Percentage error, and the denominator you choose

You forecast 1,200. The real figure came in at 1,000. How far off were you?

Two hundred, obviously. As a percentage it depends on a choice nobody announces out loud: which number goes on the bottom.

• Against the true value: 200 ÷ 1,000 × 100 = 20%
• Against your own forecast: 200 ÷ 1,200 × 100 = 16.67%

Both are arithmetically fine. Only the first is percentage error, and the gap between them is not small.

The definition

percentage error = |measured − actual| ÷ |actual| × 100

The true value goes on the bottom. Always. That is what makes one measurement's error comparable with another's: the reference has to be the thing you were aiming at, not the thing you produced. The vertical bars mean absolute value — the size of the gap, sign discarded.

Dividing by your own estimate flatters you whenever you overshoot, because overshooting makes the denominator bigger and the error look smaller. That is exactly the wrong incentive to build into a measurement.

A worked example

A scale reads 48 grams for something that truly weighs 50.

|48 − 50| = 2, then 2 ÷ 50 × 100 = 4%.

Use the reading as the denominator instead and you get 2 ÷ 48 × 100 = 4.17%. Close enough here to look like a rounding difference, which is why the habit survives. On the forecast above, the same swap moved 20% down to 16.67%.

Absolute value throws away the direction

Measure 48 against a true 50 and you get 4%. Measure 52 against a true 50 and you get |52 − 50| ÷ 50 × 100 = 4% as well. Two opposite failures, one number.

If direction matters, drop the bars and keep the sign: (48 − 50) ÷ 50 × 100 = −4% against (52 − 50) ÷ 50 × 100 = +4%. An instrument that always reads low has a bias, which you can correct. One that scatters either side has noise, which you cannot. The absolute form cannot tell you which you have.

Averaging errors, where the small denominator takes over

Three days of estimates against what actually happened:

• Actual 200, measured 210: 10 ÷ 200 × 100 = 5%
• Actual 50, measured 56: 6 ÷ 50 × 100 = 12%
• Actual 1,000, measured 970: 30 ÷ 1,000 × 100 = 3%

Average the three percentages: (5 + 12 + 3) ÷ 3 = 6.67%.

Or total the errors and divide by the total actual: (10 + 6 + 30) ÷ (200 + 50 + 1,000) × 100 = 46 ÷ 1,250 × 100 = 3.68%.

Nearly double, from the same three days. Neither is a mistake. The first treats each day as one vote, so the day with 50 units counts as much as the day with 1,000 and its 12% drags the average up. The second weights each day by its size. Both are defensible; publishing one without saying which is not.

Where percentage error breaks down

• A true value of zero. The formula divides by it, so there is no answer. There is no percentage of nothing — report the absolute difference instead.
• A true value near zero. Defined, but violently unstable. True 2, measured 3 is an error of one unit and 1 ÷ 2 × 100 = 50%. Percentages stop being informative when the base is tiny.
• Negative true values. The absolute bars sit on the denominator too, so −48 against a true −50 gives 2 ÷ 50 × 100 = 4%, the same as the positive pair. That is deliberate: it keeps the error positive rather than flipping its sign with the units.

What percentage error is not

It compares one measurement against a value you already know to be true. It is not an instrument's stated accuracy, which is a manufacturer's claim about a range of conditions. It is not a confidence interval, which is about uncertainty rather than a known miss. And if you do not have a true value, you cannot compute it at all — what you can compute is how far two measurements disagree, which is a weaker claim and should be worded as one.

Doing it without the paper

The free Percentage Error tool takes Actual and Measured, in that order, and returns the absolute percentage error — dividing by the actual value, as the definition requires. Which of your two numbers is the true one is the judgement; the division is the easy part.