How to calculate a square-root grade curve

Imagine an exam score of sixty-four percent. A square-root curve can turn it into eighty percent. Why? It does not add the same number of points to every student. It reshapes the entire zero-to-one-hundred scale, while keeping scores in the same order.

First write sixty-four percent as zero point six four of one. Picture a square whose area is zero point six four of a unit square. If each side is zero point eight, then zero point eight squared is zero point six four. The square root recovers that side length.

Rescale the side back to a percent: curved grade equals one hundred times the square root of the raw fraction. If raw score is already written as a number from zero to one hundred, the equivalent shortcut is ten times the square root of that raw number.

For sixty-four percent, the square root of sixty-four is eight. Multiply eight by ten to get eighty percent. The gain is sixteen percentage points: eighty minus sixty-four. Percentage points compare two percent grades; this is not a sixteen percent relative increase.

Reverse the calculation to check it. Divide the curved eighty by ten to get eight, then square eight to recover the raw sixty-four. In fraction form, zero point eight squared is zero point six four. Both paths lead back to the original grade.

Plot raw score across the bottom and curved score up the side. The square-root line rises from zero to one hundred, but it bends toward a flatter slope near the top. It stays above the equal-score diagonal between the endpoints, so grades rise without changing their rank order.

Compare three raw scores. Twenty-five becomes fifty, a gain of twenty-five points. Sixty-four becomes eighty, a gain of sixteen. Ninety becomes about ninety-four point eight seven, a gain of about four point eight seven. The increase is clearly not a flat bonus.

There is a boundary worth noticing: zero stays zero, and one hundred stays one hundred. The largest absolute boost actually occurs at raw twenty-five, not at zero. This curve can change low scores substantially, so a course policy should decide whether and how to apply it.

To use this square-root curve, take the square root of the raw percentage number and multiply by ten. Compare the answer with the raw grade in percentage points. For sixty-four, the result is eighty percent. To work backward from a target curved score, divide by ten and square.