACHINE LEARN     F23        150
   PSTAT 132 DATABASES FOR STATS     F23         NA
   PSTAT 134   STAT DATA SCIENCE     F23         80
   PSTAT 135  BIG DATA ANALYTICS     F23         NA
   PSTAT 140 STATISTICAL PROCESS     F23         NA
  PSTAT 160A  STOCHASTIC PROCESS     F23        132
  PSTAT 160B  STOCHASTIC PROCESS     F23         43
   PSTAT 170  INTRO MATH FINANCE     F23         59
   PSTAT 171   MATH COMPOUND INT     F23        102
  PSTAT 172A    ACTUARIAL STAT I     F23         NA
  PSTAT 172B   ACTUARIAL STAT II     F23         NA
   PSTAT 173         RISK THEORY     F23         35
   PSTAT 174         TIME SERIES     F23         NA
   PSTAT 175   SURVIVAL ANALYSIS     F23         18
   PSTAT 176    ADV MATH FINANCE     F23         NA
   PSTAT 177      FINANCIAL RISK     F23         NA
  PSTAT 182T  ACTUARIAL TUTORIAL     F23          4
   PSTAT 183  ACTUARIAL CONCEPTS     F23         NA
   PSTAT 190        ULA TRAINING     F23          5
   PSTAT 191       ULA PRACTICUM     F23         NA
   PSTAT 193 INTERNSHIP IN STATS     F23          8
   PSTAT 194  GRP STDY ADV STDNT     F23         NA
   PSTAT 195      SPECIAL TOPICS     F23         NA
   PSTAT 196   RESEARCH IN ACTSC     F23         NA
  PSTAT 197A    DS CAPSTONE PREP     F23         58
  PSTAT 197B DS CAPSTONE PROJECT     F23         NA
  PSTAT 197C DS CAPSTONE PROJECT     F23         NA
   PSTAT 199 INDEPENDENT STUDIES     F23         23
 PSTAT 199RA INDEP RESEARCH ASST     F23         NA
    PSTAT 5A  UNDERSTANDING DATA     M23        110
    PSTAT 5H          STATISTICS     M23         NA
   PSTAT 5LS  STAT LIFE SCIENCES     M23         39
     PSTAT 8  TRANS DS PROB STAT     M23         18
    PSTAT 10  DATA SCIENCE PRINC     M23         83
    PSTAT 99 INDEPENDENT STUDIES     M23         NA
   PSTAT 100        DS_CONC&ANLS     M23         48
   PSTAT 105  INTRO N-PARAM METH     M23         NA
   PSTAT 109 STATISTICS FOR ECON     M23         NA
   PSTAT 115 BAYES DATA ANALYSIS     M23         15
  PSTAT 120A   PROB & STATISTICS     M23         86
  PSTAT 120B   PROB & STATISTICS     M23        102
  PSTAT 120C   PROB & STATISTICS     M23         81
   PSTAT 122 DESIGN OF EXPERMNTS     M23         NA
   PSTAT 123  SAMPLING TECHNIQUE     M23         NA
   PSTAT 126 REGRESSION ANALYSIS     M23         84
   PSTAT 127    ADV STATS MODELS     M23         NA
   PSTAT 130    SAS BASE PROGRAM     M23         95
   PSTAT 131  STAT MACHINE LEARN     M23         55
   PSTAT 132 DATABASES FOR STATS     M23         NA
   PSTAT 134   STAT DATA SCIENCE     M23         NA
   PSTAT 135  BIG DATA ANALYTICS     M23         NA
   PSTAT 140 STATISTICAL PROCESS     M23         NA
  PSTAT 160A  STOCHASTIC PROCESS     M23         NA
  PSTAT 160B  STOCHASTIC PROCESS     M23          7
   PSTAT 170  INTRO MATH FINANCE     M23         17
   PSTAT 171   MATH COMPOUND INT     M23         32
  PSTAT 172A    ACTUARIAL STAT I     M23         NA
  PSTAT 172B   ACTUARIAL STAT II     M23         NA
   PSTAT 173         RISK THEORY     M23         NA
   PSTAT 174         TIME SERIES     M23         NA
   PSTAT 175   SURVIVAL ANALYSIS     M23         NA
   PSTAT 176    ADV MATH FINANCE     M23         NA
   PSTAT 177      FINANCIAL RISK     M23         NA
  PSTAT 182T  ACTUARIAL TUTORIAL     M23         NA
   PSTAT 183  ACTUARIAL CONCEPTS     M23         NA
   PSTAT 190        ULA TRAINING     M23         NA
   PSTAT 191       ULA PRACTICUM     M23         NA
   PSTAT 193 INTERNSHIP IN STATS     M23         20
   PSTAT 194  GRP STDY ADV STDNT     M23         NA
   PSTAT 195      SPECIAL TOPICS     M23         NA
   PSTAT 196   RESEARCH IN ACTSC     M23         NA
  PSTAT 197A    DS CAPSTONE PREP     M23         NA
  PSTAT 197B DS CAPSTONE PROJECT     M23         NA
  PSTAT 197C DS CAPSTONE PROJECT     M23         NA
   PSTAT 199 INDEPENDENT STUDIES     M23          6
 PSTAT 199RA INDEP RESEARCH ASST     M23         NA
    PSTAT 5A  UNDERSTANDING DATA     S23        212
    PSTAT 5H          STATISTICS     S23         31
   PSTAT 5LS  STAT LIFE SCIENCES     S23        299
     PSTAT 8  TRANS DS PROB STAT     S23         86
    PSTAT 10  DATA SCIENCE PRINC     S23        240
    PSTAT 99 INDEPENDENT STUDIES     S23         NA
   PSTAT 100        DS_CONC&ANLS     S23        125
   PSTAT 105  INTRO N-PARAM METH     S23         NA
   PSTAT 109 STATISTICS FOR ECON     S23         NA
   PSTAT 115 BAYES DATA ANALYSIS     S23         38
  PSTAT 120A   PROB & STATISTICS     S23        237
  PSTAT 120B   PROB & STATISTICS     S23        256
  PSTAT 120C   PROB & STATISTICS     S23         65
   PSTAT 122 DESIGN OF EXPERMNTS     S23        194
