Έλεγχος κανονικότητας
Κώστας Κούδας
2025-08-25
1 Anderson-Darling έλεγχος κανονικότητας
Πολλές διεργασίες στη Στατιστική έχουν προϋπόθεση να αντλούμε δείγματα από κανονικό πληθυσμό. Υπάρχουν διάφοροι στατιστικοί έλεγχοι αυτού του πράγματος, όπως οι \(\mathcal{X}^2\), Kolmogorov-Smirnov, Shapiro-Wilk και Anderson-Darling. Εμείς εδώ, επενδύουμε στη λιτότητα των επιλογών με στόχο την ευκολία χρήσης. Έτσι, μετά από κάποια περισυλλογή, θ’ ασχοληθούμε με τους δύο τελευταίους.
Αντλήσαμε δεδομένα από τα στατιστικά στοιχεία του Freie Universität και τα αποθηκεύσαμε σε ένα αρχείο ονόματι students.xlsx, του οποίου παρουσιάζουμε μόνο τις πρώτες 100 γραμμές για λόγους ταχύτητας εκτέλεσης του κώδικα. Θα εξετάσουμε κατά πόσον τα βάρη των φοιτητών είναι κανονικά κατανεμημένα. Η μηδενική υπόθεση θα είναι ότι ακολουθούν την κανονική κατανομή, και θα δούμε κατά πόσον αυτή απορρίπτεται ή όχι.
| stud.id | name | gender | age | height | weight | religion | nc.score | semester | major | minor | score1 | score2 | online.tutorial | graduated | salary |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 833917 | Gonzales, Christina | Female | 19 | 160 | 64.8 | Muslim | 1.91 | 1st | Political Science | Social Sciences | NA | NA | 0 | 0 | NA |
| 898539 | Lozano, T’Hani | Female | 19 | 172 | 73.0 | Other | 1.56 | 2nd | Social Sciences | Mathematics and Statistics | NA | NA | 0 | 0 | NA |
| 379678 | Williams, Hanh | Female | 22 | 168 | 70.6 | Protestant | 1.24 | 3rd | Social Sciences | Mathematics and Statistics | 45 | 46 | 0 | 0 | NA |
| 807564 | Nem, Denzel | Male | 19 | 183 | 79.7 | Other | 1.37 | 2nd | Environmental Sciences | Mathematics and Statistics | NA | NA | 0 | 0 | NA |
| 383291 | Powell, Heather | Female | 21 | 175 | 71.4 | Catholic | 1.46 | 1st | Environmental Sciences | Mathematics and Statistics | NA | NA | 0 | 0 | NA |
| 256074 | Perez, Jadrian | Male | 19 | 189 | 85.8 | Catholic | 1.34 | 2nd | Political Science | Mathematics and Statistics | NA | NA | 0 | 0 | NA |
| 754591 | Clardy, Anita | Female | 21 | 156 | 65.9 | Protestant | 1.11 | 2nd | Political Science | Social Sciences | NA | NA | 0 | 0 | NA |
| 146494 | Allen, Rebecca Marie | Female | 21 | 167 | 65.7 | Other | 2.03 | 3rd | Political Science | Economics and Finance | 58 | 62 | 0 | 0 | NA |
| 723584 | Tracy, Robert | Male | 18 | 195 | 94.4 | Other | 1.29 | 3rd | Economics and Finance | Environmental Sciences | 57 | 67 | 0 | 0 | NA |
| 314281 | Nimmons, Laura | Female | 18 | 165 | 66.0 | Orthodox | 1.19 | 2nd | Environmental Sciences | Mathematics and Statistics | NA | NA | 0 | 0 | NA |
| 200803 | Lang, Mackenzie | Female | 22 | 162 | 66.8 | Other | 1.04 | 4th | Economics and Finance | Environmental Sciences | 62 | 61 | 1 | 1 | 45254.108020687003 |
| 444907 | Rodriguez, Brianna | Female | 18 | 172 | 66.8 | Other | 3.81 | 3rd | Environmental Sciences | Economics and Finance | 76 | 82 | 0 | 0 | NA |
| 354271 | Covar Orendain, Christopher | Male | 23 | 185 | 84.6 | Orthodox | 1.00 | 4th | Environmental Sciences | Mathematics and Statistics | 71 | 76 | 1 | 1 | 40552.790242531002 |
| 317812 | Lopez, Monique | Female | 20 | 158 | 64.4 | Catholic | 2.50 | 6th | Environmental Sciences | Social Sciences | 66 | 70 | 1 | 1 | 27007.030294282002 |
| 604115 | Davis, Shagun | Female | 19 | 157 | 66.3 | Orthodox | 1.92 | 2nd | Economics and Finance | Political Science | NA | NA | 0 | 0 | NA |
| 889551 | Adams, Jose | Male | 20 | 172 | 73.9 | Other | 3.61 | 4th | Mathematics and Statistics | Political Science | 87 | 91 | 1 | 0 | NA |
| 350040 | Hines, Haileigh | Female | 22 | 156 | 61.7 | Other | 2.27 | 6th | Political Science | Biology | 57 | 54 | 0 | 1 | 33969.159268048999 |
| 240279 | Daugherty, Jesus | Male | 22 | 182 | 82.1 | Catholic | 1.42 | 1st | Economics and Finance | Environmental Sciences | NA | NA | 0 | 0 | NA |
| 865835 | Roybal, Ebony | Female | 21 | 162 | 69.2 | Catholic | 1.32 | 3rd | Political Science | Environmental Sciences | 69 | 46 | 1 | 0 | NA |
| 137196 | Baysinger, Tanisha | Female | 22 | 168 | 70.9 | Protestant | 2.33 | 2nd | Environmental Sciences | Political Science | NA | NA | 0 | 0 | NA |
| 708242 | Phillips, Laiba | Female | 20 | 167 | 68.5 | Other | 1.79 | 4th | Biology | Economics and Finance | 77 | 80 | 1 | 0 | NA |
| 499002 | Culbertson, Deshawn | Male | 37 | 175 | 70.4 | Protestant | 1.97 | 2nd | Political Science | Environmental Sciences | NA | NA | 0 | 0 | NA |
| 873149 | O Reilly, Joshua | Male | 19 | 164 | 70.3 | Protestant | 1.68 | 2nd | Political Science | Environmental Sciences | NA | NA | 0 | 0 | NA |
| 807361 | Johnson, Stephanie | Female | 38 | 155 | 67.0 | Catholic | 2.30 | 2nd | Environmental Sciences | Biology | NA | NA | 0 | 0 | NA |
