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 %