   PSTAT 123  SAMPLING TECHNIQUE     S23         NA
   PSTAT 126 REGRESSION ANALYSIS     S23        149
   PSTAT 127    ADV STATS MODELS     S23         NA
   PSTAT 130    SAS BASE PROGRAM     S23        119
   PSTAT 131  STAT MACHINE LEARN     S23        108
   PSTAT 132 DATABASES FOR STATS     S23         NA
   PSTAT 134   STAT DATA SCIENCE     S23         87
   PSTAT 135  BIG DATA ANALYTICS     S23         NA
   PSTAT 140 STATISTICAL PROCESS     S23         NA
  PSTAT 160A  STOCHASTIC PROCESS     S23        108
  PSTAT 160B  STOCHASTIC PROCESS     S23         82
   PSTAT 170  INTRO MATH FINANCE     S23         77
   PSTAT 171   MATH COMPOUND INT     S23         64
  PSTAT 172A    ACTUARIAL STAT I     S23         NA
  PSTAT 172B   ACTUARIAL STAT II     S23         23
   PSTAT 173         RISK THEORY     S23         NA
   PSTAT 174         TIME SERIES     S23        170
   PSTAT 175   SURVIVAL ANALYSIS     S23         NA
   PSTAT 176    ADV MATH FINANCE     S23         13
   PSTAT 177      FINANCIAL RISK     S23         NA
  PSTAT 182T  ACTUARIAL TUTORIAL     S23          6
   PSTAT 183  ACTUARIAL CONCEPTS     S23         46
   PSTAT 190        ULA TRAINING     S23         NA
   PSTAT 191       ULA PRACTICUM     S23         NA
   PSTAT 193 INTERNSHIP IN STATS     S23          6
   PSTAT 194  GRP STDY ADV STDNT     S23         28
   PSTAT 195      SPECIAL TOPICS     S23         NA
   PSTAT 196   RESEARCH IN ACTSC     S23         NA
  PSTAT 197A    DS CAPSTONE PREP     S23         NA
  PSTAT 197B DS CAPSTONE PROJECT     S23         NA
  PSTAT 197C DS CAPSTONE PROJECT     S23         46
   PSTAT 199 INDEPENDENT STUDIES     S23         21
 PSTAT 199RA INDEP RESEARCH ASST     S23         NA
    PSTAT 5A  UNDERSTANDING DATA     W23        279
    PSTAT 5H          STATISTICS     W23         22
   PSTAT 5LS  STAT LIFE SCIENCES     W23        294
     PSTAT 8  TRANS DS PROB STAT     W23        156
    PSTAT 10  DATA SCIENCE PRINC     W23        282
    PSTAT 99 INDEPENDENT STUDIES     W23         NA
   PSTAT 100        DS_CONC&ANLS     W23        127
   PSTAT 105  INTRO N-PARAM METH     W23         NA
   PSTAT 109 STATISTICS FOR ECON     W23         NA
   PSTAT 115 BAYES DATA ANALYSIS     W23         70
  PSTAT 120A   PROB & STATISTICS     W23        246
  PSTAT 120B   PROB & STATISTICS     W23        256
  PSTAT 120C   PROB & STATISTICS     W23         31
   PSTAT 122 DESIGN OF EXPERMNTS     W23        187
   PSTAT 123  SAMPLING TECHNIQUE     W23         NA
   PSTAT 126 REGRESSION ANALYSIS     W23        175
   PSTAT 127    ADV STATS MODELS     W23         73
   PSTAT 130    SAS BASE PROGRAM     W23         98
   PSTAT 131  STAT MACHINE LEARN     W23        197
   PSTAT 132 DATABASES FOR STATS     W23         NA
   PSTAT 134   STAT DATA SCIENCE     W23         42
   PSTAT 135  BIG DATA ANALYTICS     W23         93
   PSTAT 140 STATISTICAL PROCESS     W23         NA
  PSTAT 160A  STOCHASTIC PROCESS     W23        161
  PSTAT 160B  STOCHASTIC PROCESS     W23         81
   PSTAT 170  INTRO MATH FINANCE     W23         57
   PSTAT 171   MATH COMPOUND INT     W23         69
  PSTAT 172A    ACTUARIAL STAT I     W23         43
  PSTAT 172B   ACTUARIAL STAT II     W23         NA
   PSTAT 173         RISK THEORY     W23         NA
   PSTAT 174         TIME SERIES     W23        118
   PSTAT 175   SURVIVAL ANALYSIS     W23         NA
   PSTAT 176    ADV MATH FINANCE     W23         NA
   PSTAT 177      FINANCIAL RISK     W23         NA
  PSTAT 182T  ACTUARIAL TUTORIAL     W23          4
   PSTAT 183  ACTUARIAL CONCEPTS     W23         NA
   PSTAT 190        ULA TRAINING     W23         NA
   PSTAT 191       ULA PRACTICUM     W23         NA
   PSTAT 193 INTERNSHIP IN STATS     W23          3
   PSTAT 194  GRP STDY ADV STDNT     W23         NA
   PSTAT 195      SPECIAL TOPICS     W23         NA
   PSTAT 196   RESEARCH IN ACTSC     W23         NA
  PSTAT 197A    DS CAPSTONE PREP     W23         NA
  PSTAT 197B DS CAPSTONE PROJECT     W23         53
  PSTAT 197C DS CAPSTONE PROJECT     W23         NA
   PSTAT 199 INDEPENDENT STUDIES     W23         17
 PSTAT 199RA INDEP RESEARCH ASST     W23         NA
    PSTAT 5A  UNDERSTANDING DATA     F22        499
    PSTAT 5H          STATISTICS     F22          4
   PSTAT 5LS  STAT LIFE SCIENCES     F22         NA
     PSTAT 8  TRANS DS PROB STAT     F22         70
    PSTAT 10  DATA SCIENCE PRINC     F22        257
    PSTAT 99 INDEPENDENT STUDIES     F22         NA
   PSTAT 100        DS_CONC&ANLS     F22         NA
   PSTAT 105  INTRO N-PARAM METH     F22         79
   PSTAT 109 STATISTICS FOR ECON     F22         NA
