Math 7 · N.Q.A.3

Measurement error

No ruler, scale or stopwatch gives the exact amount. How far off can a reading be, and when does being off actually matter?

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Measurement error

Why this exists

Count the eggs in a box and you can be exactly right: there are 6, not 6.02. Measure the box with a ruler and you cannot. The edge of the box falls somewhere between two marks, and you write down the nearest one. The real length is a little more or a little less.

That is not a mistake by the person measuring. It is built into every measurement of a length, a mass, a time, a volume or a temperature, because these amounts can take any value and a tool can only show finitely many. So the honest questions are “how far off could it be?” and “is that much off a lot, for this job?” This lesson answers both, and separates two words that sound alike, accuracy and precision, which name two different ways a reading can be good.

Definition

The true value t is the amount actually there. The measured value m is the reading. The measurement error is the gap between them.

Absolute error is the size of that gap, in the same unit as the measurement:

\text{absolute error} = |m - t|

The bars are absolute value: the absolute error is never negative. Whether the reading was too high or too low is said separately, in words.

Greatest possible error. When a reading is taken to the nearest mark of a scale, and the marks are u apart, the true value is at most half a space from the reading. That largest possible gap, \tfrac{1}{2}u, is the greatest possible error of the reading. A digital display works the same way, with u the place value of its last digit: a stopwatch showing hundredths has u = 0.01\text{ s}. How finely a tool is marked is its precision, and reading a scale tells you what u is.

Relative error is the absolute error as a fraction of the true value:

\text{relative error} = \frac{|m - t|}{t}

It has no unit, because the units on top and bottom cancel. Written as a percent, it is the percent error.

Accuracy and precision. A measurement is accurate when it is close to the true value, that is, when its error is small. A measurement is precise when it is fine and repeatable: the tool reads in small units, and measuring the same thing again gives readings that agree closely with each other. Accuracy can only be judged against the true value. Precision can be judged without it, from the tool’s unit or from the spread of repeated readings.

Intuition

Lay a pencil along a ruler marked in whole centimeters, as in Figure 1. Its end sits a little past the 7 mark, closer to 7 than to 8, so you read 7\text{ cm}.

Figure 1The pencil reads 7\text{ cm} to the nearest mark, so its true length is somewhere in the bracketed stretch.

Any length from 6.5\text{ cm} to 7.5\text{ cm} would have given the same reading, because every point in that stretch is nearer to the 7 mark than to the 6 or the 8. The reading names the stretch, not a point. So a measurement is often written with its greatest possible error attached: 7 \pm 0.5\text{ cm}, read “seven plus or minus a half”.

Whether 0.5\text{ cm} is a lot depends on what is being measured. Half a centimeter off on a pencil is noticeable; half a centimeter off on the length of a room is nothing. Relative error is that judgment in numbers: the same gap divided by a bigger amount gives a smaller fraction.

Accuracy and precision are easiest to see on repeated readings. In Figure 2 each dot is one reading of the same amount and the tall tick is the true value.

Figure 2Accuracy is how close the readings sit to the true value. Precision is how close they sit to each other.

Bunched dots are precise, wherever they sit. Dots centered on the tick are accurate, however spread out they are. A tool can be one without the other: a kitchen scale that was never set to zero gives the same wrong mass every time, which is precise and inaccurate (top right).

Results

Greatest possible error. A reading r taken to the nearest mark of a scale whose marks are u apart has greatest possible error \tfrac{1}{2}u, and the true value t satisfies

r - \tfrac{1}{2}u \le t \le r + \tfrac{1}{2}u

which is written t = r \pm \tfrac{1}{2}u. Halving the space between marks halves the greatest possible error.

Relative error. The same absolute error is a smaller relative error on a bigger amount. When the true value is not known, the greatest possible error divided by the reading, \dfrac{\tfrac{1}{2}u}{r}, estimates the largest relative error the reading could have.

