Lottery tickets to Danish Schools

Last week the weekly magazine Weekend Avisen brought a story on the first year results of a governmental initiative to incentivise danish schools with poor academic performance1. A schools performance is measured by the percentage of graduating students who get at least a ‘D’ (4) in both danish and mathematics2. By that measure the 121 schools that performed the worst over the last 3 years were eligible to be part of the initiative. If a school is able to improve by 5 percentage points the first year it will receive a bonus between 1.3 and 1.5 M DKK.

The angle in Weekend Avisen was that the government had it sound like they would spend around 160 M DKK on the initiative, but with only a little more than half the schools improving sufficiently, that amount dwindled to 90 M DKK. Here I will focus on another angle, which is briefly touched upon in the article, namely the rocky statistical basis the bonus relies on.

Most schools have between 20 and 60 students graduating each year. This means a single student by herself can have an impact of up to 5 percentage point, obviously this leaves a lot of room for random chance. Furthermore the schools in the initiative are selected for having performed poorly over the last three years. In order to estimate the chance of the schools making their target by random chance we must account for the regression towards mean (RTM) effect3. In short regression towards mean is an effect that dictates that if we have just seen an extreme observation the next observation is more likely to be closer to the average: If you throw a die and it comes up 1, there is a great chance that the next throw will be higher than 1. RTM is strongest when the outcome is highly stochastic and the first observation was very extreme. For the schools in the initiative, this means that small schools4 and schools with a poor baseline, have the greatest chance of making the target just by chance.

Without further ado; here is the list of schools and the probability that they will make their target performance by random chance. The expected number of schools that make the target without implementing any changes is 43, without accounting for RTM we would underestimated this number to 29.