   PSTAT 115 BAYES DATA ANALYSIS     F22         63
  PSTAT 120A   PROB & STATISTICS     F22        419
  PSTAT 120B   PROB & STATISTICS     F22        165
  PSTAT 120C   PROB & STATISTICS     F22         87
   PSTAT 122 DESIGN OF EXPERMNTS     F22        182
   PSTAT 123  SAMPLING TECHNIQUE     F22         NA
   PSTAT 126 REGRESSION ANALYSIS     F22        182
   PSTAT 127    ADV STATS MODELS     F22         NA
   PSTAT 130    SAS BASE PROGRAM     F22         97
   PSTAT 131  STAT MACHINE LEARN     F22        178
   PSTAT 132 DATABASES FOR STATS     F22         NA
   PSTAT 134   STAT DATA SCIENCE     F22         33
   PSTAT 135  BIG DATA ANALYTICS     F22         NA
   PSTAT 140 STATISTICAL PROCESS     F22         NA
  PSTAT 160A  STOCHASTIC PROCESS     F22        184
  PSTAT 160B  STOCHASTIC PROCESS     F22         58
   PSTAT 170  INTRO MATH FINANCE     F22         55
   PSTAT 171   MATH COMPOUND INT     F22        108
  PSTAT 172A    ACTUARIAL STAT I     F22         NA
  PSTAT 172B   ACTUARIAL STAT II     F22         NA
   PSTAT 173         RISK THEORY     F22         43
   PSTAT 174         TIME SERIES     F22         69
   PSTAT 175   SURVIVAL ANALYSIS     F22         73
   PSTAT 176    ADV MATH FINANCE     F22         NA
   PSTAT 177      FINANCIAL RISK     F22         NA
  PSTAT 182T  ACTUARIAL TUTORIAL     F22          7
   PSTAT 183  ACTUARIAL CONCEPTS     F22         NA
   PSTAT 190        ULA TRAINING     F22         20
   PSTAT 191       ULA PRACTICUM     F22         NA
   PSTAT 193 INTERNSHIP IN STATS     F22          2
   PSTAT 194  GRP STDY ADV STDNT     F22         NA
   PSTAT 195      SPECIAL TOPICS     F22         NA
   PSTAT 196   RESEARCH IN ACTSC     F22         NA
  PSTAT 197A    DS CAPSTONE PREP     F22         59
  PSTAT 197B DS CAPSTONE PROJECT     F22         NA
  PSTAT 197C DS CAPSTONE PROJECT     F22         NA
   PSTAT 199 INDEPENDENT STUDIES     F22         21
 PSTAT 199RA INDEP RESEARCH ASST     F22         NA
    PSTAT 5A  UNDERSTANDING DATA     M22        116
    PSTAT 5H          STATISTICS     M22         NA
   PSTAT 5LS  STAT LIFE SCIENCES     M22         63
     PSTAT 8  TRANS DS PROB STAT     M22         23
    PSTAT 10  DATA SCIENCE PRINC     M22         87
    PSTAT 99 INDEPENDENT STUDIES     M22         NA
   PSTAT 100        DS_CONC&ANLS     M22         NA
   PSTAT 105  INTRO N-PARAM METH     M22         NA
   PSTAT 109 STATISTICS FOR ECON     M22         NA
   PSTAT 115 BAYES DATA ANALYSIS     M22         NA
  PSTAT 120A   PROB & STATISTICS     M22         65
  PSTAT 120B   PROB & STATISTICS     M22        174
  PSTAT 120C   PROB & STATISTICS     M22        109
   PSTAT 122 DESIGN OF EXPERMNTS     M22         NA
   PSTAT 123  SAMPLING TECHNIQUE     M22         NA
   PSTAT 126 REGRESSION ANALYSIS     M22         67
   PSTAT 127    ADV STATS MODELS     M22         NA
   PSTAT 130    SAS BASE PROGRAM     M22        138
   PSTAT 131  STAT MACHINE LEARN     M22         NA
   PSTAT 132 DATABASES FOR STATS     M22         NA
   PSTAT 134   STAT DATA SCIENCE     M22         22
   PSTAT 135  BIG DATA ANALYTICS     M22         NA
   PSTAT 140 STATISTICAL PROCESS     M22         NA
  PSTAT 160A  STOCHASTIC PROCESS     M22         51
  PSTAT 160B  STOCHASTIC PROCESS     M22         54
   PSTAT 170  INTRO MATH FINANCE     M22         41
   PSTAT 171   MATH COMPOUND INT     M22         49
  PSTAT 172A    ACTUARIAL STAT I     M22         NA
  PSTAT 172B   ACTUARIAL STAT II     M22         NA
   PSTAT 173         RISK THEORY     M22         NA
   PSTAT 174         TIME SERIES     M22         NA
   PSTAT 175   SURVIVAL ANALYSIS     M22         NA
   PSTAT 176    ADV MATH FINANCE     M22         NA
   PSTAT 177      FINANCIAL RISK     M22         NA
  PSTAT 182T  ACTUARIAL TUTORIAL     M22         NA
   PSTAT 183  ACTUARIAL CONCEPTS     M22         NA
   PSTAT 190        ULA TRAINING     M22         NA
   PSTAT 191       ULA PRACTICUM     M22         NA
   PSTAT 193 INTERNSHIP IN STATS     M22         35
   PSTAT 194  GRP STDY ADV STDNT     M22         NA
   PSTAT 195      SPECIAL TOPICS     M22         NA
   PSTAT 196   RESEARCH IN ACTSC     M22         NA
  PSTAT 197A    DS CAPSTONE PREP     M22         NA
  PSTAT 197B DS CAPSTONE PROJECT     M22         NA
  PSTAT 197C DS CAPSTONE PROJECT     M22         NA
   PSTAT 199 INDEPENDENT STUDIES     M22         10
 PSTAT 199RA INDEP RESEARCH ASST     M22         NA
    PSTAT 5A  UNDERSTANDING DATA     S22        194
    PSTAT 5H          STATISTICS     S22         NA
   PSTAT 5LS  STAT LIFE SCIENCES     S22        294
     PSTAT 8  TRANS DS PROB STAT     S22         59
    PSTAT 10  DATA SCIENCE PRINC     S22        278
