Math 7 · 7.RP.A.3

Percent error

A guess, a prediction or an experiment gives one number, and the true value turns out to be another. How far off was it, in a way that means the same thing whether the true value is 40 or 4,000?

You need

Percent increase and decreaseMath 7

Where you are 4 in Percent change

Fractions, ratio and percent3

Percent8

Percent change4

Percent increase and decrease

Percent error

The multiplier for a percent change Two percent changes one after the other

When you reach for it

Someone produced a number before the true one was known, and now the true one is known. Leocadia guessed how many marbles were in a jar, and then the jar was counted. An organizer predicted how many people would come to a concert, and then the tickets at the door were counted. A class worked out the time a pendulum takes to swing, and then compared it with the accepted value printed in a book.

Subtracting tells you how far off the number was. As with any change, that amount alone cannot say whether the miss was large. Being 10 marbles out is a poor guess for a jar of 40 and an excellent one for a jar of 4{,}000. To judge the miss, and to compare misses on things of different sizes, you measure it against the true value. That measurement, written as a percent, is the percent error.

The arithmetic is the arithmetic of percent increase and decrease: a gap, divided by the amount it is measured against. What changes is the question. There is no “before” and “after” here, only a value that was supposed to be right and the value that actually is.

The procedure

Two numbers go in: the estimate (a guess, a prediction, an experimental result) and the true value (the actual count, what really happened, or the accepted value). Take a playing field that is truly 50 meters long, which Noor paces out as 40 meters.

Find the size of the error. The error is how far the estimate sits from the true value:

|40 - 50| = |-10| = 10

So Noor was 10 meters out. The bars |\;| are the absolute value, which keeps the size of a difference and drops its sign; |a - b| is the distance between a and b on the number line, as in absolute value as a distance. The bars are there because an error is a distance, and a distance is never negative. Had Noor paced 60 meters, the error would be |60 - 50| = 10 meters all the same.

Figure 1Estimates of 40 and 60 meters sit the same distance, 10 meters, from the true 50, so both are off by \displaystyle \frac{10}{50} = 20\%.

Write the error as a percent of the true value. Now ask what percent 10 is of 50, the question from finding what percent one number is of another:

\frac{10}{50} = \frac{20}{100} = 20\%

Noor’s estimate has a percent error of 20\%. Put together, the two steps are one formula:

\begin{aligned} \text{error} &= |\text{estimate} - \text{true value}| \\ \text{percent error} &= \frac{\text{error}}{\text{true value}} \times 100\% \end{aligned}

Why the true value is the whole. A fraction means nothing until its whole is named, and here the question names it: “how far off was the estimate, for something this size?” The size of the thing is the true value. The estimate is the one number that might be wrong, so it is the wrong number to measure against. Two people pacing the same field are judged against the same 50 meters, whatever they paced.

This means the two numbers cannot be swapped. If the field were truly 40 meters and someone paced it as 50, the error would still be 10 meters, but measured against 40:

\frac{10}{40} = \frac{1}{4} = 25\%

Same two numbers, same gap, a different percent, because a different one of them is the truth.

The size of the answer. An estimate that is too low lies somewhere between 0 and the true value, so its error is at most the whole true value, and its percent error is at most 100\%. An estimate that is too high has no such ceiling: guessing 150 meters for a 50 meter field is 100 meters out, a percent error of 200\%, in the territory of percents above one hundred.

Keeping the direction. The absolute value throws away whether the estimate was too high or too low. When that matters, say it in words (“20\% too low”), or drop the bars and let the sign carry it: \displaystyle \frac{40 - 50}{50} = -20\%. Some science books use this signed form; a negative percent error then means the estimate was low.

Worked examples

Example 1

Marbles in a jar

Leocadia guesses that a jar holds 228 marbles. When the marbles are counted, there are 285. Find the percent error of her guess.

Solution

The error is the distance between the guess and the count:

|228 - 285| = 57 \text{ marbles}

The true value is the count, 285, so that is the whole. Both 57 and 285 are multiples of 57:

\frac{57}{285} = \frac{1}{5} = 20\%

Leocadia’s guess has a percent error of 20\%; it was 20\% too low.

Figure 2The true count, 285, is the whole and lines up with 100\%. The guess falls 57 marbles short, which is the last 20\% of the bar.

Check with the figure: the guess is 80\% of the true count, since 0.8 \times 285 = 228, so the part it falls short by is the remaining 100\% - 80\% = 20\%.