Greatest possible errors add. If one length is known to within e_1 and another to within e_2, their sum is known to within e_1 + e_2. This is the worst case. The actual errors may partly cancel, so the actual error of the sum can be smaller than e_1 + e_2, but never larger.

Why it is true

Half the unit. A reading of r to the nearest mark means the true value was closer to the mark r than to either neighbor, r - u or r + u. The point halfway between r and r + u is r + \tfrac{1}{2}u; everything below it is closer to r, everything above it closer to r + u. In the same way, everything above r - \tfrac{1}{2}u is closer to r than to r - u. So the true value lies between r - \tfrac{1}{2}u and r + \tfrac{1}{2}u, and can be as much as \tfrac{1}{2}u from r but no more. This is exactly rounding to the nearest mark, and the half comes from the midpoint.

Errors add. Suppose two lengths are measured as a and b, with true values within e_1 of a and within e_2 of b. The smallest the true total can be is (a - e_1) + (b - e_2) = (a + b) - (e_1 + e_2), and the largest is (a + e_1) + (b + e_2) = (a + b) + (e_1 + e_2). So the true total is within e_1 + e_2 of a + b. Those extremes happen only when both readings are off by the most possible in the same direction; nothing rules that out, so the bound cannot be made smaller.

Worked examples

Example 1

Reading a measuring jug

Florentyna pours water into a measuring jug marked every 50\text{ mL}. The water level is closest to the 650\text{ mL} mark. What is the greatest possible error, and what range must the true volume lie in?

Solution

The marks are u = 50\text{ mL} apart, and the reading is taken to the nearest mark, so the greatest possible error is half a space:

\tfrac{1}{2} \times 50 = 25\text{ mL}

The true volume is within 25\text{ mL} of the reading on either side:

650 - 25 = 625 \qquad 650 + 25 = 675

So the true volume is between 625\text{ mL} and 675\text{ mL}, written 650 \pm 25\text{ mL}.

Example 2

Reading a digital display

A digital thermometer shows 17.3^\circ\text{C}, and a stopwatch shows 25.43\text{ s}. Find the greatest possible error and the range of the true value for each.

Solution

A display has no marks to look at, but its last digit plays the same part. The thermometer shows tenths, so it rounds to the nearest u = 0.1^\circ\text{C}, and the greatest possible error is \tfrac{1}{2} \times 0.1 = 0.05^\circ\text{C}. The true temperature is between 17.3 - 0.05 = 17.25^\circ\text{C} and 17.3 + 0.05 = 17.35^\circ\text{C}.

The stopwatch shows hundredths, so u = 0.01\text{ s} and the greatest possible error is 0.005\text{ s}. The true time is between 25.425\text{ s} and 25.435\text{ s}. More decimal places mean a smaller unit and a narrower range.

Example 3

Small error, big error

Gwilym measures a classroom wall with a tape measure and gets 7.96\text{ m}. Its true length is 8\text{ m}. He also measures a pencil and gets 13.3\text{ cm}, when its true length is 14\text{ cm}. Find the absolute and relative error of each. Which measurement was better?

Solution

Absolute error is the gap between reading and true value. For the wall,

|7.96 - 8| = 0.04\text{ m}

which is 4\text{ cm}. For the pencil,

|13.3 - 14| = 0.7\text{ cm}

The wall’s error is more than five times the pencil’s. But relative error divides each by the size of the thing, with top and bottom in the same unit:

\begin{aligned} \text{wall:}\quad & \frac{0.04}{8} = 0.005 \\ \text{pencil:}\quad & \frac{0.7}{14} = 0.05 \end{aligned}

The wall reading is wrong by five thousandths of the wall; the pencil reading is wrong by five hundredths of the pencil, ten times as large a share. So the wall measurement was the better one, even though its error in centimeters was bigger.

Example 4

A fine scale and a coarse one

Henrike weighs flour on a kitchen scale marked every 5\text{ g} and reads 125\text{ g}. She then stands on a bathroom scale marked every 0.5\text{ kg} and reads 50\text{ kg}. Compare the greatest possible errors, and the relative errors they allow.