    PSTAT 99 INDEPENDENT STUDIES     S22         NA
   PSTAT 100        DS_CONC&ANLS     S22         95
   PSTAT 105  INTRO N-PARAM METH     S22         NA
   PSTAT 109 STATISTICS FOR ECON     S22         NA
   PSTAT 115 BAYES DATA ANALYSIS     S22         NA
  PSTAT 120A   PROB & STATISTICS     S22        244
  PSTAT 120B   PROB & STATISTICS     S22        216
  PSTAT 120C   PROB & STATISTICS     S22        119
   PSTAT 122 DESIGN OF EXPERMNTS     S22        137
   PSTAT 123  SAMPLING TECHNIQUE     S22         NA
   PSTAT 126 REGRESSION ANALYSIS     S22        250
   PSTAT 127    ADV STATS MODELS     S22         78
   PSTAT 130    SAS BASE PROGRAM     S22        122
   PSTAT 131  STAT MACHINE LEARN     S22         75
   PSTAT 132 DATABASES FOR STATS     S22         NA
   PSTAT 134   STAT DATA SCIENCE     S22         NA
   PSTAT 135  BIG DATA ANALYTICS     S22         NA
   PSTAT 140 STATISTICAL PROCESS     S22         NA
  PSTAT 160A  STOCHASTIC PROCESS     S22        130
  PSTAT 160B  STOCHASTIC PROCESS     S22        109
   PSTAT 170  INTRO MATH FINANCE     S22         74
   PSTAT 171   MATH COMPOUND INT     S22         70
  PSTAT 172A    ACTUARIAL STAT I     S22         NA
  PSTAT 172B   ACTUARIAL STAT II     S22         26
   PSTAT 173         RISK THEORY     S22         NA
   PSTAT 174         TIME SERIES     S22         61
   PSTAT 175   SURVIVAL ANALYSIS     S22         NA
   PSTAT 176    ADV MATH FINANCE     S22         15
   PSTAT 177      FINANCIAL RISK     S22         NA
  PSTAT 182T  ACTUARIAL TUTORIAL     S22         12
   PSTAT 183  ACTUARIAL CONCEPTS     S22         68
   PSTAT 190        ULA TRAINING     S22         NA
   PSTAT 191       ULA PRACTICUM     S22         NA
   PSTAT 193 INTERNSHIP IN STATS     S22          5
   PSTAT 194  GRP STDY ADV STDNT     S22         66
   PSTAT 195      SPECIAL TOPICS     S22         NA
   PSTAT 196   RESEARCH IN ACTSC     S22         NA
  PSTAT 197A    DS CAPSTONE PREP     S22         NA
  PSTAT 197B DS CAPSTONE PROJECT     S22         NA
  PSTAT 197C DS CAPSTONE PROJECT     S22         47
   PSTAT 199 INDEPENDENT STUDIES     S22         19
 PSTAT 199RA INDEP RESEARCH ASST     S22         NA
    PSTAT 5A  UNDERSTANDING DATA     W22        278
    PSTAT 5H          STATISTICS     W22         NA
   PSTAT 5LS  STAT LIFE SCIENCES     W22        284
     PSTAT 8  TRANS DS PROB STAT     W22         NA
    PSTAT 10  DATA SCIENCE PRINC     W22        287
    PSTAT 99 INDEPENDENT STUDIES     W22         NA
   PSTAT 100        DS_CONC&ANLS     W22         88
   PSTAT 105  INTRO N-PARAM METH     W22         93
   PSTAT 109 STATISTICS FOR ECON     W22         NA
   PSTAT 115 BAYES DATA ANALYSIS     W22         81
  PSTAT 120A   PROB & STATISTICS     W22        229
  PSTAT 120B   PROB & STATISTICS     W22        241
  PSTAT 120C   PROB & STATISTICS     W22         NA
   PSTAT 122 DESIGN OF EXPERMNTS     W22        119
   PSTAT 123  SAMPLING TECHNIQUE     W22         NA
   PSTAT 126 REGRESSION ANALYSIS     W22        123
   PSTAT 127    ADV STATS MODELS     W22         91
   PSTAT 130    SAS BASE PROGRAM     W22         98
   PSTAT 131  STAT MACHINE LEARN     W22         76
   PSTAT 132 DATABASES FOR STATS     W22         NA
   PSTAT 134   STAT DATA SCIENCE     W22         66
   PSTAT 135  BIG DATA ANALYTICS     W22         NA
   PSTAT 140 STATISTICAL PROCESS     W22         NA
  PSTAT 160A  STOCHASTIC PROCESS     W22         77
  PSTAT 160B  STOCHASTIC PROCESS     W22         72
   PSTAT 170  INTRO MATH FINANCE     W22         69
   PSTAT 171   MATH COMPOUND INT     W22         68
  PSTAT 172A    ACTUARIAL STAT I     W22         60
  PSTAT 172B   ACTUARIAL STAT II     W22         NA
   PSTAT 173         RISK THEORY     W22         50
   PSTAT 174         TIME SERIES     W22         74
   PSTAT 175   SURVIVAL ANALYSIS     W22         NA
   PSTAT 176    ADV MATH FINANCE     W22         NA
   PSTAT 177      FINANCIAL RISK     W22         NA
  PSTAT 182T  ACTUARIAL TUTORIAL     W22          5
   PSTAT 183  ACTUARIAL CONCEPTS     W22         NA
   PSTAT 190        ULA TRAINING     W22         NA
   PSTAT 191       ULA PRACTICUM     W22         NA
   PSTAT 193 INTERNSHIP IN STATS     W22          4
   PSTAT 194  GRP STDY ADV STDNT     W22         NA
   PSTAT 195      SPECIAL TOPICS     W22         NA
   PSTAT 196   RESEARCH IN ACTSC     W22         NA
  PSTAT 197A    DS CAPSTONE PREP     W22         NA
  PSTAT 197B DS CAPSTONE PROJECT     W22         46
  PSTAT 197C DS CAPSTONE PROJECT     W22         NA
   PSTAT 199 INDEPENDENT STUDIES     W22          5
 PSTAT 199RA INDEP RESEARCH ASST     W22         NA