| 531029 | Mix, Aziel | Male | 23 | 183 | 81.8 | Catholic | 2.11 | 4th | Economics and Finance | Environmental Sciences | 69 | 65 | 0 | 0 | NA |
| 970589 | Gonzalez, Dixie | Female | 26 | 145 | 54.0 | Other | 1.14 | 1st | Biology | Environmental Sciences | NA | NA | 0 | 0 | NA |
| 250298 | Clark, Janelle | Female | 25 | 161 | 66.8 | Other | 1.45 | 3rd | Social Sciences | Economics and Finance | 45 | 37 | 1 | 0 | NA |
| 763393 | Woolsey, Bronson | Male | 24 | 182 | 80.1 | Protestant | 1.09 | 5th | Economics and Finance | Social Sciences | 61 | 73 | 1 | 1 | 50617.641869938998 |
| 544433 | Diawara, Erica | Female | 54 | 169 | 71.4 | Protestant | 1.75 | 2nd | Political Science | Environmental Sciences | NA | NA | 0 | 0 | NA |
| 252935 | Lord, Benjamin | Male | 22 | 172 | 69.6 | Protestant | 3.94 | 3rd | Mathematics and Statistics | Social Sciences | 89 | 90 | 1 | 0 | NA |
| 453762 | Cordova, Justin | Male | 23 | 172 | 73.0 | Protestant | 1.26 | 1st | Economics and Finance | Mathematics and Statistics | NA | NA | 0 | 0 | NA |
| 759365 | Hayes, Vamshi | Male | 19 | 192 | 88.9 | Other | 1.19 | 3rd | Social Sciences | Mathematics and Statistics | 44 | 36 | 0 | 0 | NA |
| 662678 | French, Sidney | Male | 22 | 175 | 72.6 | Protestant | 3.03 | 4th | Economics and Finance | Biology | 65 | 70 | 1 | 0 | NA |
| 188703 | Torres, Andrew | Male | 18 | 173 | 75.7 | Other | 1.19 | 4th | Environmental Sciences | Mathematics and Statistics | 74 | 74 | 0 | 0 | NA |
| 806369 | Medina, Diamond | Female | 20 | 170 | 71.6 | Protestant | 2.27 | 2nd | Environmental Sciences | Political Science | NA | NA | 0 | 0 | NA |
| 679521 | Pascua, Burphy | Male | 21 | 176 | 74.1 | Other | 4.00 | 3rd | Biology | Economics and Finance | 72 | 75 | 1 | 0 | NA |
| 371307 | Glenn-Bracey, Mia | Female | 20 | 180 | 75.6 | Protestant | 1.09 | 1st | Political Science | Economics and Finance | NA | NA | 0 | 0 | NA |
| 411253 | Yang, Manuel | Male | 23 | 185 | 80.7 | Orthodox | 3.15 | 5th | Political Science | Mathematics and Statistics | 54 | 46 | 0 | 1 | 37391.872365386 |
| 835757 | Lujan, Malia | Female | 22 | 172 | 68.3 | Protestant | 2.12 | 4th | Mathematics and Statistics | Environmental Sciences | 86 | 92 | 1 | 1 | 35598.881686346998 |
| 232384 | Formby, Christina | Female | 21 | 171 | 74.5 | Protestant | 1.43 | 1st | Environmental Sciences | Mathematics and Statistics | NA | NA | 0 | 0 | NA |
| 871735 | Nelson, Anddi | Female | 19 | 153 | 64.9 | Catholic | 1.15 | 2nd | Economics and Finance | Mathematics and Statistics | NA | NA | 0 | 0 | NA |
| 503879 | Lucas, Asashia | Female | 20 | 157 | 64.4 | Other | 1.66 | 3rd | Social Sciences | Mathematics and Statistics | 52 | 33 | 0 | 0 | NA |
| 186259 | Holguin, Lance | Male | 23 | 190 | 89.3 | Protestant | 3.46 | 2nd | Mathematics and Statistics | Political Science | NA | NA | 0 | 0 | NA |
| 890074 | Ontiveros, Aleena | Female | 20 | 173 | 78.3 | Orthodox | 2.67 | 1st | Political Science | Social Sciences | NA | NA | 0 | 0 | NA |
| 493398 | Clark, Marlana | Female | 19 | 170 | 68.0 | Other | 2.45 | 1st | Social Sciences | Environmental Sciences | NA | NA | 0 | 0 | NA |
| 902697 | Phan, Edeer | Male | 19 | 160 | 62.2 | Catholic | 2.92 | 5th | Environmental Sciences | Economics and Finance | 78 | 75 | 1 | 1 | 39703.744884975997 |
| 679949 | Robles Escamilla, Rosario | Female | 23 | 170 | 69.6 | Orthodox | 3.23 | 2nd | Biology | Economics and Finance | NA | NA | 0 | 0 | NA |
| 509606 | Longwolf, Addam | Male | 20 | 171 | 72.1 | Protestant | 2.03 | 1st | Environmental Sciences | Political Science | NA | NA | 0 | 0 | NA |
| 773757 | Romero, Keyanna | Female | 22 | 160 | 66.2 | Other | 1.37 | 3rd | Biology | Environmental Sciences | 83 | 73 | 1 | 0 | NA |
| 889115 | Carter, Robert | Male | 23 | 181 | 76.2 | Orthodox | 1.32 | 2nd | Economics and Finance | Social Sciences | NA | NA | 0 | 0 | NA |
| 280980 | Jumbo, Isiah | Male | 21 | 161 | 66.7 | Catholic | 1.41 | 3rd | Social Sciences | Political Science | 55 | 42 | 0 | 0 | NA |
| 879108 | Dillenberg, Soraya | Female | 25 | 164 | 65.9 | Other | 1.43 | 2nd | Social Sciences | Economics and Finance | NA | NA | 0 | 0 | NA |
| 756828 | Nguyen, Juancarlos | Male | 24 | 170 | 68.0 | Protestant | 1.83 | 5th | Economics and Finance | Environmental Sciences | 71 | 69 | 1 | 0 | NA |
| 873225 | Shanahan, Carlos | Male | 20 | 174 | 72.8 | Protestant | 1.89 | 1st | Economics and Finance | Biology | NA | NA | 0 | 0 | NA |
| 267025 | Brown, Nhan | Male | 21 | 171 | 65.7 | Catholic | 2.77 | >6th | Political Science | Mathematics and Statistics | 57 | 58 | 1 | 1 | 30086.764812451998 |
