It’s a busy time for Satish Pakki, a telecoms solutions architect, and his daughter, Vanshika, as they prepare for her 11-plus test in September.

Preparations entail weekly tuition in maths and English, and monthly mock tests at a local tuition centre. Vanshika is applying to Kendrick school in Berkshire – one of 16 grammars given funds to expand under a flagship government policy aimed at improving social mobility.

As a result, Kendrick, a girls’ school, is prioritising pupils from postcodes where it says levels of disadvantage are higher – and they don’t include Lower Earley in Reading, where the family live.

Satish Pakki, a Berkshire parent, says it is not a level playing field.

Pakki is not convinced his daughter is competing on a level playing field. “When they talk about poverty, I know there are some areas in Reading which are poorer, but if you go to the areas that are prioritised they already have a good school, and none of the houses there are below £500,000,” he says.

The housing website Zoopla suggests the average house price in Lower Earley is £377,000, while in nearby Earley, which is being prioritised, it is £592,000.

“I’m not saying we should have preference, but they should treat everyone equally,” Pakki says. “If my daughter doesn’t get in to Kendrick because children from more disadvantaged homes are being prioritised, then that’s fine. But I don’t think they are.”

The figures support his fears that schools will still fail to reach out to children from disadvantaged homes. Theresa May promised in September 2016 that the extra money to expand grammar schools and improve social mobility would help create “a truly meritocratic Britain”. Grammars would have to “contribute meaningfully”, she said. “This could mean taking a proportion of pupils from lower-income households, so that selective education is not reserved for those with the means to move into a catchment area or pay for tuition to pass the test.”

The government has provided an extra £50m a year for the expansion, creating about 30 extra places in each year group in each of the first 16 schools to receive the funding – a total of 2,700 extra places.

However, according to the data scientist James Coombs, from Comprehensive Future, a campaigning group for fair school admissions, at some schools only two or three of the 30 new places will go to pupil premium students – those looked after by local authorities or from poor homes. His analysis is based on a summary of schools’ plans to ensure new places go to pupils from disadvantaged backgrounds, recently published by the government.

Coombs concludes that none of the schools have reserved a proportion of places for pupils from poorer backgrounds, though some have said that if these pupils pass the entrance test, they will be admitted ahead of their better-off peers, and some have lowered the pass mark for poorer pupils. Just one, Altrincham grammar school for boys, has a target – its aim is for 15% of its year 7 students to be entitled to pupil premium by 2023.

Using free school meals’ data alongside test scores provided by the University of Durham, which sets the 11-plus tests, Coombs calculates that at Kendrick – his local school – just three out of 32 additional places are likely to go to girls eligible for free school meals. That’s despite the fact Kendrick has lowered its pass mark by five points for poorer pupils.

James Coombs has conducted a detailed analysis and found no school has reserved a specific proportion of places for children from poorer backgrounds. Photograph: Jill Mead/The Guardian

Two other expanding grammars, Sir William Borlase’s and John Hampden in Buckinghamshire, have respectively prioritised “up to” 10 and 12 places for disadvantaged children. Coombs calculates only two or three will go to such pupils.

He says that instead of manipulating the pass mark by lowering it to “catch” poorer students, schools should have followed May’s original suggestion and simply earmarked a set number of places for such pupils.

“Damian Hinds has repeatedly promised schools must have ‘realistic and ambitious plans’ to admit more disadvantaged children. He seems to assume these targets can be achieved with a sprinkling of fairy dust,” Coombs says.

Kendrick school and Sir William Borlase’s grammar school did not respond to requests for comment. But the head of John Hampden grammar school, Tracey Hartley, said in a written statement that the response to its public consultation had been overwhelmingly positive.

“We have lowered the qualifying mark for these 12 places to encourage applications from pupil premium and service premium students and this step has been very favourably received by our partner primary schools as it will be by many parents in our catchment area,” she said. “The awarding of the funding will benefit the young people in our community for years to come.”

Sir William Borlase’s grammar school in Buckinghamshire. Photograph: Felix Clay/The Guardian

Concern over the lack of social diversity in selective schools is heightened by new figures obtained by Lucy Powell, the Labour MP and former shadow education secretary, on the proportion of pupils on free school meals at the 16 expanding grammar schools. While the national average proportion of secondary pupils on free meals stood at 13.8%, the average in those selective schools stood at well below 3%. The lowest was at Sir William Borlase’s, which had just 0.45% of pupils – three – on free meals in January 2017.

Powell said none of the schools should be allowed to expand until they had improved those figures. “Grammar schools do nothing for social mobility, they do the opposite – they entrench it,” she said. “I just can’t see how the government can justify allowing schools to expand when they have such poor records in reflecting their communities.”

Jim Skinner, chief executive of the Grammar School Heads Association, said campaigners complaining there were too few poorer pupils in grammar schools were missing the point. He said official figures showed that while 12% of pupils who were not disadvantaged achieved a high level of attainment in reading, writing and maths, just 4% of those from disadvantaged backgrounds did so.

“We are fully supportive of the long-term goal of getting as many disadvantaged children in grammar schools as there are in the areas where those schools are. The work grammar schools are doing will help to close that national gap, but you can’t overnight have the same percentage of disadvantaged children in grammar schools as you have nationally,” he said.

A spokesman for the Department for Education said this was the first time grammar school expansion had been linked to a requirement to widen access. “We want schools to go further so that families who may not have previously thought about applying to a selective school consider it as an option. We have already seen innovative ideas from successful schools that are taking action such as setting up help desks in partner primary schools to assist parents registering children for the test, or holding information sessions in community libraries,” he said.