Data

Structure vs. Semantics

Recap

Data has both semantics and structure (though when most people say “data” they are typically refering to the semantics).

Both the wide and long layouts of the enrollments dataset contain the semantics, but have different structures.
Loosely speaking, the structure of a given dataset refers to the way the values in the dataset are actually displayed.
- Specification of rows, columns, and number of tables (yes, sometimes we need multiple tables to express a particular dataset!)
- Decision of how to encode particular values (e.g. should we use high, medium, low, or 3, 2, 1?)

Data

Structure vs. Semantics

There is some variability “in the wild” when it comes to the structure of datasets, and there isn’t always a single “best” way to structure a given dataset.
However, it is important to note that computer will only ever be able to understand the structure of a dataset: it can only read rows, columns, and entries.
So, how can we get a computer to understand the semantics of a dataset?
Idea: use a structure that maps to the semantics in some way.
- Indeed, one popular framework of structuring data that achieves this is the so-called Tidy framework of data.

Tidy Data

Definition

Definition

Developed by Hadley Wickham, the tidy standard of data seeks to map the semantics of a dataset to its structure, by proposing a series of structural constraints:

Each variable forms a column
Each observation forms a row
Each type of observational unit forms a table

Tidy or Not?

`Enrollments`: Wide Layout

      Course               Title S25 W25 F24 M24 S24 W24 F23 M23 S23 W23 F22
    PSTAT 5A  UNDERSTANDING DATA 167 222 213  76 258 283 549 110 212 279 499
    PSTAT 5H          STATISTICS  11  14  30  NA  31  24  25  NA  31  22   4
   PSTAT 5LS  STAT LIFE SCIENCES 355 340 348  59 304 299  NA  39 299 294  NA
     PSTAT 8  TRANS DS PROB STAT  70 105 137  22  83 134 132  18  86 156  70
    PSTAT 10  DATA SCIENCE PRINC 235 241 286  93 234 260 285  83 240 282 257
    PSTAT 99 INDEPENDENT STUDIES   1   2  NA  NA  NA  NA  NA  NA  NA  NA  NA
   PSTAT 100        DS_CONC&ANLS 109 131 108  NA  95  NA 104  48 125 127  NA
   PSTAT 105  INTRO N-PARAM METH  NA  79  NA  NA  NA  NA  39  NA  NA  NA  79
   PSTAT 109 STATISTICS FOR ECON  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA
   PSTAT 115 BAYES DATA ANALYSIS  81  56  74  25  74  96  16  15  38  70  63
  PSTAT 120A   PROB & STATISTICS 310 383 345  75 367 372 419  86 237 246 419
  PSTAT 120B   PROB & STATISTICS 312 329 236 136 382 241 261 102 256 256 165
  PSTAT 120C   PROB & STATISTICS 163  NA  37 129 131  39  19  81  65  31  87
   PSTAT 122 DESIGN OF EXPERMNTS 240 210 201  NA 177 162 167  NA 194 187 182
   PSTAT 123  SAMPLING TECHNIQUE  NA  NA  NA  NA  NA 137  NA  NA  NA  NA  NA
   PSTAT 126 REGRESSION ANALYSIS 197 214 272  94 147 200 123  84 149 175 182
   PSTAT 127    ADV STATS MODELS  NA  NA  72  NA  NA  NA  57  NA  NA  73  NA
   PSTAT 130    SAS BASE PROGRAM 116 120 121 108 122  97 120  95 119  98  97
   PSTAT 131  STAT MACHINE LEARN  96 156  64  80  93 138 150  55 108 197 178
   PSTAT 132 DATABASES FOR STATS  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA
   PSTAT 134   STAT DATA SCIENCE  46  91  78  NA  70  78  80  NA  87  42  33
   PSTAT 135  BIG DATA ANALYTICS  96  NA  NA  NA  NA  99  NA  NA  NA  93  NA
   PSTAT 140 STATISTICAL PROCESS  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA
  PSTAT 160A  STOCHASTIC PROCESS 132 217 126  NA 100 153 132  NA 108 161 184
  PSTAT 160B  STOCHASTIC PROCESS 116  77  86  NA 106  73  43   7  82  81  58
   PSTAT 170  INTRO MATH FINANCE 116  65  92  19 125  65  59  17  77  57  55
   PSTAT 171   MATH COMPOUND INT  73  77  96  47  84  68 102  32  64  69 108
  PSTAT 172A    ACTUARIAL STAT I  NA  49  NA  NA  NA  60  NA  NA  NA  43  NA
  PSTAT 172B   ACTUARIAL STAT II  14  NA  NA  NA  20  NA  NA  NA  23  NA  NA
   PSTAT 173         RISK THEORY  NA  NA  64  NA  NA  NA  35  NA  NA  NA  43
   PSTAT 174         TIME SERIES  NA  89  83  NA 104  63  NA  NA 170 118  69
   PSTAT 175   SURVIVAL ANALYSIS  71  NA  NA  NA  55  NA  18  NA  NA  NA  73
   PSTAT 176    ADV MATH FINANCE  17  NA  NA  NA  15  NA  NA  NA  13  NA  NA
   PSTAT 177      FINANCIAL RISK  31  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA
  PSTAT 182T  ACTUARIAL TUTORIAL  12  20  15  NA  17   6   4  NA   6   4   7
   PSTAT 183  ACTUARIAL CONCEPTS  28  NA  NA  NA  44  NA  NA  NA  46  NA  NA
   PSTAT 190        ULA TRAINING  NA  NA  14  NA  NA  NA   5  NA  NA  NA  20
   PSTAT 191       ULA PRACTICUM  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA
   PSTAT 193 INTERNSHIP IN STATS   4   3   4  28   6   4   8  20   6   3   2
   PSTAT 194  GRP STDY ADV STDNT  NA  NA  NA  11  29  NA  NA  NA  28  NA  NA
   PSTAT 195      SPECIAL TOPICS  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA
   PSTAT 196   RESEARCH IN ACTSC  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA
  PSTAT 197A    DS CAPSTONE PREP  NA  NA  59  NA  NA  NA  58  NA  NA  NA  59
  PSTAT 197B DS CAPSTONE PROJECT  NA  58  NA  NA  NA  56  NA  NA  NA  53  NA
  PSTAT 197C DS CAPSTONE PROJECT  52  NA  NA  NA  53  NA  NA  NA  46  NA  NA
   PSTAT 199 INDEPENDENT STUDIES  23  22  18   6  19  17  23   6  21  17  21
 PSTAT 199RA INDEP RESEARCH ASST  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA  NA
 M22 S22 W22
 116 194 278
  NA  NA  NA
  63 294 284
  23  59  NA
  87 278 287
  NA  NA  NA
  NA  95  88
  NA  NA  93
  NA  NA  NA
  NA  NA  81
  65 244 229
 174 216 241
 109 119  NA
  NA 137 119
  NA  NA  NA
  67 250 123
  NA  78  91
 138 122  98
  NA  75  76
  NA  NA  NA
  22  NA  66
  NA  NA  NA
  NA  NA  NA
  51 130  77
  54 109  72
  41  74  69
  49  70  68
  NA  NA  60
  NA  26  NA
  NA  NA  50
  NA  61  74
  NA  NA  NA
  NA  15  NA
  NA  NA  NA
  NA  12   5
  NA  68  NA
  NA  NA  NA
  NA  NA  NA
  35   5   4
  NA  66  NA
  NA  NA  NA
  NA  NA  NA
  NA  NA  NA
  NA  NA  46
  NA  47  NA
  10  19   5
  NA  NA  NA

Semantics

Observations:	Enrollment in a Course in a given Quarter
Variables:	Course number, title, quarter, and enrollment count
Observational Units:	(UCSB PSTAT Undergrad) Courses

Structure

Rows:	Course-by-Course Records
Columns:	Value of Quarter
# of Tables:	1

Rules 1 and 2 violated; not tidy

Tidy or Not?

`Enrollments`: Long Layout

    Course              Title Quarter Enrollment
  PSTAT 5A UNDERSTANDING DATA     S25        167
  PSTAT 5H         STATISTICS     S25         11
 PSTAT 5LS STAT LIFE SCIENCES     S25        355

Semantics

Observations:	Enrollment in a Course in a given Quarter
Variables:	Course number, title, quarter, and enrollment count
Observational Units:	(UCSB PSTAT Undergrad) Courses

Structure

Rows:	Enrollment in a Course in a given Quarter
Columns:	Course number, title, quarter, and enrollment count
# of Tables:	1

All rules satisfied; tidy

Tidy or Not?