School Approximate number of graduating students Percentage of "weak students" the last 3 years Probability of making the target by chance
Lindeskovskolen 21 50 % 53.9 %
Dalmose skole 18 54 % 52.5 %
Firkløverskolen 15 33 % 51.9 %
Hanstholm Skole 25 46 % 51.8 %
Maribo Skole - Borgerskoleafdelingen 25 46 % 51.8 %
Arenaskolen 31 62 % 51.5 %
Eskilstrup Skole 30 45 % 51.5 %
Stubbekøbing Skole 27 47 % 49.9 %
Tovshøjskolen 25 61 % 49.8 %
Nordvestskolen, Ravnsborg afdeling 22 49 % 49.1 %
H. C. Andersen Skolen 32 61 % 47.3 %
Tingbjerg heldagsskole 46 60 % 47.3 %
Gadehaveskolen 36 60 % 46.3 %
Særslev-Hårslev-Skolen 28 42 % 46.2 %
Læssøesgades Skole 37 46 % 46.1 %
Grønhøjskolen 41 52 % 46.1 %
Skælskør Skole 41 47 % 44.2 %
Svend Gønge-Skolen, Lundby 36 48 % 44 %
Lille Næstved Skole, afd. Fuglebjerg 33 46 % 43.7 %
Møn Skole, Stege 39 44 % 43.7 %
Sydfalster Skole 26 42 % 43.6 %
Tagensbo Skole 50 60 % 43.6 %
Abildgårdskolen 45 55 % 43.4 %
Blågård Skole 56 59 % 43.3 %
Søndre Skole, Rødby 39 50 % 42.9 %
Hirtshals undervisningssted 49 48 % 42.9 %
Mørkhøj Skole 40 46 % 42.2 %
Midtskolen, Sofiendal afdeling 42 41 % 41.9 %
Brændgårdskolen 34 35 % 41.1 %
Marienhoffskolen 38 37 % 41 %
Rådmandsgades Skole 62 57 % 40.4 %
Amagerskolen 35 44 % 40.4 %
Vejlebroskolen 42 47 % 39.3 %
Ikast Vestre Skole 41 40 % 39.3 %
Skærbæk Distriktsskole 53 45 % 38.8 %
Ålholm Skole 58 56 % 38.4 %
Nysted Skole 40 47 % 38.4 %
Åboulevarden 27 33 % 38.2 %
Høje Kolstrup Skole 45 39 % 37.8 %
Strandgårdskolen 50 46 % 37.5 %
Urbanskolen, Urban P 37 36 % 37.2 %
Skolen i Midten 24 33 % 37.2 %
Ullerup Bæk Skolen, Nr. Alle 38 38 % 37.2 %
Signaturskolen, Kvaglund 44 35 % 37 %
Ulstrup Skole 36 34 % 36.7 %
Nørreskov-Skolen 36 34 % 36.7 %
Kolind Centralskole 37 33 % 36.7 %
Vejle Midtbyskole 57 46 % 36.4 %
Baltorpskolen - Rugvænget 48 46 % 36.4 %
Dalumskolen 56 41 % 36.4 %
Nymarkskolen 51 41 % 36.1 %
Korsvejens Skole 47 40 % 36.1 %
Toftegårdsskolen 30 34 % 35.8 %
Baltorpskolen - Grantoften 55 48 % 35.4 %
Nørrevangsskolen 55 54 % 35.3 %
Nr Alslev Skole 36 41 % 34.8 %
Vesterbro Ny Skole 75 52 % 34.6 %
Brædstrup Skole 58 38 % 34.5 %
Breum Skole 26 31 % 34.3 %
Gildbroskolen 52 47 % 34.2 %
Roslev Skole 34 30 % 34.1 %
Frederiksværk skole, Enghave 58 40 % 34 %
Nordskolen - afdeling Højby 42 43 % 34 %
Distriktsskolen 43 34 % 33.9 %
Ejerslykkeskolen 53 42 % 33.7 %
Herstedlund Skole 60 46 % 33.6 %
Seden Skole 60 46 % 33.6 %
Kirstinebjergskolen, Havepladsvej 45 35 % 33.5 %
Avedøre Skole 67 41 % 33.5 %
Nørrebro Park Skole 72 53 % 33.5 %
Skovvangskolen 47 31 % 33.4 %
Nordals skolen 38 36 % 33.3 %
Nordvestskolen 36 35 % 33.1 %
Hannæs-Østerild Skole 29 32 % 33 %
Hendriksholm Skole 64 38 % 32.9 %
Charlotteskolen 43 37 % 32.6 %
Engstrandskolen 56 38 % 32.1 %
Sct. Jørgens Skole 44 31 % 32.1 %
Hammerum Skole 44 39 % 31.4 %
Langmarkskolen 52 37 % 31.2 %
NOVAskolen 47 34 % 31 %
Ulkebøl Skole 58 33 % 31 %
Birkhovedskolen 48 38 % 30.9 %
Munkebo Skole - mellem nor og fjord 58 35 % 30.7 %
Ørum Skole 62 38 % 30.6 %
Farsø Skole 69 34 % 30.5 %
Risingskolen 74 39 % 30.5 %
Højmarkskolen 55 33 % 30.4 %
Skolen ved Rønnebær Allé 49 35 % 30.4 %
Arresø Skole, Magleblik 65 43 % 30.3 %
Skolen ved Gurrevej 60 34 % 29.6 %
Gungehusskolen 60 34 % 29.6 %
Thorning Skole 39 30 % 29.5 %
Glostrup Skole, Skovvang 51 36 % 29.4 %
Ikast Østre Skole 68 38 % 29.3 %
Dagnæsskolen 54 36 % 29.3 %
Gribskolen 57 38 % 29 %
Østerbyskolen 57 38 % 29 %
Mølleholmskolen 76 34 % 28.6 %
Seminarieskolen 46 33 % 28.2 %
Sønderskov-Skolen 68 35 % 28.2 %
Distrikt Nord, Skagen skoleafdeling 44 32 % 28.1 %
Tungelundskolen 37 29 % 27.9 %
Herlev byskole 84 36 % 27.7 %
Borgerskolen 78 39 % 27 %
Hasle Skole 61 34 % 27 %
Herstedvester Skole 93 43 % 26.8 %
Søndervangskolen 103 47 % 26.8 %
Skolen ved Bulowsvej 92 40 % 26.8 %
Sophieskolen 81 43 % 26.3 %
Rønneskolen, afdeling Åvang 74 40 % 25.6 %
Bakkegårdsskolen 76 39 % 25.2 %
Sofiendalskolen 64 31 % 25.1 %
Horsens Byskole, folkeskoleafdeling 106 50 % 24.8 %
Tarm Skole 62 34 % 24.5 %
Brøndby Strand Skole 104 41 % 23.6 %
Tingstrup Skole 58 29 % 22.9 %
Auning Skole 74 32 % 21.3 %
Dueholmskolen 91 32 % 21.2 %
Distrikt Øst, Nordstjernenskoleafdeling 92 32 % 19.5 %
Distrikt Øst, Frydenstrandskoleafdeling 79 29 % 19.1 %
  1. Karakterer for millioner 

  2. The Ministry of Education seems to think that the negation of getting at least 4 in both Danish and Mathematics is getting less than 4 in both Danish and Mathematics. At one place the hope is to reduce the percentage of students who get less than 4 in Danish and Mathematics (“For at få del i puljen skal skolerne i hvert af de tre år reducere andelen af elever, der får under karakteren 4 i dansk og matematik”) at another place it is the percentage of students who doesn’t get at least 4 in danish and mathematics (“[…] forstået som andelen af elever, der ikke opnår mindst 4 i dansk og matematik i 9.-klasseprøverne”) 

  3. I assume that each school has an inherent probability that a graduating student will get at least 4 in danish and mathematics. The prior distribution for this probability is modelled by a beta distribution. The parameters of this distribution is chosen such that the mean is .22 (the national average) and the probability of being larger than .41 is 0.0431 (I assume all schools performing worse than .41 are part of the initiative). The posterior distribution for each school is again a beta distribution calculated from the baseline performance and the number of students graduating in 3 years. 

  4. The sizes of the schools are approximate and based on this list of school sizes supplemented with information from individual schools websites. 

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