Solution

The kitchen scale’s greatest possible error is \tfrac{1}{2} \times 5 = 2.5\text{ g}. The bathroom scale’s is \tfrac{1}{2} \times 0.5 = 0.25\text{ kg}, which is 250\text{ g}, a hundred times as much.

The true values are not known, so divide each greatest possible error by its reading:

\begin{aligned} \text{flour:}\quad & \frac{2.5}{125} = 0.02 \\ \text{Henrike:}\quad & \frac{0.25}{50} = 0.005 \end{aligned}

The coarse bathroom scale gives the smaller relative error, a quarter of the kitchen scale’s, because 50\text{ kg} is so much larger than its half-unit. A scale’s marks have to suit the size of what it weighs, and both of these do.

Example 5

Timing a race: accuracy against precision

A race is timed by an electronic gate at 11.40\text{ s}, which is taken as the true time. Two people time a video of the same race three times each. Ignatius uses a stopwatch showing hundredths and gets 11.64, 11.66 and 11.65\text{ s}. Jolene uses one showing tenths and gets 11.2, 11.6 and 11.4\text{ s}. Whose timing is more precise, and whose is more accurate?

Solution

Precision is about the readings among themselves. Ignatius’s three times are only 11.66 - 11.64 = 0.02\text{ s} apart, and his stopwatch reads in hundredths. Jolene’s are 11.6 - 11.2 = 0.4\text{ s} apart. Ignatius is far more precise.

Accuracy is about distance from the true time. Ignatius’s errors are

\begin{aligned} |11.64 - 11.40| &= 0.24 \\ |11.66 - 11.40| &= 0.26 \\ |11.65 - 11.40| &= 0.25 \end{aligned}

so he is about a quarter of a second slow every time, as he would be if he pressed the button a little late at the finish. Jolene’s errors are 0.2, 0.2 and 0 seconds, and the Mean of her times is

\frac{11.2 + 11.6 + 11.4}{3} = \frac{34.2}{3} = 11.4\text{ s}

exactly the true time. Every one of Jolene’s readings is closer to the truth than every one of Ignatius’s, so she is more accurate. Extra digits show precision; they do not promise accuracy.

Example 6

Adding up errors: a perimeter

Eamon measures a rectangular tabletop to the nearest centimeter: 110\text{ cm} by 70\text{ cm}. What is the perimeter, and how far off could it be?

Solution

From the readings, the perimeter is 2(110 + 70) = 360\text{ cm}.

Each side is known only to within 0.5\text{ cm}. The shortest the table could be is 109.5\text{ cm} by 69.5\text{ cm}, and the longest 110.5\text{ cm} by 70.5\text{ cm}:

\begin{aligned} \text{least perimeter} &= 2(109.5 + 69.5) = 358\text{ cm} \\ \text{greatest perimeter} &= 2(110.5 + 70.5) = 362\text{ cm} \end{aligned}

So the perimeter is 360 \pm 2\text{ cm}. The greatest possible error grew from 0.5\text{ cm} to 2\text{ cm} because the perimeter counts each measured side twice, so the greatest possible errors add up to 2 \times 0.5 + 2 \times 0.5 = 2. That is the worst case, with both readings off the most in the same direction. The actual error may well be smaller, since a length read too long and a width read too short partly cancel.

Example 7

Choosing a tool to suit the job

Henrike wants to measure a gap about 2.5\text{ m} wide with a relative error of at most 0.001. Will a tape measure marked in centimeters do? What is the widest spacing of marks that would?

Solution

Turn the relative error into an absolute one by multiplying by the size, as in finding a part of a quantity:

0.001 \times 2.5 = 0.0025\text{ m} = 2.5\text{ mm}

So the greatest possible error must be at most 2.5\text{ mm}. It is half the spacing of the marks, so the spacing can be at most 2 \times 2.5 = 5\text{ mm}.