A Brief Interlude

Before we proceed, I’d like to admit that there’s some subjectivity going on behind the scenes.
For example, you might ask: with regards to the enrollments dataset, what if we adopted the following variables:
- Course
- Title
- Spring 2025 Enrollment
- Winter 2025 Enrollment
- Etc.
Now the wide layout becomes tidy - what’s going on?!

Tidy or Not?

A Brief Interlude

Firstly, I’ll stress this again: I view the main point of the tidy framework as a way to link the semantics and the structure of a dataset.
From that point of view, there is nothing inherently “incorrect” about modifying our variables and observations as we did on the previous slide to make the wide layout tidy.
However, I believe it “puts the cart before the horse” - in the field, we typically start with the semantics, and then structure the data around the semantics. Changing the semantics to match the structure (though not incorrect) is a little backwards.

Tidy or Not?

A Brief Interlude

There’s another reason adopting these second set of variables is not the best idea, and it has to do with the notion of updating a data frame.
Suppose we gain additional data, from Summer 2025.
- In the second set of semantics, we’d have to add an entirely new variable to incorporate this new information.
- In the first set of semantics (where we simply have a Quarter variable and an Enrollment variable), we can update out dataset by simply adding a new observation (which is much more efficient.)

“[…] it is surprisingly difficult to precisely define variables and observations in general.” – Wickham, 2004; page 4

Tidy Data

Pros and Cons

So, yes, there is some subjectivity regarding what we classify as variables.
- But, to reiterate a point from yesterday: a data dictionary often takes the guesswork out of things.
Finally, I’d like to make another note: messy (the opposite of tidy) datasets sometimes have their benefits!
For example, messy datasets can actually appear (visually) more succinct than their tidy counterparts, which is why you’ll sometimes see messy datasets presented in reports or presentations.
But, tidy data is usually preferred in the analysis stage of the DSL, as computers are a bit better-equipped at handling tidy data than they are at handling messy data.

Tidy Data

Example: Blood Pressures

Data Description: The systolic and diastolic blood pressures of 3 individuals was recorded.

Data Dictionary:
Data
Tidy Constraints

Name: Name of the individual
Systolic BP: Systolic blood pressure (in mm HG)
Diastolic BP: Systolic blood pressure (in mm HG)

Name	Type	Measurement
Anurag	Systolic	100
Anurag	Diastolic	70
Biyonka	Systolic	101
Biyonka	Diastolic	72
Chae	Systolic	99
Chae	Diastolic	68

Each variable forms a column
Each observation forms a row
Each type of observational unit forms a table

Based on the provided data dictionary, is this data frame tidy?

Tidying Data

Interlude

Line Plots

I’d like to motivate our discussion on tidying data by way of statistical visualizations
- We’ll talk more about these next lecture.
For now, I’d like to introduce the notion of a line plot, which is a common plot used to display the change in a numerical quantity over time.
- Time is represented on the horizontal axis, and the quantity in question is displayed on the vertical axis.

Interlude

Line Plots

Source: Google Trends

Interlude

Line Plots

Source

Line Plot

Enrollments Over Time

Let’s say we want to make a line graph, plotting how the different courses’ enrollments changed over time (perhaps even using colors to separate the different courses).

    Course              Title S25 W25 F24 M24 S24 W24 F23 M23 S23 W23 F22 M22
  PSTAT 5A UNDERSTANDING DATA 167 222 213  76 258 283 549 110 212 279 499 116
  PSTAT 5H         STATISTICS  11  14  30  NA  31  24  25  NA  31  22   4  NA
 PSTAT 5LS STAT LIFE SCIENCES 355 340 348  59 304 299  NA  39 299 294  NA  63
   PSTAT 8 TRANS DS PROB STAT  70 105 137  22  83 134 132  18  86 156  70  23
 S22 W22
 194 278
  NA  NA
 294 284
  59  NA
 [ reached 'max' / getOption("max.print") -- omitted 43 rows ]

How do we tell R what to put on the x-axis, when the “time” values are embedded as column headers?

Line Plot

Enrollments Over Time

We do not have this problem with the long (tidy) format:

    Course              Title Quarter Enrollment
  PSTAT 5A UNDERSTANDING DATA     S25        167
  PSTAT 5H         STATISTICS     S25         11
 PSTAT 5LS STAT LIFE SCIENCES     S25        355
   PSTAT 8 TRANS DS PROB STAT     S25         70
  PSTAT 10 DATA SCIENCE PRINC     S25        235
 [ reached 'max' / getOption("max.print") -- omitted 653 rows ]

Now the Quarter variable clearly shows up as its own column, making it much easier to access using R. This is one of the benefits of tidiness.
The question we now turn to is: how to we transform the wide layout into the long layout?

Dataframe Transformations

Melting

The answer: an operation known as melting.
To illustrate the melting operation, let’s consider a simpler mock dataset.
We imagine 3 subjects were each administered two dosages of a drug. Their heart rate (in bpm) after each dose was recorded, and stored in a dataframe.

Dataframe Transformations

	A	B
Jane	62	65
John	71	75
Jack	64	70

Name	Dosage	Heartrate
Jane	A	62
John	A	71
Jack	A	64
Jane	B	65
John	B	75
Jack	B	70

The original dataframe is “wide” whereas the molten dataframe is “long”.