| 865072 | Tak, Ruben | Male | 22 | 184 | 83.3 | Catholic | 2.87 | 3rd | Economics and Finance | Biology | 63 | 67 | 1 | 0 | NA |
| 180057 | Hussainy, Luis | Male | 19 | 173 | 73.2 | Other | 2.80 | 1st | Biology | Mathematics and Statistics | NA | NA | 0 | 0 | NA |
| 350628 | Maven, Yee Joo | Female | 20 | 167 | 73.3 | Other | 1.13 | 2nd | Political Science | Biology | NA | NA | 0 | 0 | NA |
| 809000 | Payte, Alexander | Male | 21 | 182 | 83.5 | Protestant | 2.23 | 1st | Biology | Economics and Finance | NA | NA | 0 | 0 | NA |
| 286216 | Tate, Shelby | Female | 23 | 159 | 63.0 | Catholic | 2.01 | 1st | Biology | Economics and Finance | NA | NA | 0 | 0 | NA |
| 373120 | Livingston, Sarah | Female | 44 | 167 | 69.0 | Catholic | 2.51 | 4th | Political Science | Environmental Sciences | 56 | 52 | 0 | 1 | 31494.320815455001 |
| 797140 | Vang, Austin | Male | 18 | 183 | 81.8 | Other | 3.14 | 2nd | Economics and Finance | Social Sciences | NA | NA | 0 | 0 | NA |
| 201130 | Cebrun, Linda | Female | 21 | 157 | 62.9 | Catholic | 2.59 | >6th | Environmental Sciences | Social Sciences | 64 | 75 | 0 | 1 | 32519.224081044998 |
| 655450 | Meza Ramirez, Maricela | Female | 20 | 173 | 78.1 | Protestant | 2.08 | 2nd | Political Science | Mathematics and Statistics | NA | NA | 0 | 0 | NA |
| 401840 | Nicholson, Willow | Female | 32 | 171 | 67.6 | Orthodox | 1.53 | 3rd | Mathematics and Statistics | Economics and Finance | 88 | 93 | 1 | 0 | NA |
| 358005 | Eydi Shahroodi, Erik | Male | 20 | 182 | 77.4 | Other | 3.82 | 4th | Economics and Finance | Political Science | 65 | 63 | 1 | 1 | 42593.260382113003 |
| 680483 | Truong, Patrick | Male | 20 | 169 | 71.6 | Other | 3.80 | 4th | Economics and Finance | Environmental Sciences | 73 | 73 | 0 | 1 | 60407.317266003003 |
| 275470 | Vasquez, Timothy | Male | 22 | 167 | 65.9 | Other | 3.96 | 3rd | Mathematics and Statistics | Economics and Finance | 89 | 93 | 1 | 0 | NA |
| 258961 | Warren, Justin | Male | 23 | 174 | 70.7 | Orthodox | 1.46 | 1st | Economics and Finance | Social Sciences | NA | NA | 0 | 0 | NA |
| 495076 | Frost, Yuvorn | Male | 41 | 194 | 94.9 | Other | 1.78 | 2nd | Environmental Sciences | Political Science | NA | NA | 0 | 0 | NA |
| 156460 | Minor, Alex | Male | 23 | 182 | 82.7 | Other | 1.22 | 4th | Social Sciences | Political Science | 38 | 49 | 1 | 0 | NA |
| 249571 | Arkadie, Houa | Female | 24 | 152 | 64.9 | Other | 2.73 | 4th | Biology | Political Science | 75 | 83 | 1 | 0 | NA |
| 962284 | Iverson, Sierra | Female | 20 | 154 | 63.0 | Other | 2.30 | 1st | Biology | Environmental Sciences | NA | NA | 0 | 0 | NA |
| 870282 | Bustamante-Harrison, Sierra | Female | 24 | 159 | 60.6 | Catholic | 1.88 | 4th | Social Sciences | Mathematics and Statistics | 42 | 42 | 0 | 0 | NA |
| 308092 | Treto, Teonna | Female | 23 | 153 | 63.2 | Other | 3.18 | 3rd | Biology | Economics and Finance | 76 | 75 | 1 | 0 | NA |
| 694127 | Gonzales, Brianna | Female | 20 | 150 | 63.3 | Other | 2.62 | 2nd | Political Science | Biology | NA | NA | 0 | 0 | NA |
| 841076 | Pacheco, Cydney | Female | 19 | 167 | 72.8 | Other | 2.73 | >6th | Biology | Political Science | 74 | 81 | 0 | 1 | 49111.427488433997 |
| 794058 | Castaneda-Amaya, Tyandra | Female | 25 | 149 | 56.2 | Catholic | 1.02 | 3rd | Economics and Finance | Environmental Sciences | 76 | 61 | 1 | 0 | NA |
| 518947 | Garcia, Yamel | Female | 21 | 155 | 62.1 | Catholic | 1.36 | 2nd | Biology | Mathematics and Statistics | NA | NA | 0 | 0 | NA |
| 216224 | Ho, Javier | Male | 22 | 177 | 77.5 | Protestant | 2.17 | 2nd | Economics and Finance | Political Science | NA | NA | 0 | 0 | NA |
| 980136 | Munoz, Derrick | Male | 24 | 180 | 77.1 | Orthodox | 1.63 | 5th | Environmental Sciences | Political Science | 69 | 68 | 1 | 1 | 33068.848645060003 |
| 253731 | Castillo, Anastasia | Female | 24 | 159 | 62.2 | Other | 2.27 | 6th | Environmental Sciences | Biology | 63 | 75 | 1 | 1 | 37803.513208607001 |
| 416538 | Serrano, Shanae | Female | 19 | 174 | 71.1 | Catholic | 1.42 | 3rd | Biology | Environmental Sciences | 83 | 79 | 1 | 0 | NA |
| 888830 | Park, Damien | Male | 23 | 174 | 74.9 | Other | 2.49 | 3rd | Economics and Finance | Social Sciences | 71 | 74 | 0 | 0 | NA |
| 306032 | Lesley, Tiffany | Female | 21 | 159 | 62.4 | Protestant | 1.76 | 1st | Environmental Sciences | Political Science | NA | NA | 0 | 0 | NA |
| 536813 | Turner, Savannah | Female | 19 | 160 | 65.3 | Orthodox | 1.38 | 4th | Biology | Environmental Sciences | 78 | 76 | 0 | 0 | NA |
| 732709 | Begay-Watson, Anthony | Male | 22 | 169 | 68.0 | Orthodox | 3.06 | 4th | Political Science | Economics and Finance | 53 | 46 | 1 | 0 | NA |