A centimeter tape has a greatest possible error of 5\text{ mm}, giving a relative error of \dfrac{0.005}{2.5} = 0.002, twice what is allowed, so it will not do. A millimeter tape gives \dfrac{0.0005}{2.5} = 0.0002, well inside the limit.

Boundaries

Half a unit assumes a well-made tool read correctly. The greatest possible error measures only the rounding to the nearest mark. A kitchen scale that reads 9\text{ g} with nothing on it, or a tape with its first centimeter broken off, is off by more than half a unit on every reading, and the marks give no warning. The greatest possible error describes precision, not accuracy.

Not every display rounds. If a display drops the digits it cannot show instead of rounding, a reading can be off by up to a whole unit, all in one direction. The half-unit rule is for readings taken to the nearest mark.

Exactly halfway. A true value exactly at r + \tfrac{1}{2}u is as close to one mark as the other, and which one it is read as is a matter of convention. That is why the range is written with “at most”: the error never exceeds \tfrac{1}{2}u.

Relative error needs a true zero. Dividing by the true value only makes sense when zero means “none of it”, as it does for length, mass, time and volume. A temperature of 0^\circ\text{C} is not “no temperature”, so the same pair of readings gives different relative errors in Celsius and in Fahrenheit, and the ratio means nothing; percent error works one through. For the same reason relative error cannot be found at a true value of zero.

Dividing by the reading is an estimate. When the true value is unknown, the greatest possible error divided by the reading is slightly under the largest relative error: the true value could be smaller than the reading, and dividing by a smaller number gives a bigger result. For 125 \pm 2.5\text{ g} the worst case is \tfrac{2.5}{122.5}, a little over 0.02. The estimate is close whenever the error is small next to the reading.

Where this goes next

Relative error is most often written as a percent, and percent error does exactly that, including comparing a measured value with an accepted one.

The greatest possible error also decides how many digits of a reading mean anything. Significant figures are the digits a tool actually supported, so a length read as 7\text{ cm} on a centimeter ruler should never be copied out as 7.00\text{ cm}, which would claim a much finer tool.

Example 6 points further on: any quantity worked out from measurements, such as a perimeter, an area or a volume, carries their errors with it, and Exercise 18 shows that for an area the errors do not simply add.

Exercises

Greatest possible error

A ruler is marked in millimeters, and a screw reads 84\text{ mm} to the nearest mark. Find the greatest possible error and the range the true length lies in.
A measuring jug is marked every 25\text{ mL}, and the level is closest to the 425\text{ mL} mark. Find the greatest possible error and the range of the true volume.
A bathroom scale is marked every 0.2\text{ kg} and reads 58.4\text{ kg}. Find the greatest possible error and the range of the true mass.
A digital thermometer shows 21.7^\circ\text{C}. Find the greatest possible error and the range of the true temperature.
One length is recorded as 3\text{ m} to the nearest meter, another as 3.00\text{ m} to the nearest centimeter. Give the greatest possible error and the range for each.

Absolute and relative error

A kitchen scale reads 515\text{ g} for a bag of rice that holds exactly 500\text{ g}. Find the absolute and relative error, and say whether the reading is too high or too low.
A window is 1.25\text{ m} tall, and a tape measure gives 1.20\text{ m}. Find the absolute and relative error.
A 400\text{ m} run is timed with a hand stopwatch at 75.6\text{ s}, and the timing gate gives 72.0\text{ s}. Taking the gate’s time as true, find the absolute and relative error.
A screw 40\text{ mm} long is measured as 42\text{ mm}, and a board 4\text{ m} long is measured as 3.98\text{ m}. Which has the bigger absolute error, and which the bigger relative error?
A ruler marked in millimeters reads 25\text{ mm}, and a tape marked in centimeters reads 250\text{ cm}. Estimate the largest relative error of each reading by dividing its greatest possible error by the reading.
A bathroom scale has a relative error of 0.004 when it weighs someone whose true mass is 75\text{ kg}. What is its absolute error?
A crumb 2\text{ mm} across is read as 5\text{ mm} on a ruler held at a slant. Find the absolute and relative error. Can a relative error be more than 1?

