Modelling Kendrick School’s proposed admission policy

Background

Kendrick Grammar School is described as ‘super selective’ school.  Essentially, candidates sit an entrance test, the results are then standardised to an arbitrary level of precision to ensure each applicant can be uniquely ranked and places are awarded to the highest scoring candidates. 

The school are now proposing to change their admissions to give preference to disadvantaged girls and one quarter of the places for local applicants who meet a Qualifying Score (QS).  The school’s full proposals can be found on their website but remarkably, don’t explain how the Qualifying Score (QS) is determined other than it might be ‘up to 5 points lower’ for disadvantaged applicants.  These two values are absolutely critical in determining how admissions operate and the lack of an objective definition of them is akin to playing a game of football and deciding afterwards where the goal posts should be.

The model

This model predicts how the proposed changes would affect the two targeted demographics, disadvantaged and ‘local’ applicants, for a range of different combinations of Qualifying Scores based on the two following datasets.

SourceDescription
DfEHome locations of those attending the school in 2015 is used to model geographical segmentation of applicants
2017 test resultsThe full 2017 test results are used to provide a model distribution

The same test is used in a neighbouring county and the results included in the ranking so it’s not possible to just take the n highest scoring results as those which would normally secure a place.  Fortunately the school published an FAQ which explains, “The lowest qualifying score admitted in September 2017 was 114.38” (excluding disadvantaged applicants.)  This enables a quantile of those who ‘passed’ the test to be defined after taking into consideration that about one third of the ranked scores relate to candidates not even applying for a place at the school. 

Assumptions

In creating the model the following assumptions were made:

  • The proportion of disadvantaged girls getting a place before being afforded any priority is 2.0%.  DfE data for 2015-16 shows 2.7% of the schools pupils in receipt of Pupil Premium (PP) but over a third (5 out of 13) of the pupils which made up that figure started Y7 in 2012.  Data for 2017-18 shows when that cohort moved on, the overall PP actually dropped to 2.1% even though the school had by then taken steps to prioritise disadvantaged applicants.  The figure of 2.0% chosen here is probably over generous.
  • Disadvantaged applicants are unlikely to live in the wider P2 catchment area as they cannot afford the cost of commuting.  (Eg. Newbury to Reading annual child rail season ticket is £878.)  This assumption allows the following geographic/socioeconomic model of applicants:
    • 2% disadvantaged living in P1
    • 38% non-disadvantaged living in P1
    • 60% non-disadvantaged living in P2
  • The scores for these geographic and socioeconomic segments are evenly distributed through the distribution.

Constants

Coefficient Value Description
PP 2% Proportion of disadvantaged (Pupil Premium) applicants who would succeed in getting a place without any preference being afforded them.
P1 38% The school have defined an inner “priority area”.  P1 is the proportion of (non-disadvantaged) candidates who live in this new inner priority area.   
P2 60% P2 is the proportion of applicants who are neither disadvantaged nor living in the inner priority area.
P 96 The number of existing places
NP 32 The number of new places prioritised for applicants in group P1.

Parameters

Parameter Description
QS The Qualifying Score.  This is not fixed value so is a variable to the model.  A range of values between 106 and 111 are modelled
PPQS The DfE define disadvantaged children as those in receipt of Pupil Premium (PP) funding hence PPQS is used to signify the The Qualifying Score for this group.  This is also not fixed and therefore defined as a variable to the model.  For each QS this will be modelled as 1, 3 and 5 marks below QS. 
MSSThe Minimum Successful Score (MSS) can be thought of as the effective cut-off point if the school simply filled all places with the highest scoring in the tests after accounting for the fact that about one third of the scores related to candidates applying to other schools. For 2017 this was 114.38 but there were only 96 places so for the expanded 128 form entry needs to be scaled:

 151 \times \frac {128}{96} \approxeq 201

The 201st ranked candidate scored 111.52 giving:

 MSS=111.52

Quantiles

Within the distribution three quantiles are defined based on the input parameters

QuantileDescription
MSS \leqq  \textbf{a} The count of scores at or above the Minimum Successful Score (MSS)
 QS \leqq \textbf{b} < MSS The count of scores at or above the Qualifying Score but below the MSS.  This can be thought of as the ‘benefit zone’ for both local and disadvantaged applicants
 PPQS \leqq \textbf{c} < QS The count of scores at or above the lower QS for disadvantaged applicants but below the QS for others.  This can be thought of as the ‘benefit zone’ for both disadvantaged applicants only.

These quantiles and their relationship to the model’s parameters are shown here:

Kernel Density Plot of Kendrick 2017 test results

Formulae

The predicted numbers of applicants from each socio-economic and demographic segment likely to be admitted to the school can be modelled using the following formulae based on the constants, parameters and quantiles defined above, The total number of disadvantaged girls getting a place is given by

 PP_t_o_t_a_l = (P +NP) \times \frac{a+b+c}{a} \times PP

The number of applicants living inside the inner catchment area (P1) who get one of the 96 ‘original’ places is given by

 P_p_1 = (P - PP_t_o_t_a_l) \times \frac{P1}{P1+2}

The number of applicants living outside the inner catchment area who get one of the 96 ‘original’ places is given by

 P_p_2 = (P - PP_t_o_t_a_l) \times \frac{P2}{P1+2}

The number of inner catchment area (P1) applicants who get one of the 32 places for which they are prioritised is given by

 NP_p_1 = NP \times \frac {a+b}{a} \times \frac{P1}{P1+2}

The remainder of the 32 places prioritised to those living in the inner catchment which go to those living outside this area is given by

 NP_p_2 = NP - NP_p_1

‘Preference’

40% of Kendrick’s girls already come from what they call the ‘local Reading area’ so if, for example, the school were to promise to ensure that the 25% new places all went to local girls this would be meaningless. In the context of a public consultation the key question which needs to be answered is what additional benefits are being offered to the targeted segments. 

Two further formulae can be defined which can quantify this for both groups.  The number of additional disadvantaged girls who would only be admitted as a result of the proposed changes is given by

 PP_p_r_e_f = (P + NP) \times \frac {b+c}{a} \times PP

The number of additional local applicants who would be admitted as a results of the proposed changes to give them priority over the additional 32 places is

 LP_P_1_p_r_e_f = NP \times \frac {b}{a} \times \frac {P1}{P1 + P2}

Conclusions

This graph models the number of additional places going ‘local’ applicants for a range of Qualifying Scores between 106 and 111 and additional places for disadvantaged applicants where this is set a further 1, 3 and 5 marks lower. 

The school’s expansion is being funded by the Selective Schools Expansion Fund (SSEF) which is predicated on their demonstrating ‘realistic and ambitious’ plans to admit more disadvantaged children, although setting the Pupil Premium Qualifying Score (PPQS) a full 10 marks below the Minimum Successful Score (MSS) is predicted to only result in three additional disadvantaged applicants being admitted. 

The school have promised that the 32 new places will be prioritised for local applicants, which itself appears to be contrary to the terms of the SSEF funding.  Setting the Qualifying Score a full 5 marks below the MSS would only result in one third (8) of these new places going to local applicants who wouldn’t already get a place. 

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