| 514389 | Picard, Brittney | Female | 21 | 166 | 67.9 | Muslim | 2.26 | 5th | Environmental Sciences | Mathematics and Statistics | 72 | 77 | 1 | 1 | 35191.662764719003 |
| 367719 | Wilson, Navya | Female | 21 | 155 | 67.5 | Other | 2.03 | 3rd | Economics and Finance | Biology | 71 | 74 | 1 | 0 | NA |
| 362637 | Dinh, Fidel | Male | 26 | 178 | 76.7 | Other | 2.45 | 3rd | Biology | Social Sciences | 76 | 79 | 0 | 0 | NA |
| 409917 | Huddleston, Tina | Female | 23 | 162 | 61.8 | Orthodox | 1.21 | 2nd | Mathematics and Statistics | Economics and Finance | NA | NA | 0 | 0 | NA |
| 977593 | Gonzalez, Merissa | Female | 21 | 181 | 79.0 | Protestant | 1.53 | 3rd | Economics and Finance | Biology | 65 | 65 | 1 | 0 | NA |
| 884302 | Norris, Lauren | Female | 20 | 168 | 69.4 | Orthodox | 2.45 | 1st | Mathematics and Statistics | Social Sciences | NA | NA | 0 | 0 | NA |
| 689045 | Kim, Eaba | Male | 20 | 181 | 80.0 | Other | 2.83 | 2nd | Mathematics and Statistics | Environmental Sciences | NA | NA | 0 | 0 | NA |
| 706060 | Miranda Quintero, Olivia | Female | 20 | 158 | 60.6 | Orthodox | 1.46 | 4th | Social Sciences | Biology | 43 | 52 | 1 | 0 | NA |
| 168184 | Briseno, Davie | Male | 23 | 177 | 79.4 | Muslim | 2.02 | 4th | Mathematics and Statistics | Social Sciences | 93 | 87 | 1 | 1 | 39342.522668919002 |
| 890431 | Garcia, Kush | Male | 18 | 169 | 73.8 | Catholic | 1.41 | 5th | Mathematics and Statistics | Social Sciences | 78 | 90 | 1 | 0 | NA |
| 345772 | Chandramouli, Nelson | Male | 19 | 176 | 76.4 | Orthodox | 2.56 | 2nd | Environmental Sciences | Political Science | NA | NA | 0 | 0 | NA |
| 167649 | Edwards, Linh | Female | 21 | 166 | 64.3 | Protestant | 1.99 | 1st | Biology | Economics and Finance | NA | NA | 0 | 0 | NA |
| 372924 | Cook, Macallyson | Female | 23 | 173 | 71.6 | Protestant | 1.41 | 3rd | Biology | Political Science | 80 | 79 | 1 | 0 | NA |
Αρχικά θα εγκαταστήσουμε και θα χρησιμοποιήσουμε το πακέτο
nortest:
## Loading required package: nortestΑκολούθως θα χρησιμοποιήσουμε τη συνάρτηση ad.test(),
όπου σαν όρισμα θα βάλουμε τη λίστα με τους αριθμούς των οποίων θέλουμε
να ελέγξουμε την κανονικότητά τους (δηλαδή την
students$weight). Οπότε γράφουμε:
##
## Anderson-Darling normality test
##
## data: students$weight
## A = 56.014, p-value < 2.2e-16Οπότε βρίσκουμε p-τιμή πάρα πολύ μικρή. Αυτό σημαίνει ότι η μηδενική υπόθεση απορρίπτεται, άρα τα βάρη των φοιτητών δεν ακολουθούν την κανονική κατανομή.
2 Γραφικός έλεγχος
2.1 Καμπύλη κανονικής κατανομής πάνω από ιστόγραμμα
Πέραν των διαφόρων ελέγχων κανονικότητας μπορούμε να εξακριβώσουμε
και γραφικά το αν έχουμε κανονικά κατανεμημένους αριθμούς ή όχι. Σε αυτό
θα μας βοηθήσει να προσαρμόσουμε στα ιστογράμματα και την αντίστοιχη
καμπύλη κανονικής κατανομής χρησιμοποιώντας τη συνάρτηση
curve(). Για την ακρίβεια θα χρησιμοποιήσουμε μια πιο
εξειδικευμένη εκδοχή της, την
curve(dnorm(x, mean=🎻🎻🎻, sd=⏲⏲⏲), add=TRUE), όπου:
στο
🎻🎻🎻γράφουμε τη μέση τιμή του δείγματός μας, δηλαδή τηνmean(students$height),στο
⏲⏲⏲γράφουμε την τυπική απόκλιση του δείγματός μας, δηλαδή τηνsd(students$height)καιγράφουμε
add=TRUEγια να μπει επιπλέον η καμπύλη στο ιστόγραμμα.
hist(students$weight, freq = F)
curve(dnorm(x, mean=mean(students$weight), sd=sd(students$weight)), add=TRUE)
hist(students$height, freq = F)
curve(dnorm(x, mean=mean(students$height), sd=sd(students$height)), add=TRUE)
Βλέπουμε λοιπόν ότι η κατανομή των υψών είναι λίγο πιο ομοιομορφη από την κανονική, οι τιμές τους είναι αρκετά απλωμένες γύρω από τη μέση τιμή. Και η κατανομή των βαρών φαίνεται να διαφέρει από την κανονική, μιας και αυτά είναι συσσωρευμένα γύρω από μικρές τιμές.
2.2 Γράφημα QQ
Η αλήθεια είναι πως δεν ήταν τόσο εύκολο να δούμε γιατί τα ύψη δεν είναι κανονικά κατανεμημένα. Πολλοί δεν θα πείθονταν από τα λεγόμενά μας περί αρκετά απλωμένων τιμών. Αυτοί ίσως να έπρεπε να εξετάσουν κι ένα άλλου τύπου γράφημα, το γράφημα Q-Q.
Το γράφημα Q-Q είναι απλά ένα διάγραμμα διασποράς ανάμεσα στα ποσοστημόρια του δείγματος και της κανονικής κατανομής με ίδια μέση τιμή και τυπική απόκλιση. Έτσι, αν έχουμε κανονικά κατανεμημένους αριθμούς, τότε το γράφημα Q-Q θα είναι μια ευθεία γραμμή.
Αντίθετα, αν η έλλειψη κανονικότητας αντικατοπτρίζεται στην ασυμμετρία, δηλαδή έχουμε τιμές μαζεμένες στη μία ή στην άλλη μεριά, τότε θα ξεφεύγει η μία ή η άλλη «ουρά» του γραφήματος Q-Q.
Απόκλιση από την κανονικότητα ενδέχεται να έχουμε είτε επειδή οι τιμές μας είναι πολύ συγκεντρωμένες γύρω από τη μέση τιμή ή, στους αντίποδες αυτής της περίπτωσης, αν είναι πολύ διασκορπισμένες. Και αυτό το γεγονός αντικατοπτρίζεται σε ένα γράφημα Q-Q.