Accuracy and precision

A kitchen scale reads 9\text{ g} with nothing on it. Weighing a block of butter three times, it reads 259\text{ g} each time. (a) What is the butter’s true mass, assuming the scale is otherwise correct? (b) Is the scale precise? Is it accurate?
Three thermometers are put in a mixture of ice and water, whose true temperature is 0.0^\circ\text{C}, three times each. Thermometer A reads 2.1, 2.0 and 2.2; B reads -0.6, 0.5 and 0.1; C reads 0.1, 0.0 and -0.1, all in degrees Celsius. Find the mean and the spread (largest minus smallest) of each, and describe each as precise or not and accurate or not.
A tape measure has its first centimeter broken off, so its end is at the 1\text{ cm} mark. A shelf reads 63.4\text{ cm} on it. What is the shelf’s true length, and has the break changed the tape’s precision, its accuracy, or both?

Combining measurements

The side of a square tile is measured as 30\text{ cm} to the nearest centimeter. Between what values does its perimeter lie, and what is the perimeter’s greatest possible error?
Two pieces of string measure 45\text{ cm} and 38\text{ cm}, each to the nearest centimeter. They are laid end to end. Between what values does the total length lie?
A rectangle measures 12\text{ cm} by 8\text{ cm}, each to the nearest centimeter. Find the least and greatest possible area, and how far each is from the area worked out from the readings.
A length of about 4\text{ m} must be measured with a relative error of at most 0.0005. What is the widest spacing of marks that will do? Will a centimeter tape do? A millimeter tape?

Reasoning

Explain why no reading from a marked scale can be exact, while the number of eggs in a box can be.
Explain why the greatest possible error of a reading to the nearest mark is half the space between marks, and not the whole space.
One stopwatch shows hundredths of a second and another shows tenths. Which is more precise? Can the one showing tenths give more accurate times? Explain.
A time of 20\text{ s} is measured 0.1\text{ s} off, and a mass of 1000\text{ g} is measured 4\text{ g} off. Explain why these two errors cannot be compared directly, then compare them with relative error and say which measurement was better.