Example 2

A pendulum experiment

Kasimir’s group makes a pendulum from a weight and a string and finds that one full swing, out and back, takes 2.31 seconds. The accepted value for a pendulum of that length is 2.20 seconds. Find the percent error of the group’s result.

Solution

The error is

|2.31 - 2.20| = 0.11 \text{ seconds}

and the accepted value, 2.20 seconds, is the true value. Decimals in a fraction are easier to see in hundredths of a second: 0.11 seconds is 11 hundredths and 2.20 seconds is 220 hundredths, and measuring both in the same unit does not change their ratio.

\frac{11}{220} = \frac{1}{20} = 5\%

The group’s result is 5\% too high.

Example 3

A concert prediction

Pilar, organizing a concert, predicts that 3{,}150 people will come. On the night, 2{,}870 come. Find the percent error of her prediction, to the nearest tenth of a percent.

Solution

The error is

|3{,}150 - 2{,}870| = 280 \text{ people}

What actually happened is the true value, so the whole is the 2{,}870 people who came, not the 3{,}150 that were predicted:

\frac{280}{2{,}870} = 0.097560\ldots

As a percent that is 9.7560\ldots\%. The hundredths digit is 5, so the tenths digit goes up:

\frac{280}{2{,}870} \approx 9.8\%

Pilar’s prediction was about 9.8\% too high.

The whole is the 2{,}870 because the question is how far off the prediction was for a crowd of the size that actually came. The fraction \displaystyle \frac{280}{3{,}150} \approx 8.9\% measures something else: the share of the predicted crowd that never turned up.

Example 4

Two estimates of different things

Noor paces out the straight of a running track and gets 38 meters; it is really 40 meters. Oleander estimates a crowd watching a street parade at 4{,}200 people; a count from photographs gives 3{,}500. Who was closer, for the size of the thing they were estimating?

Solution

The two errors cannot be compared directly. Noor was |38 - 40| = 2 meters out and Oleander was |4{,}200 - 3{,}500| = 700 people out, and meters and people are different units. A percent has no unit, so measure each error against its own true value.

Noor:

\frac{2}{40} = \frac{5}{100} = 5\%

Oleander:

\frac{700}{3{,}500} = \frac{1}{5} = 20\%

Noor’s estimate was closer for its size: 5\% off against 20\%. Oleander’s crowd estimate was still a fair one for a crowd of thousands, but it was off by a fifth of the true count, and Noor’s was off by a twentieth.

Example 5

A prediction more than double

Mungo predicts that the school bake sale will raise 275 dollars. It raises 110 dollars. Find the percent error of his prediction.

Solution

The error is

|275 - 110| = 165 \text{ dollars}

measured against the amount actually raised, 110 dollars:

\frac{165}{110} = \frac{3}{2} = 1.5 = 150\%

The percent error is 150\%. That is over 100\% because the error, 165 dollars, is bigger than the true amount itself. Only an estimate that is too high can do that. Had Mungo erred the other way, the lowest he could have predicted was 0 dollars, an error of 110 dollars and a percent error of exactly 100\%, since an amount of money raised cannot be negative.

Example 6

Working back to the true value

Pilar guesses that a jar holds 660 marbles. She is told only that her guess has a percent error of 20\%. How many marbles could the jar hold?

Solution

The 20\% is a percent of the true count, which is the number not yet known, so call it T. There are two cases, because the guess may be too high or too low.

If the guess is too high, it is the true count plus 20\% of the true count, which is 120\% of it:

1.2\,T = 660, \qquad T = \frac{660}{1.2} = 550

If the guess is too low, it is the true count minus 20\% of it, which is 80\% of it:

0.8\,T = 660, \qquad T = \frac{660}{0.8} = 825

The jar holds either 550 or 825 marbles.

Check each: \displaystyle \frac{|660 - 550|}{550} = \frac{110}{550} = 20\%, and \displaystyle \frac{|660 - 825|}{825} = \frac{165}{825} = 20\%. The two errors, 110 and 165, are different, because each is 20\% of a different true count.

Taking 20\% of the guess, 132, and going that far either side gives 528 and 792, and neither works: \displaystyle \frac{132}{528} = 25\% and \displaystyle \frac{132}{792} \approx 16.7\%. The percent is of the true value, so the guess cannot be the starting point. Writing “120\% of T” as the single multiplication 1.2\,T is the idea of the multiplier for a percent change.