Dataframe Transformations

Melting

We still have a Name variable / column

Dataframe Transformations

Melting

The values from previously separate columns have now been concatenated into a single column

Dataframe Transformations

Melting

The old column headers (which were values!) now appear as values in a new column.

Dataframe Transformations

Melting

As such, we can think of melting as an operation parametrized by the following:
- Colvars (i.e. columns that are already variables)
- A new variable containing the concatenated data values from the previously separate columns
- A new variable/column containing the old column headers

Dataframe Transformations

Melting

As such, we can think of melting as an operation parametrized by the following:
- Colvars (i.e. columns that are already variables)
- A new variable containing the concatenated data values from the previously separate columns
- A new variable/column containing the old column headers

Dataframe Transformations

Melting

As such, we can think of melting as an operation parametrized by the following:
- Colvars (i.e. columns that are already variables)
- A new variable containing the concatenated data values from the previously separate columns
- A new variable/column containing the old column headers

Dataframe Transformations

Melting

As such, we can think of melting as an operation parametrized by the following:
- Colvars (i.e. columns that are already variables)
- A new variable containing the concatenated data values from the previously separate columns
- A new variable/column containing the old column headers

Melting in `R`

Via the `reshape2::melt()` Function

melt(
  data,
  id.vars,
  measure.vars,
  variable.name = "variable",
  ...,
  na.rm = FALSE,
  value.name = "value",
  factorsAsStrings = TRUE
)

1: The name of the dataframe
2: The name of the colvars
3: The name of the column whose values will be split post-melt
4: An optional specification of what you want the new variable column to be called (e.g. Dosage, in our previous example)

Hover over the bubbled numbers for a description of what each argument represents

Melting in `R`

Via the `tidyverse::pivot_longer()` Function

pivot_longer(
  data,
  cols,
  ...,
  names_to = "name",
  ...,
  values_to = "value",
  ...
)

1: The name of the dataframe
2: The name of the non-colvars
3: The name of the new column containing old column names
4: The name of the new column containing values post-split

You’ll likely see me use melt() more often than pivot_longer(), however that is mainly personal preference!

Melting in `R`

Comparison

Original Dataframe
Using melt()
Using pivot_longer()

Expand Code

dosage_df <- data.frame(
  Name = c("Jane", "John", "Jack"),
  A = c(62, 71, 64),
  B = c(65, 75, 70)
)

dosage_df %>% pander()

Name	A	B
Jane	62	65
John	71	75
Jack	64	70

Expand Code

dosage_df %>%
  melt(
    id.vars = "Name",
    variable.name = "Dosage",
    value.name = "Heart_Rate"
  ) %>% pander()

Name	Dosage	Heart_Rate
Jane	A	62
John	A	71
Jack	A	64
Jane	B	65
John	B	75
Jack	B	70

Expand Code

dosage_df %>%
  pivot_longer(
    cols = !c("Name"),
    names_to = "Dosage",
    values_to = "Heart_Rate",
    cols_vary = "slowest"         ## optional
  ) %>% pander()

Name	Dosage	Heart_Rate
Jane	A	62
John	A	71
Jack	A	64
Jane	B	65
John	B	75
Jack	B	70

The inverse of melting is called pivoting, and will be discussed further in Lab 02.

Live Demo!

Live Demo

Time for our first live demo! Feel free to boot up your laptops and follow along; I’ve uploaded the enrollments datset to our course computing server. In this demo we’ll:

Melt the enrollments dataset
Generate a line graph displaying the change in enrollments over time within lower-division undergraduate PSTAT courses
Interpret the results

Next Time

And Coming Up

Next lecture, we’ll explore statistical visualizations further.
During our first Lab session this afternoon, you’ll get practice with some dataframe manipulation commands from the tidyverse
- The last problem on the first homework will also give you a chance to practice some of these
Start thinking about who you want to work with on the Mid-Quarter Project!
- As a reminder, you are required to work in groups of 3-5 on the project.
- You must finalize (and submit) your groups by Monday of next week.

PSTAT 100: Lecture 02

Review: Data Semantics

Definition

Review: Data Semantics

Another Illustration

Review: Course Enrollments Dataset

Some Sample Observations

Data

Structure vs. Semantics

Data

Structure vs. Semantics

Tidy Data

Definition

Tidy or Not?

Enrollments: Wide Layout

Tidy or Not?

Enrollments: Long Layout

Tidy or Not?

A Brief Interlude

Tidy or Not?

A Brief Interlude

Tidy or Not?

A Brief Interlude

Tidy Data

Pros and Cons

Tidy Data

Example: Blood Pressures

Tidying Data

Interlude

Line Plots

Interlude

Line Plots

Interlude

Line Plots

Line Plot

Enrollments Over Time

Line Plot

Enrollments Over Time

Dataframe Transformations

Melting

Dataframe Transformations

Melting

Dataframe Transformations

Melting

Dataframe Transformations

Melting

Dataframe Transformations

Melting

Dataframe Transformations

Melting

Dataframe Transformations

Melting

Dataframe Transformations

Melting

Dataframe Transformations

Melting

Melting in R

Via the reshape2::melt() Function

Melting in R

Via the tidyverse::pivot_longer() Function

Melting in R

Comparison

Live Demo!

Next Time

And Coming Up

`Enrollments`: Wide Layout

`Enrollments`: Long Layout

Melting in `R`

Via the `reshape2::melt()` Function

Melting in `R`

Via the `tidyverse::pivot_longer()` Function

Melting in `R`