Answers

Greatest possible error 0.5\text{ mm}; the true length is between 83.5\text{ mm} and 84.5\text{ mm}. The marks are 1\text{ mm} apart, and half of that is 0.5\text{ mm}.
Greatest possible error 12.5\text{ mL}; the true volume is between 412.5\text{ mL} and 437.5\text{ mL}, from 425 \pm 12.5.
Greatest possible error 0.1\text{ kg}; the true mass is between 58.3\text{ kg} and 58.5\text{ kg}. Half of 0.2 is 0.1.
Greatest possible error 0.05^\circ\text{C}; the true temperature is between 21.65^\circ\text{C} and 21.75^\circ\text{C}. The display shows tenths, so u = 0.1.
To the nearest meter: 0.5\text{ m}, so between 2.5\text{ m} and 3.5\text{ m}. To the nearest centimeter: 0.005\text{ m}, so between 2.995\text{ m} and 3.005\text{ m}. The two zeros in 3.00 say the length was read a hundred times more finely.
Absolute error 15\text{ g}, relative error 0.03, too high. |515 - 500| = 15 and \tfrac{15}{500} = 0.03.
Absolute error 0.05\text{ m} (too low), relative error 0.04. |1.20 - 1.25| = 0.05 and \tfrac{0.05}{1.25} = 0.04.
Absolute error 3.6\text{ s}, relative error 0.05. |75.6 - 72.0| = 3.6 and \tfrac{3.6}{72} = 0.05.
The board has the bigger absolute error, 0.02\text{ m} = 20\text{ mm} against the screw’s 2\text{ mm}. The screw has the bigger relative error: \tfrac{2}{40} = 0.05 against \tfrac{0.02}{4} = 0.005.
Ruler: \tfrac{0.5}{25} = 0.02. Tape: \tfrac{0.5}{250} = 0.002. The tape’s greatest possible error is ten times the ruler’s, but its reading is a hundred times the ruler’s, so its relative error is ten times smaller.
0.3\text{ kg}. The absolute error is the relative error times the true value: 0.004 \times 75 = 0.3.
Absolute error 3\text{ mm}, relative error 1.5. |5 - 2| = 3 and \tfrac{3}{2} = 1.5. Yes: a relative error above 1 means the error is bigger than the thing itself.
(a) 250\text{ g}, since every reading is 9\text{ g} too high and 259 - 9 = 250. (b) It is precise, since the three readings agree exactly, but not accurate, since every reading is 9\text{ g} off.
A: mean 2.1^\circ\text{C}, spread 0.2; precise but not accurate. B: mean 0.0^\circ\text{C}, spread 1.1; accurate on average but not precise. C: mean 0.0^\circ\text{C}, spread 0.2; precise and accurate. For A, \tfrac{2.1 + 2.0 + 2.2}{3} = 2.1; for B, \tfrac{-0.6 + 0.5 + 0.1}{3} = 0; for C, \tfrac{0.1 + 0.0 - 0.1}{3} = 0.
62.4\text{ cm}. The tape starts at 1\text{ cm}, so every reading is 1\text{ cm} too long: 63.4 - 1 = 62.4. The marks are as fine as before, so its precision is unchanged; only its accuracy is lost.
Between 118\text{ cm} and 122\text{ cm}, a greatest possible error of 2\text{ cm}. The side is between 29.5 and 30.5, and 4 \times 29.5 = 118, 4 \times 30.5 = 122.
Between 82\text{ cm} and 84\text{ cm}. Each piece is within 0.5\text{ cm}, so the total, 83\text{ cm}, is within 0.5 + 0.5 = 1\text{ cm}: 44.5 + 37.5 = 82 and 45.5 + 38.5 = 84.
Least 86.25\text{ cm}^2, greatest 106.25\text{ cm}^2. The readings give 12 \times 8 = 96\text{ cm}^2. The least is 11.5 \times 7.5 = 86.25, which is 9.75 below; the greatest is 12.5 \times 8.5 = 106.25, which is 10.25 above. The error is not the same both ways, because each side’s half centimeter is multiplied by the other side: 0.5 \times 8 + 0.5 \times 12 + 0.5 \times 0.5 = 10.25.
Marks at most 4\text{ mm} apart. The greatest possible error may be 0.0005 \times 4 = 0.002\text{ m} = 2\text{ mm}, so the spacing may be 2 \times 2 = 4\text{ mm}. A centimeter tape gives \tfrac{0.005}{4} = 0.00125, too big, so it will not do; a millimeter tape gives \tfrac{0.0005}{4} = 0.000125, so it will.
A length, mass or time can be any amount at all, but a scale has only finitely many marks, so the true amount almost always falls between two of them and the reading rounds it. Eggs come in whole numbers, and counting them leaves nothing to round.
A reading to the nearest mark means the true value was closer to that mark than to its neighbors. The points closer to it than to the next mark are those before the midpoint, which is half a space away. A whole space off would put the true value on the next mark, which would have been the reading.
The one showing hundredths is more precise: its unit is 0.01\text{ s} against 0.1\text{ s}. But yes, the tenths one can be more accurate. Precision is how finely and repeatably a tool reads; accuracy is how close the readings are to the truth. A finely marked stopwatch pressed late every time is precise and inaccurate, as Ignatius’s was in Example 5.
The errors are in different units, seconds and grams, so neither number is bigger than the other in any meaningful way. As fractions of the amounts: \tfrac{0.1}{20} = 0.005 and \tfrac{4}{1000} = 0.004. The mass measurement was better, by a little.