Boundaries

The true value must not be 0. The percent error divides by the true value, and division by zero is not defined. Guessing 3 marbles are left in a jar that is empty is 3 marbles out, but by no percent at all.
There has to be a true value. The formula needs a count, an outcome or an accepted value to measure against. Two guesses with no true value between them can be compared by subtraction, but neither has a percent error.
The amounts should be measured from a true zero. Counts, lengths, heights and times all start at a real “none at all”, and percent error is built for them. Temperatures in degrees Celsius or Fahrenheit do not: their zero is a chosen point, and the same experiment gives a different percent error on each scale (Exercise 19). The kelvin scale used in science starts at the coldest possible temperature, so its zero is a true one.
The bars hide the direction. A percent error of 20\% may be too high or too low. Say which in words, or use the signed form, when it matters.
Too low stops at 100\%; too high does not stop. An estimate of an amount cannot go below 0, which is 100\% out. An estimate that is too high can be any percent out.
The two numbers do not swap. The same gap gives a different percent depending on which number is the true one.
Percent error is not the whole story of a measurement. How large a reading’s error can be, how precise an instrument is, and the error as a plain fraction rather than a percent are the subject of Measurement error.

Where this goes next

Every reading taken with an instrument carries some error, and Measurement error gives the gap and its relatives their full names: the error as a plain fraction of the true value, the greatest error a scale can make, and the difference between being accurate and being precise. In a science laboratory, a percent error is the standard way to report how close an experiment came to the accepted value.

Working back from a percent error to the true value, as in Example 6, is a percent change run backwards, and it is quickest with the multiplier for a percent change. And the bars in the formula are the first everyday use of absolute value as a distance: the error is how far apart two numbers sit on the number line, with no regard for which side.

Exercises

Finding the percent error

Find the percent error of the estimate, given the true value.

Estimate 45, true value 50.
Estimate 63, true value 60.
Estimate 18, true value 24.
Estimate 2.7, true value 2.5.
Estimate 130, true value 80.

In context

Kasimir guesses that a jar holds 150 marbles. It holds 120. Find the percent error of his guess.
Oleander paces out a garden fence and estimates it at 27 meters. It is 24 meters long. Find the percent error.
Leocadia estimates a crowd at a summer fair at 1{,}020 people. The gate count is 1{,}200. Find the percent error, and say whether her estimate was too high or too low.
The organizers of a concert predict an audience of 900, and 740 people come. Find the percent error of the prediction, to the nearest tenth of a percent.
Noor predicts her bean plant will climb to 1.6 meters. It climbs to 2.5 meters. Find the percent error of her prediction.
A class times an echo and works out the speed of sound as 326.4 meters per second. The accepted value for the air in the room is 340 meters per second. Find the percent error.
Pilar’s group finds that one full swing of their pendulum takes 2.05 seconds. The accepted value is 2.00 seconds. Find the percent error.
Mungo paces out a garden path as 45 meters; it is 50 meters. Pilar guesses 345 marbles in a jar that holds 300. Whose estimate was closer for the size of the thing estimated?

Working back

A jar holds 500 marbles. A guess has a percent error of 6\%. What could the guess have been?
A swimming pool is 25 meters long. Which of the paced estimates 23 meters, 26.5 meters and 27.5 meters have a percent error of at most 8\%?
Mungo guesses 330 marbles, and his guess has a percent error of 10\%. Find the two values the formula allows for the true count, and say which one the jar must actually hold.

Reasoning

Explain why an estimate of a crowd that is too low can never have a percent error above 100\%, while an estimate that is too high can.
Kasimir says that guessing 90 when the true count is 100 has the same percent error as guessing 100 when the true count is 90. Find both percent errors and explain why he is wrong.
Leocadia’s class finds that water boils at 97 degrees Celsius, against an accepted 100 degrees Celsius. In degrees Fahrenheit, the same two temperatures are 206.6 and 212. Find the percent error on each scale, to the nearest tenth of a percent where needed, and explain why they differ.
Kasimir guesses there are 3 marbles left in a jar. The jar is empty. Explain why the percent error of his guess cannot be found, and say how the error could be described instead.
A jar holds 500 marbles. Oleander guesses 400 and Noor guesses 600. Show that their guesses have the same percent error. Then show that dividing by the guess instead of the true count would wrongly make one of them look better than the other.

