The National Association of Head Teachers has been bombarded with letters and telephone calls from heads deeply sceptical about the accuracy of the results of the curriculum tests for 11-year-olds which they sent to parents at the end of the summer term.

The union is now analysing results for a full report in September, and says the credibility of the tests is in doubt. This could embarrass the Government which has fought union boycotts of the key stage 2 tests, the results of which might eventually be used for value-added league tables.

John Kenward, chair of the NAHT’s curriculum and assessment committee, has studied nearly 60 letters from angry heads and heard dozens of complaints.

“The whole credibility of the SATs exercise must be called into question now,” he said. “If our analysis of the test results backs up our initial findings we will have to ask whether the SATs road is the right one to go down. It does appear that the primary curriculum is so vast that to try to take a snapshot pencil-and-paper test at the end of a key stage is almost impossible. Our current view is that you either have to have a comprehensive series of tests which would be extortionate in terms of time or you go back to moderated teacher assessment.”

Last month it was revealed that Government officials were investigating allegations that several schools had cheated in the KS2 tests. These new complaints may be an even bigger embarrassment to the School Curriculum and Assessment Authority because the protests are being voiced from Cumbria to Cornwall.

Unrealistically high scores achieved by some SEN children entitled to “support” from ancillary staff is one of many anomalies that have raised serious doubts.

Heads believe maths scores have also been distorted, partly because schools found it impossible to monitor the use of calculators which were permitted for some questions but not for others. Another complaint was that children were given insufficient time to finish the first maths paper and were demoralised before starting the second.

Crawley Ridge junior school in Camberley, Surrey, believed the maths test prevented pupils from demonstrating their true capability, and discovered most of its 67 pupils did as well or better on the KS3 level 3-5 paper that 14-year-olds took this year.

Science papers were criticised for being easier than maths or English as they tested children’s retention of facts rather than their ability to carry out an investigation. One Hertfordshire primary reported that only 3 per cent of its 11-year-olds had achieved level 5 (above average) in English whereas 67 per cent had obtained level 5 in science.

Many schools reported unrealistically low English scores, largely because insufficient time was allowed for the comprehension test. One Surrey head said several of her pupils earned level 5 in writing and handwriting but only obtained a level 4 in comprehension. “That means they were given a level 4 overall,” she said. “But we’ve told the parents not to worry. You can’t be a level 4 reader and a level 5 writer. It’s one of the many inconsistencies in these tests.”

Although schools were supposed to be relieved of the responsibility for marking KS2 tests many had to moderate the English markers’ work because there were so many errors. One school said that more than half of the 64 papers it submitted were incorrectly marked.

Oxfordshire primary heads sent a 10-point list of grievances to David Hart, the NAHT general secretary. They complain that the tests damaged pupils’ self-esteem and disrupted schools, and that the reporting system enabled parents of children in small schools to identify individual pupils’ scores.

Geoff Clifford, the Oxfordshire NAHT branch secretary, said: “The marking of the tests has been clearly inaccurate in some cases, and unfair in others. ”

A spokeswoman for SCAA said only 500 appeals had been lodged out of nearly 2 million scripts. “Even so, we will take seriously any feedback we get from schools.”