Τέλος, ακόμα μεγαλύτερη απόκλιση από την κανονικότητα έχουμε όταν οι τιμές των δεδομένων μας είναι συγκεντρωμένες σε δύο ή περισσότερες περιοχές. Σε αυτή την περίπτωση το γράφημα Q-Q θα παρουσιάζει χάσματα κι αυτό είναι μια ουσιαστική διαφορά σε σχέση με το [γράφημα ασυμμετρίας](asymmetria.html#22_Γράφημα_ασυμμετρίας.
par(mfrow = c(1,2))
nn <- 1e3
set.seed(1)
betas <- rbeta(nn,2,2)
Bimodal <- c(betas[1:(nn/2)]*2+1,
betas[(nn/2+1):nn]*2+3.3)
hist(Bimodal, freq=F, ylim=c(0, 0.4))
curve(dnorm(x, mean(Bimodal), sd(Bimodal)), add=TRUE, col="red", lwd=2)
qqnorm(Bimodal)
qqline(Bimodal)
mtext("Δίκορφο ιστόγραμμα", side = 3, line = - 1.3, outer = TRUE, cex = 1.5)
Πώς όμως σχηματίζουμε το εν λόγω γράφημα; Θα χρειαστούμε τις
συναρτήσεις qqnorm() και qqline(), όπου σαν όρισμα θα
βάλουμε τη λίστα με τους αριθμούς των οποίων θέλουμε να ελέγξουμε την
κανονικότητά τους (δηλαδή την students$weight). Οπότε
γράφουμε:
Έτσι προκύπτουν τα γραφήματα Q-Q των βαρών και των υψών των φοιτητών αντίστοιχα, όπως και οι ευθείες που θα έπρεπε να σχηματίζουν αν ήταν κανονικά κατανεμημένα.
Έτσι βλέπουμε και πάλι ότι ούτε τα βάρη, αλλά ούτε και τα ύψη είναι κανονικά κατανεμημένα,
Όλος ο σημερινός κώδικας παρατίθεται αμέσως μετά:
if(!require(nortest)){
install.packages("nortest")
library(nortest)
}
ad.test(students$weight)
ad.test(students$height)
hist(students$weight, freq = F)
curve(dnorm(x, mean=mean(students$weight), sd=sd(students$weight)), add=T)
hist(students$height, freq = F)
curve(dnorm(x, mean=mean(students$height), sd=sd(students$height)), add=TRUE)
qqnorm(students$weight)
qqline(students$weight)
qqnorm(students$height)
qqline(students$height)3 Shapiro-Wilk έλεγχος κανονικότητας
Ο έλεγχος Anderson-Darling δεν δίνει αξιόπιστα αποτελέσματα, αν το δείγμα μας είναι μικρό. Σε αυτή την περίπτωση θα πρέπει να χρησιμοποιήσουμε τον έλεγχο Shapiro-Wilk. Και αυτός δέχεται ως μηδενική υπόθεση την παραδοχή ότι έχουμε κανονικά κατανεμημένους αριθμούς.
Από τη σελίδα αυτή αντλήσαμε στοιχεία για τα έξοδα του Εθνικού Μουσείου Φυσικής Ιστορίας κατά τον Ιούνιο του 2023, τα οποία τα αποθηκεύσαμε σε ένα αρχείο ονόματι foi-expenditure-over-25000-june-2023.xlsx.
| Department Famil | |Entity | |Date | |Expense Type | |Expense Area | |Supplier | |Transaction Nu | ber | |
|---|---|---|---|---|---|---|---|
| DCMS | Natural History Museum | 2023-06-08 | Building & Engineering PPM | Estates Buildings Projects | DBR (London) Limited | 03586 | 140251.48 |
| DCMS | Natural History Museum | 2023-06-08 | Estates Property Rates | Estates Management | Royal Borough of Kensington & Chelse | 629799352 JULY 23/24 | 167574.00 |
| DCMS | Natural History Museum | 2023-06-08 | Accommodation | NHM_Unlocked Programme (Estates) | Key Travel Ltd | 51903560 | 309.62 |
| DCMS | Natural History Museum | 2023-06-08 | Accommodation | NHM_Unlocked Programme (Science) | Key Travel Ltd | 51903560 | 309.62 |
| DCMS | Natural History Museum | 2023-06-08 | Accommodation | NHM_Unlocked Programme (Science) | Key Travel Ltd | 51903560 | 309.62 |
| DCMS | Natural History Museum | 2023-06-08 | Accommodation | NHM_Unlocked (Programme Mgmt) | Key Travel Ltd | 51903560 | 309.62 |
| DCMS | Natural History Museum | 2023-06-08 | Estates Outsourced Services | Estates Buildings Ops | CBRE Managed Services Limited | S01083700 | 169886.29 |
| DCMS | Natural History Museum | 2023-06-08 | Agency Staff | Human Resources - Ops | Adecco UK Ltd | 29558369 | 771.51 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | Vital Energi Solutions Ltd | SI-13706 | 3875.83 |
| DCMS | Natural History Museum | 2023-06-08 | Gas | Estates Utilities | Corona Energy Retail 4 Limited | 17579678 | 20970.20 |
| DCMS | Natural History Museum | 2023-06-08 | Gas | Estates Utilities | Corona Energy Retail 4 Limited | 17579679 | 9504.71 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Estates Buildings Projects | Vital Energi Solutions Ltd | SI-13705 | 87011.55 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Estates Buildings Projects | Vital Energi Solutions Ltd | SI-13708 | 14814.00 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086641 | 3114.75 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086640 | 3012.57 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086635 | 2461.76 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086632 | 2401.67 |
| DCMS | Natural History Museum | 2023-06-08 | Travel-Air Passenger Transport | Science Research | Key Travel Ltd | 51951318 | 741.26 |
| DCMS | Natural History Museum | 2023-06-08 | Travel-Air Passenger Transport | Science Research | Key Travel Ltd | 51951318 | 741.26 |
| DCMS | Natural History Museum | 2023-06-08 | Grant Sub Contr./ award payments | Science Research | University of Southampton | 44268943 | 68342.91 |
| DCMS | Natural History Museum | 2023-06-08 | Energy Services Unitary Charge | SS CHP Boilerhouse | Vital Energi Solutions Ltd | SI-13707 | 19530.92 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086451 | 720.22 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086460 | 917.26 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086463 | 944.26 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086481 | 1147.54 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086488 | 1158.47 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086579 | 1488.42 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086594 | 1510.25 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086508 | 1319.47 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086624 | 1999.61 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086653 | 4918.60 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086651 | 4342.99 |
| DCMS | Natural History Museum | 2023-06-08 | Accommodation | Science Research | Key Travel Ltd | 51950236 | 1136.10 |
| DCMS | Natural History Museum | 2023-06-08 | Travel-Air Passenger Transport | Science Research | Key Travel Ltd | 51950521 | 831.21 |
| DCMS | Natural History Museum | 2023-06-08 | Travel-Air Passenger Transport | Science Research | Key Travel Ltd | 51950521 | 889.10 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086627 | 2010.93 |
| DCMS | Natural History Museum | 2023-06-08 | Energy Services Unitary Charge | Estates Utilities | Veolia ES Energy Performance (UK) Ltd | INV0018460 | 27232.31 |
| DCMS | Natural History Museum | 2023-06-08 | Energy Services Unitary Charge | Estates Utilities | Veolia ES Energy Performance (UK) Ltd | INV0018460 | 6096.79 |
| DCMS | Natural History Museum | 2023-06-08 | Energy Services Unitary Charge | Estates Utilities | Veolia ES Energy Performance (UK) Ltd | INV0018460 | 7316.14 |
| DCMS | Natural History Museum | 2023-06-08 | Energy Efficiency Savings | Estates Utilities | Veolia ES Energy Performance (UK) Ltd | INV0018461 | 3949.00 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101028272 | 5168.71 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101028273 | 1722.90 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101028274 | 2322.17 |
