Schoolchildren from Scotland’s poorest neighbourhoods are falling further behind pupils from the wealthiest homes, new data on exam results has revealed.

The latest figures for all 32 Scottish councils show that children in the least well-off areas are seven times less likely to get the right grades for university than those from the most affluent neighbourhoods.

In four local authority areas, including wealthy East Lothian, no children from the poorest 20% achieved the benchmark for university entrance of three or more As from up to five Highers – Scotland’s equivalent to the A-level – in 2014.

The statistics, from a freedom of information request by the Scottish Conservatives, which replicated an investigation by the Guardian in 2012, shows that the gap between the least wealthy 20% of pupils and the richest 20% has widened sharply.

They caused a fresh row at Holyrood over the Scottish government’s record after eight years in power, in which Ruth Davidson, the Scottish Conservative leader, accused Nicola Sturgeon of neglecting a long-running, obvious problem. The first minister insisted she was taking action.

The gap between the least well-off and better-off pupils gaining at least three As grew by more than three percentage points between the 2011 results disclosed by the Guardian and the 2014 data published on Thursday by the Scottish Tories.

In 2011, 17.4% of the best-off children achieved that result, compared to 2.5% of the least well-off. Last year, that gap had jumped to 21.13% of the wealthiest getting three or more As, compared to 2.99% of the poorest pupils.

In some of Scotland’s richest areas, the gulf in performance was significant. In East Renfrewshire, the authority with Scotland’s best-performing state schools, there was a 31.5-point gap between the best and least well-off pupils.

Davidson raised the figures at first minister’s questions, where she told Sturgeon they did not make pretty reading.

“We knew that the Scottish National party government was not closing the attainment gap,” Davidson said. “Now, from these figures, we know that the gap between the richest and poorest students is actually getting wider.

“The first minister has said that she wants to be judged on her record. In education, her record is one of failure, and the experts say that her plans will not fix it. I ask her: how bad do things have to get before we see the action that we need?”

Sturgeon said she was taking the right steps, including releasing £11m for the Scottish government’s attainment challenge fund, which was focusing on improving standards and education at 300 primary schools with the poorest outcomes.

Concentrating on primary education was the most productive way of building up skills, she said. Additional measures were being introduced, including a new national attainment framework to measure performance, Sturgeon added.

She said there were already small signs of improvement for pupils from the poorest backgrounds. When the SNP took power in 2007, 23% of pupils from the least wealthy areas achieved one Higher, but this has risen to 40%. Achievement gaps in other exam areas are also improving, the first minister said.

“I have made it very clear how serious I am about improving education in Scotland and closing the attainment gap,” Sturgeon told MSPs, as she brushed off Scottish Liberal Democrat complaints about a new national testing regime leading to league tables by the back door. “We are seeing evidence of the attainment gap in Scotland closing. That is not far enough or fast enough for my liking, which is why I am determined that we go further and faster.

“As Ruth Davidson knows, we are taking action, and we will continue to take action. I am not standing here saying that there is not more work to do – and I never have stood here and said that. That is why we have taken the action around the attainment challenge that I have already talked about.”