Answers

10\%. The error is |45 - 50| = 5, and \displaystyle \frac{5}{50} = \frac{10}{100} = 10\%.
5\%. The error is |63 - 60| = 3, and \displaystyle \frac{3}{60} = \frac{5}{100} = 5\%.
25\%. The error is |18 - 24| = 6, and \displaystyle \frac{6}{24} = \frac{1}{4} = 25\%.
8\%. The error is |2.7 - 2.5| = 0.2, and \displaystyle \frac{0.2}{2.5} = \frac{2}{25} = \frac{8}{100} = 8\%.
62.5\%. The error is |130 - 80| = 50, and \displaystyle \frac{50}{80} = \frac{5}{8} = 0.625 = 62.5\%.
25\%. The error is |150 - 120| = 30 marbles, measured against the true 120: \displaystyle \frac{30}{120} = \frac{1}{4} = 25\%.
12.5\%. The error is |27 - 24| = 3 meters, and \displaystyle \frac{3}{24} = \frac{1}{8} = 12.5\%.
15\%, too low. The error is |1{,}020 - 1{,}200| = 180 people, and \displaystyle \frac{180}{1{,}200} = \frac{15}{100} = 15\%.
About 21.6\%. The error is |900 - 740| = 160 people, measured against the 740 who came: \displaystyle \frac{160}{740} = 0.216216\ldots \approx 21.6\%.
36\%. The error is |1.6 - 2.5| = 0.9 meters, and \displaystyle \frac{0.9}{2.5} = \frac{9}{25} = 36\%.
4\%. The error is |326.4 - 340| = 13.6 meters per second, and \displaystyle \frac{13.6}{340} = \frac{136}{3{,}400} = \frac{4}{100} = 4\%.
2.5\%. The error is |2.05 - 2.00| = 0.05 seconds, and \displaystyle \frac{0.05}{2.00} = \frac{5}{200} = 2.5\%.
Mungo. His error is 5 meters and \displaystyle \frac{5}{50} = 10\%; Pilar’s is 45 marbles and \displaystyle \frac{45}{300} = 15\%. The errors themselves are in different units and cannot be compared, but the percents can.
470 or 530. The error is 6\% of the true 500, which is 0.06 \times 500 = 30 marbles, so the guess was 500 - 30 = 470 or 500 + 30 = 530.
23 meters and 26.5 meters. The errors are 2, 1.5 and 2.5 meters, so the percent errors are \displaystyle \frac{2}{25} = 8\%, \displaystyle \frac{1.5}{25} = 6\% and \displaystyle \frac{2.5}{25} = 10\%. Equivalently, 8\% of 25 is 2, so the estimates allowed run from 23 to 27 meters.
300 or 366\tfrac{2}{3}, and the jar holds 300. Too high: 1.1\,T = 330 gives T = 300. Too low: 0.9\,T = 330 gives T = 366\tfrac{2}{3}. A number of marbles is a whole number, so only 300 is possible. Check: \displaystyle \frac{|330 - 300|}{300} = \frac{30}{300} = 10\%.
The percent error is the error divided by the true count. An estimate that is too low lies between 0 and the true count (a crowd cannot have fewer than 0 people), so its error is at most the true count itself, and the fraction is at most \displaystyle \frac{\text{true count}}{\text{true count}} = 1 = 100\%. An estimate that is too high can be any number above the true count, so its error can be any number of times the true count, and the percent error can be any size.
10\% and about 11.1\%. Guessing 90 for 100 is \displaystyle \frac{10}{100} = 10\% out. Guessing 100 for 90 is \displaystyle \frac{10}{90} = \frac{1}{9} = 0.111\ldots \approx 11.1\% out. The gap is 10 both times, but the percent error measures it against the true count, and the true count is different in the two cases.
3\% in Celsius and about 2.5\% in Fahrenheit. In Celsius the error is 3 degrees and \displaystyle \frac{3}{100} = 3\%. In Fahrenheit it is 212 - 206.6 = 5.4 degrees and \displaystyle \frac{5.4}{212} = 0.02547\ldots \approx 2.5\%. They differ because the percent compares the error with how far the true value sits from 0, and the two scales put their 0 at different temperatures, neither of which is “no heat at all”. The accepted boiling point is 100 degrees above the Celsius zero but 212 degrees above the Fahrenheit zero. On the kelvin scale, whose zero is the coldest possible temperature, the accepted value is about 373 kelvins and the percent error is \displaystyle \frac{3}{373.15} \approx 0.8\%.
The percent error is the error divided by the true count, here \displaystyle \frac{3}{0}, and dividing by zero is not defined: there is no true amount to measure the error against. The error can still be given as an amount: the guess was 3 marbles too high.
Each guess is 100 marbles out, since |400 - 500| = |600 - 500| = 100, and each is measured against the true 500: \displaystyle \frac{100}{500} = 20\% for both. Dividing by the guesses instead gives \displaystyle \frac{100}{400} = 25\% for Oleander and \displaystyle \frac{100}{600} \approx 16.7\% for Noor, which makes Noor look closer although both missed the same jar by the same amount.