| DCMS | Natural History Museum | 2023-06-08 | Scientific Outsourced Services | CRL Imaging & Analytical | CBRE Managed Services Limited | S01086453 | 810.93 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101028278 | 550.00 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 6866.15 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 57762.02 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 13545.47 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101027513 | 1964.00 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101027513 | 11611.11 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101027513 | 8800.16 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101027513 | 2585.94 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101027513 | 2030.00 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101027513 | 240.00 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101027513 | 781.01 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101027513 | 1400.61 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 2499.86 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 53846.35 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 22734.57 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 95437.66 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 4716.15 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 10718.16 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 36628.53 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 27910.21 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 2748.33 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 178.64 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 1161.13 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 42520.94 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 44150.34 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 19961.13 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 9485.28 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 38299.97 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 10650.04 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 26921.58 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 63769.91 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 14355.25 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 1708.85 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 1848.63 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 65163.68 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 516.90 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 49614.49 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 17085.07 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 1112.80 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 351.76 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 5052.92 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 10856.67 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 68203.53 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 115116.26 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 81368.82 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 39653.63 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/012 | 69003.66 |
| DCMS | Natural History Museum | 2023-06-08 | Agency Staff | Human Resources - Ops | Adecco UK Ltd | 29519952 | 577.39 |
| DCMS | Natural History Museum | 2023-06-08 | Exhibition Construction / Hire | Public Programme (Temporary Exhibitions) | CBRE Managed Services Limited | S01086636 | 2476.85 |
| DCMS | Natural History Museum | 2023-06-08 | Travel-Air Passenger Transport | ES Mineral & Planetary Sciences Division | Key Travel Ltd | 51951453 | 352.00 |
| DCMS | Natural History Museum | 2023-06-08 | Accommodation | ES Mineral & Planetary Sciences Division | Key Travel Ltd | 51951453 | 211.32 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086474 | 1116.07 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086615 | 1593.44 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086622 | 1817.36 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086644 | 3891.80 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086642 | 3348.63 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086638 | 2579.23 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086631 | 2329.23 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086630 | 2077.95 |
| DCMS | Natural History Museum | 2023-06-08 | Grant Sub Contr./ award payments | Science Research | University of Aberdeen | 3023721 | 61206.20 |
| DCMS | Natural History Museum | 2023-06-08 | Agency Staff | Visitor Experience Donations & Maps | Adecco UK Ltd | 29538782 | 1571.67 |
| DCMS | Natural History Museum | 2023-06-08 | Assets under Construction | Estates Buildings Projects | DBR (London) Limited | 2260/AP00000021 | 515540.77 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086443 | 653.55 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086447 | 710.38 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086455 | 893.99 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086458 | 910.93 |
| DCMS | Natural History Museum | 2023-06-08 | Build & Eng Reactive Maintenance | Estates Buildings Ops | CBRE Managed Services Limited | S01086494 | 1219.00 |
| DCMS | Natural History Museum | 2023-06-08 | Agency Staff | Visitor Experience Cloakroom | Adecco UK Ltd | 29538782 | 1774.23 |
| DCMS | Natural History Museum | 2023-06-08 | Agency Staff | Visitor Experience Operations | Adecco UK Ltd | 29538782 | 226.89 |
| DCMS | Natural History Museum | 2023-06-08 | Agency Staff | Public Programme - Exhibition Sales | Adecco UK Ltd | 29538782 | 2588.65 |
| DCMS | Natural History Museum | 2023-06-08 | Agency Staff | Public Programme - Exhibition Sales | Adecco UK Ltd | 29538782 | 3378.18 |
| DCMS | Natural History Museum | 2023-06-08 | Accommodation | LS Diversity & Informatics Division | Key Travel Ltd | 51946329 | 1873.00 |
| DCMS | Natural History Museum | 2023-06-08 | Electricity | SS HV Ringmain | EDF Energy Cust Field Services | 000015343870 | 439678.30 |
| DCMS | Natural History Museum | 2023-06-08 | Travel-Air Passenger Transport | Science Research | Key Travel Ltd | 51943988 | 3680.43 |
| DCMS | Natural History Museum | 2023-06-19 | Grant Sub Contr./ award payments | Science Research | Universiti Malaya | 1800038839 | 30595.92 |
| DCMS | Natural History Museum | 2023-06-21 | Equip. Maint. & Repairs (non IT) | CRL Imaging & Analytical | Cameca | FASE2023.06-006 | 44489.91 |
| DCMS | Natural History Museum | 2023-06-22 | Prof. Fees excl Legal & Audit | NHM_Unlocked Programme (Estates) | CPC Project Services LLP |
|
23538.00 |
| DCMS | Natural History Museum | 2023-06-22 | Accommodation | Science Collections | Key Travel Ltd | 51960727 | 1100.72 |
| DCMS | Natural History Museum | 2023-06-22 | Travel-Air Passenger Transport | Sci Directorate | Key Travel Ltd | 51964320 | 1181.70 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Human Resources - Ops | Adecco UK Ltd | 29541515 | 577.39 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Olympic Construction Ltd | 14698 | 20251.50 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Olympic Construction Ltd | 14698 | 11098.50 |
| DCMS | Natural History Museum | 2023-06-22 | Accommodation | Science Research | Key Travel Ltd | 51961425 | 222.48 |
| DCMS | Natural History Museum | 2023-06-22 | Accommodation | Science Research | Key Travel Ltd | 51961425 | 114.71 |
| DCMS | Natural History Museum | 2023-06-22 | Accommodation | Science Research | Key Travel Ltd | 51961425 | 205.30 |
| DCMS | Natural History Museum | 2023-06-22 | Accommodation | Science Research | Key Travel Ltd | 51961427 | 779.10 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Estates Buildings Projects | CBRE Managed Services Limited | S01086597 | 2042.08 |
| DCMS | Natural History Museum | 2023-06-22 | Building & Engineering PPM | Estates Buildings Projects | CBRE Managed Services Limited | S01086588 | 1680.43 |
| DCMS | Natural History Museum | 2023-06-22 | Building & Engineering PPM | Estates Buildings Projects | CBRE Managed Services Limited | S01086580 | 6711.63 |
| DCMS | Natural History Museum | 2023-06-22 | Special Event Costs | Visitor Events | Wilson James Limited | 90085806 | 835.67 |
| DCMS | Natural History Museum | 2023-06-22 | Accommodation | Science Research | Key Travel Ltd | 51960358 | 534.10 |
| DCMS | Natural History Museum | 2023-06-22 | Special Event Costs | Visitor Events | Wilson James Limited | 90085803 | 2768.01 |
| DCMS | Natural History Museum | 2023-06-22 | Special Event Costs | Visitor Events | Wilson James Limited | 90085803 | 2728.58 |
| DCMS | Natural History Museum | 2023-06-22 | Estates Outsourced Services | Estates Security SK | Wilson James Limited | 90085786 | 12200.87 |
| DCMS | Natural History Museum | 2023-06-22 | Exceptional Costs | SS CHP Boilerhouse | The Environment Agency | NC11730-P3-10 | 4400.89 |
| DCMS | Natural History Museum | 2023-06-22 | Travel-Air Passenger Transport | Science Collections | Key Travel Ltd | 51959216 | 1173.11 |
| DCMS | Natural History Museum | 2023-06-22 | Travel-Air Passenger Transport | Science Research | Key Travel Ltd | 51958656 | 1037.81 |
| DCMS | Natural History Museum | 2023-06-22 | Estates Outsourced Services | Estates Security SK | Wilson James Limited | 90085056 | 20591.93 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101028276 | 1400.61 |
| DCMS | Natural History Museum | 2023-06-22 | Travel-Air Passenger Transport | Science Research | Key Travel Ltd | 51957230 | 2239.18 |
| DCMS | Natural History Museum | 2023-06-22 | Travel-Air Passenger Transport | Science Research | Key Travel Ltd | 51957934 | 861.56 |
| DCMS | Natural History Museum | 2023-06-22 | Travel-Air Passenger Transport | Science Research | Key Travel Ltd | 51957934 | 861.56 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101028276 | 1964.00 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101028276 | 11611.11 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101028276 | 8800.16 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101028276 | 2585.94 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101028276 | 2030.00 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101028276 | 240.00 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101028276 | 781.01 |
| DCMS | Natural History Museum | 2023-06-22 | Prof. Fees excl Legal & Audit | NHM_Unlocked Programme (Estates) | CPC Project Services LLP |
|
25054.00 |
| DCMS | Natural History Museum | 2023-06-22 | Hospitality | Comms - Media Relations | Wilson James Limited | 90085796 | 578.55 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Finance | Adecco UK Ltd | 29498118 | 694.80 |
| DCMS | Natural History Museum | 2023-06-22 | Special Event Costs | Membership | Wilson James Limited | 90085801 | 620.56 |
| DCMS | Natural History Museum | 2023-06-22 | Travel-Air Passenger Transport | LS Insects Division | Key Travel Ltd | 51965914 | 683.42 |
| DCMS | Natural History Museum | 2023-06-22 | Travel-Air Passenger Transport | LS Insects Division | Key Travel Ltd | 51965914 | 683.42 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Estates Buildings Projects | CBRE Managed Services Limited | S01086619 | 66093.96 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Estates Buildings Projects | CBRE Managed Services Limited | S01086613 | 1146.21 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Estates Buildings Projects | CBRE Managed Services Limited | S01086610 | 1559.89 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Estates Buildings Projects | CBRE Managed Services Limited | S01086558 | 5123.00 |
| DCMS | Natural History Museum | 2023-06-22 | Travel-Air Passenger Transport | LS Diversity & Informatics Division | Key Travel Ltd | 51960558 | 1207.77 |
| DCMS | Natural History Museum | 2023-06-22 | Travel-Air Passenger Transport | LS Vertebrates Division | Key Travel Ltd | 51967795 | 2238.27 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Membership | Adecco UK Ltd | 29561315 | 127.28 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Membership | Adecco UK Ltd | 29561315 | 143.19 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Membership | Adecco UK Ltd | 29561315 | 113.75 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Membership | Adecco UK Ltd | 29561315 | 127.28 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Membership | Adecco UK Ltd | 29561315 | 111.37 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Membership | Adecco UK Ltd | 29561315 | 107.87 |
| DCMS | Natural History Museum | 2023-06-22 | Accommodation | Engagement Group (MPG) | Key Travel Ltd | 51964813 | 558.97 |
| DCMS | Natural History Museum | 2023-06-22 | Accommodation | Sci Directorate | Key Travel Ltd | 51966838 | 2178.10 |
| DCMS | Natural History Museum | 2023-06-22 | Travel-Air Passenger Transport | Sci Directorate | Key Travel Ltd | 51966838 | 2260.60 |
| DCMS | Natural History Museum | 2023-06-22 | Exceptional Costs | SS CHP Boilerhouse | The Environment Agency | NC11730-P3-9 | 63684.72 |
| DCMS | Natural History Museum | 2023-06-22 | Estates Outsourced Services | Estates Security SK | Wilson James Limited | 90085787 | 20591.93 |
| DCMS | Natural History Museum | 2023-06-22 | Estates Outsourced Services | Estates Security SK | Wilson James Limited | 90085785 | 165207.61 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Visitor Experience Operations | Adecco UK Ltd | 29558371 | 463.68 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Visitor Experience Operations | Adecco UK Ltd | 29558371 | 115.92 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Visitor Experience Operations | Adecco UK Ltd | 29558371 | 231.84 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 9335.09 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 16513.96 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 10878.01 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 66019.49 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 94775.08 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 23382.16 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 79026.41 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 20802.80 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 34170.12 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 21128.44 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 17017.49 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 19640.13 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 6820.37 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 45462.48 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 12617.19 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 12515.90 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 758.26 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 4073.22 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 63161.81 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 88215.15 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 121703.99 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 4808.65 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 744.25 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 48.38 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101028277 | 576.87 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101028277 | 1964.00 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101028277 | 8180.60 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101028277 | 1233.02 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Mace Ltd | 2101028277 | 159.85 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 3734.39 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 7240.16 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 68949.03 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 7716.65 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 1846.27 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 3141.32 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 30134.06 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 7993.09 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 78834.06 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 23895.05 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 69740.92 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 3291.21 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 30947.32 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 14802.11 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Engagement Group (MPG) | Walter Lilly & Co. Ltd. | 1484/013 | 17586.38 |
| DCMS | Natural History Museum | 2023-06-22 | Grant Sub Contr./ award payments | Science Research | University of Strathclyde | 8319401 | 149447.38 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Estates Buildings Projects | Faithful+Gould Limited | 23021705 | 15636.80 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Estates Buildings Projects | Faithful+Gould Limited | 23021705 | 9432.50 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Estates Buildings Projects | Faithful+Gould Limited | 23021705 | 7488.40 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Estates Buildings Projects | Faithful+Gould Limited | 23021705 | 3200.40 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Estates Buildings Projects | Faithful+Gould Limited | 23021705 | 9410.70 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Estates Buildings Projects | Faithful+Gould Limited | 23021705 | 3773.25 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Estates Buildings Projects | Faithful+Gould Limited | 23021705 | 3255.00 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Estates Buildings Projects | Faithful+Gould Limited | 23021705 | 2488.50 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Estates Buildings Projects | Faithful+Gould Limited | 23021705 | 261.45 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Estates Buildings Projects | Faithful+Gould Limited | 23021705 | 1965.50 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Estates Buildings Projects | Faithful+Gould Limited | 23021705 | 1497.30 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Estates Buildings Projects | Faithful+Gould Limited | 23021705 | 1295.40 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Estates Buildings Projects | Faithful+Gould Limited | 23021705 | 1903.50 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | Estates Buildings Projects | Faithful+Gould Limited | 23021705 | 636.36 |
| DCMS | Natural History Museum | 2023-06-22 | Special Event Costs | Development Special Events | Event Concept Ltd | 40906 | 33850.34 |
| DCMS | Natural History Museum | 2023-06-22 | IT Costs | Technology Solutions - Operations | Open Text | 9003993372 | 48215.34 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Finance | Adecco UK Ltd | 29541516 | 550.05 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Visitor Experience Donations & Maps | Adecco UK Ltd | 29544306 | 438.23 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Visitor Experience Donations & Maps | Adecco UK Ltd | 29544307 | 1412.79 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Visitor Experience Cloakroom | Adecco UK Ltd | 29544306 | 1563.73 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Visitor Experience Operations | Adecco UK Ltd | 29544306 | 556.43 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Visitor Experience Operations | Adecco UK Ltd | 29544306 | 438.02 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Public Programme - Exhibition Sales | Adecco UK Ltd | 29544306 | 2444.10 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Public Programme - Exhibition Sales | Adecco UK Ltd | 29544306 | 3345.99 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Visitor Experience Operations | Adecco UK Ltd | 29544305 | 7707.29 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Learning: Community, Youth & Participation | Adecco UK Ltd | 29544305 | 553.58 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Visitor Experience Operations | Adecco UK Ltd | 29544308 | 8729.99 |
| DCMS | Natural History Museum | 2023-06-22 | Prof. Fees excl Legal & Audit | Risk & Assurance | Government Internal Audit Agency | 45631 | 25067.16 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Visitor Experience Cloakroom | Adecco UK Ltd | 29544307 | 1712.87 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Visitor Experience Operations | Adecco UK Ltd | 29544307 | 226.88 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Public Programme - Exhibition Sales | Adecco UK Ltd | 29544307 | 2570.41 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Public Programme - Exhibition Sales | Adecco UK Ltd | 29544307 | 3383.85 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | CRL Imaging & Analytical | Buehler Limited | INV202962 | 2026.40 |
| DCMS | Natural History Museum | 2023-06-22 | Agency Staff | Finance | Adecco UK Ltd | 29492832 | 694.80 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | CRL Imaging & Analytical | Buehler Limited | INV202276 | 600.80 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | CRL Imaging & Analytical | Buehler Limited | INV202040 | 16868.00 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | CRL Imaging & Analytical | Buehler Limited | INV202040 | 385.60 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | CRL Imaging & Analytical | Buehler Limited | INV202040 | 36.80 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | CRL Imaging & Analytical | Buehler Limited | INV202040 | 311.20 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | CRL Imaging & Analytical | Buehler Limited | INV202040 | 109.60 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | CRL Imaging & Analytical | Buehler Limited | INV202040 | 633.60 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | CRL Imaging & Analytical | Buehler Limited | INV202040 | 716.80 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | CRL Imaging & Analytical | Buehler Limited | INV202040 | 979.20 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | CRL Imaging & Analytical | Buehler Limited | INV202040 | 1468.80 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | CRL Imaging & Analytical | Buehler Limited | INV202040 | 716.80 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | CRL Imaging & Analytical | Buehler Limited | INV202040 | 716.80 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | CRL Imaging & Analytical | Buehler Limited | INV202040 | 716.80 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | CRL Imaging & Analytical | Buehler Limited | INV202040 | 158.40 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | CRL Imaging & Analytical | Buehler Limited | INV202040 | 302.40 |
| DCMS | Natural History Museum | 2023-06-22 | Assets under Construction | CRL Imaging & Analytical | Buehler Limited | INV202040 | 150.00 |
| DCMS | Natural History Museum | 2023-06-22 | Estates Outsourced Services | Estates Buildings Ops | Wilson James Limited | 90085813 | 518.40 |
| DCMS | Natural History Museum | 2023-06-22 | Exceptional Costs | Development | Wilson James Limited | 90085798 | 679.96 |
| DCMS | Natural History Museum | 2023-06-30 | Special Event Costs | Director’s Office | Kerb Food Limited | INV-2041 | 26250.00 |
| DCMS | Natural History Museum | 2023-06-30 | Grant Sub Contr./ award payments | Science Research | DE LA SALLE UNIVERSITY INC | 2000000325 | 108278.66 |
| DCMS | Natural History Museum | 2023-06-30 | Assets under Construction | CRL Imaging & Analytical | Teledyne CETAC Technologies | S020599703 | 6171.00 |
| DCMS | Natural History Museum | 2023-06-30 | Assets under Construction | CRL Imaging & Analytical | Teledyne CETAC Technologies | S020599703 | 196256.00 |
| DCMS | Natural History Museum | 2023-06-30 | Assets under Construction | CRL Imaging & Analytical | Teledyne CETAC Technologies | S020599703 | 6171.00 |
Γράφουμε λοιπόν:
if(!require(nortest)){
install.packages("nortest")
library(nortest)
}
shapiro.test(foi_expenditure_over_25000_june_2023$Amount)##
## Shapiro-Wilk normality test
##
## data: foi_expenditure_over_25000_june_2023$Amount
## W = 0.39378, p-value < 2.2e-16Φυσικά, η γραμμή 1 μπορεί να παραλειφθεί, αν έχει εγκατασταθεί ήδη το πακέτο. Βρίσκουμε, λοιπόν, p-τιμή μικρότερη του ήδη απειροελάχιστου \(2.2\cdot 10^{-16}\), όπερ σημαίνει ότι η μηδενική υπόθεση πρέπει να απορριφθεί ως απίθανη. Συνεπώς τα έξοδα του Ιουνίου του 2023 δεν είναι κανονικά κατανεμημένα.
Τα παραπάνω μπορούν να διαπιστωθούν και από το γράφημα Q-Q:
qqnorm(foi_expenditure_over_25000_june_2023$Amount)
qqline(foi_expenditure_over_25000_june_2023$Amount)
Βρίσκουμε, λοιπόν, κάτι που απέχει παρασάγγας από ευθεία γραμμή.
Συνολικά ο κώδικάς μας ήταν ο: