Approximations and Errors
Your fitness app says you walked $2.4$ miles. The route is really $2.43$ miles. Is the app's number a good approximation? In science, engineering, and even a school lab, you almost never get the exact value. You get an approximation, and you need a way to say how far off it is and whether that matters. That is what this page is about.
Truncating and Rounding
Two ways to shorten a long decimal are truncating (chopping) and rounding. Try the widget: change the number of digits and watch what happens to the last digit.
When you approximate a number $N$, the two neighbors at a given place bracket it. Truncating a positive number gives a lower approximation (less than or equal to $N$). Rounding up gives an upper approximation (greater than or equal to $N$).
Approximate the decimal number $4.8275$ to two decimal places using the truncation method and the rounding method.
Approximate the decimal number $4.8275$ to two decimal places using the truncation method and the rounding method.
Approximate the irrational number $\sqrt{7}$ to three decimal places, clearly indicating which is the lower bound approximation and which is the upper bound…
Approximate the irrational number $\sqrt{7}$ to three decimal places, clearly indicating which is the lower bound approximation and which is the upper bound approximation.
Absolute Error and Relative Error
Once you have an approximation, you want to say how far off it is. There are two ways: the plain difference and the difference compared with the size of the number.
The absolute value makes the error always positive, because we only care about how far off you are, not in which direction.
Both bars have the same absolute error, but the pink part is a much bigger share of the pencil. The relative error is what tells you whether an approximation is good.
A common use of relative error is a tolerance: a limit on how much error is allowed.
A digital thermometer shows a temperature of $38.2^\circ \text{C}$. If a high-precision instrument shows that the real temperature is $38.45^\circ \text{C}$, determine…
A digital thermometer shows a temperature of $38.2^\circ \text{C}$. If a high-precision instrument shows that the real temperature is $38.45^\circ \text{C}$, determine the absolute error and the relative error of the digital thermometer's measurement.
During a track and field competition, the head judge measures the length of the track and records $399.5$ ft. If the official and exact length of the track must be $400$…
During a track and field competition, the head judge measures the length of the track and records $399.5$ ft. If the official and exact length of the track must be $400$ ft, calculate the absolute error and the percentage relative error of the judge's measurement. Do you consider it an acceptable measurement if the regulations allow a maximum relative error of $0.2\%$?
When making the budget for a construction project, the cost of materials is estimated to be $\$15\,400…
When making the budget for a construction project, the cost of materials is estimated to be $\$15\,400$. At the end of the project, the invoices add up to a real total of $\$15\,825$. Calculate the absolute and relative error of the initial estimate. If the construction company assumes the losses if the relative error exceeds $3\%$, who will pay the difference?
Errors with Fractions
Many exact values are fractions, like $\tfrac{2}{3}$. If you turn them into decimals too early, you get decimals that are themselves approximate. It is cleaner to compute the error with fractions and keep it exact.
Calculate the absolute error made when using the approximation $0.33$ for the exact value of the fraction $\frac{1}{3}$. Express the result as a fraction.
Calculate the absolute error made when using the approximation $0.33$ for the exact value of the fraction $\frac{1}{3}$. Express the result as a fraction.
From the previous exercise, calculate the relative error made when approximating $\frac{1}{3}$ by $0.33$. Express the result as a percentage.
From the previous exercise, calculate the relative error made when approximating $\frac{1}{3}$ by $0.33$. Express the result as a percentage.
Two carpenters cut wooden slats that must measure exactly $\frac{5}{7}$ of a foot. The first carpenter cuts the slat to $0.71$ ft and the second cuts it to $0.72$ ft.…
Two carpenters cut wooden slats that must measure exactly $\frac{5}{7}$ of a foot. The first carpenter cuts the slat to $0.71$ ft and the second cuts it to $0.72$ ft. Without using decimals, operating with fractions, determine which of the two has made a smaller absolute error.
Comparing Precision and Very Large Numbers
The measurement with the smaller relative error is the more precise one. To compare, write both errors in the same form, either both as decimals or both as percents.
Huge numbers are usually written in scientific notation, and the approximation often is too. To find $E_a$, write both numbers with the same power of $10$ and subtract the front parts.
Two different measurements have been taken. In the first, a relative error of $0.015$ has been made. In the second, the relative error was $1.2\%$. Determine which of…
Two different measurements have been taken. In the first, a relative error of $0.015$ has been made. In the second, the relative error was $1.2\%$. Determine which of the two measurements is more precise and justify your answer.
In a chemistry laboratory, a student uses the value $6.02 \cdot 10^{23}$ for Avogadro's number in their calculations. If the most precise value currently accepted is…
In a chemistry laboratory, a student uses the value $6.02 \cdot 10^{23}$ for Avogadro's number in their calculations. If the most precise value currently accepted is $6.02214 \cdot 10^{23}$, determine the absolute error in scientific notation and the percentage relative error made by the student.
The mean distance from the Earth to the Sun is approximately $149\,597\,870$ miles. A high school textbook states that the distance is $1.5 \cdot 10^8$ miles. Calculate…
The mean distance from the Earth to the Sun is approximately $149\,597\,870$ miles. A high school textbook states that the distance is $1.5 \cdot 10^8$ miles. Calculate the percentage relative error introduced by the textbook when using this approximation in scientific notation.
Error Bounds When You Round
Often you do not know $V_r$ at all. All you know is that a number was rounded. You cannot find $E_a$ exactly, but you can find an error bound: a number the error cannot exceed.
State the absolute error bound assumed when stating that the weight of an object, rounded to three decimal places, is $12.458$ lb.
State the absolute error bound assumed when stating that the weight of an object, rounded to three decimal places, is $12.458$ lb.
The size of a computer file is $4567.89$ Megabytes (MB). When copying it to a hard drive, the operating system truncates the value to one decimal place, while a…
The size of a computer file is $4567.89$ Megabytes (MB). When copying it to a hard drive, the operating system truncates the value to one decimal place, while a compression program rounds it to one decimal place. Calculate the difference between the absolute error made by the operating system and that made by the compression program.
To calculate the density of a metal block, its mass is measured obtaining $m = 12.4$ lb and its volume obtaining $V = 0.004$ ft$^3$. If both measurements have been…
To calculate the density of a metal block, its mass is measured obtaining $m = 12.4$ lb and its volume obtaining $V = 0.004$ ft$^3$. If both measurements have been rounded to the last decimal digit shown, determine the absolute error bound of the mass and the volume.
Intervals and Error Bounds
An error bound is the same thing as an interval around the approximate value. If $V_a$ is the approximation and $E_a$ is the error bound, then
$$|x - V_a| \le E_a \iff V_a - E_a \le x \le V_a + E_a \iff x \in [V_a - E_a,\ V_a + E_a].$$We often write this as $V_a \pm E_a$. For example, a scale reading of $250 \pm 5$ oz means the real weight is in $[245, 255]$.
To go the other way, from an interval $[a, b]$ back to the approximation, use the midpoint and the half-width:
$$V_a = \frac{a+b}{2} \qquad E_a = \frac{b-a}{2}.$$The largest possible relative error happens when the real value is smallest, so divide by the left endpoint: $E_r^{\max} = \dfrac{E_a}{a}$.
Sometimes two different approximations give two different intervals for the same unknown $x$. Then $x$ must be in both, so you take the intersection.
A kitchen scale gives a reading of $250$ oz with an absolute error bound of $5$ oz. Write the range of values in which the real weight of the object lies in the form of…
A kitchen scale gives a reading of $250$ oz with an absolute error bound of $5$ oz. Write the range of values in which the real weight of the object lies in the form of a mathematical interval.
The following diagram shows the confidence interval of a measurement in a physics experiment, where the central point represents the approximate value $V_a$ and the…
The following diagram shows the confidence interval of a measurement in a physics experiment, where the central point represents the approximate value $V_a$ and the endpoints represent the limits given by the absolute error bound $E_a$. If the endpoints of the interval are $14.35$ and $14.65$, determine the approximate value $V_a$, the absolute error bound $E_a$ and the maximum possible relative error.
It is known that when approximating a number $x$ by $3.14$, the absolute error made is less than $0.005$. On the other hand, when approximating the same number $x$ by…
It is known that when approximating a number $x$ by $3.14$, the absolute error made is less than $0.005$. On the other hand, when approximating the same number $x$ by $3.146$, the absolute error is less than $0.002$. Express both conditions as intervals, calculate their intersection and determine the range of possible values for the exact number $x$.
Working Backward from a Relative Error
Sometimes you know the approximation and the relative error, and you must find the real value. Write the definition as an equation and solve it. Because of the absolute value, there are two cases: the real value is above the approximation, or below it.
Now a general fact. If a number is rounded, the worst relative error happens when the real number is as small as possible, because then you divide by the smallest value.
A financial analyst estimates the price of a stock on the stock market. We know that the relative error of his estimate was exactly $4\%$. If the estimated value he…
A financial analyst estimates the price of a stock on the stock market. We know that the relative error of his estimate was exactly $4\%$. If the estimated value he published was $\$26.00$, set up and solve the necessary equations to find the two possible exact real values that the stock could have.
Prove mathematically that if a positive number $x$ is rounded to the nearest whole number obtaining an approximation $A$, the relative error $E_r$ always satisfies the…
Prove mathematically that if a positive number $x$ is rounded to the nearest whole number obtaining an approximation $A$, the relative error $E_r$ always satisfies the inequality $E_r \le \frac{0.5}{A - 0.5}$. Apply this formula to find the maximum relative error bound when rounding to $A = 5$.
Errors in Sums, Products, and Quotients
When you calculate with approximate measurements (an area, a speed), the errors travel into the answer. To see how much, find the smallest and largest results by using the smallest and largest possible inputs.
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Sums: the error bounds add up. (The same is true for differences, in the worst case.)
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Products of positive numbers: smallest result = (smallest)(smallest), largest = (largest)(largest).
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Quotients of positive numbers: largest result = largest numerator $\div$ smallest denominator, smallest result = smallest numerator $\div$ largest denominator.
An architect measures the dimensions of a rectangular room, obtaining $4.5$ ft long and $3.2$ ft wide. He knows that his tape measure has an absolute error bound of…
An architect measures the dimensions of a rectangular room, obtaining $4.5$ ft long and $3.2$ ft wide. He knows that his tape measure has an absolute error bound of $0.1$ ft for each measurement. Calculate the measured area and determine between which maximum and minimum values the real area of the room lies.
The propagation speed of a signal in a fiber optic network is calculated using the formula $v = \frac{d}{t}$. A distance of $d = 1200 \pm 5$ ft and a time of…
The propagation speed of a signal in a fiber optic network is calculated using the formula $v = \frac{d}{t}$. A distance of $d = 1200 \pm 5$ ft and a time of $t = 0.006 \pm 0.0001$ s are measured. Calculate the maximum and minimum possible values for the speed $v$, and deduce from them the general absolute error bound of the speed measurement.
Powers, Roots, and How Errors Grow
What happens when a measurement is squared, or cubed, or raised to a big power? Suppose a side $s$ is measured $1\%$ too large, so you use $1.01\,s$. Then
$$(1.01\,s)^3 = 1.030301\,s^3,$$so the volume is about $3.03\%$ too large, about $3\%$. In general, writing a small relative error as $\varepsilon$,
$$(1 + \varepsilon)^n = 1 + n\varepsilon + (\text{tiny terms}),$$because $\varepsilon^2$ is much smaller than $\varepsilon$ when $\varepsilon$ is small.
For a function like $y = \sqrt{x}$ that keeps increasing, plug the endpoints of the interval for $x$ into the function. That gives the endpoints for $y$.
The most dramatic case of growth is an exponent that is large, like years in compound interest, $A = P(1 + r)^t$.
In the manufacture of a cylindrical part for a high-precision engine, the radius is required to be $r = 2.50$ in and the height $h = 8.00$ in. If the machinery has a…
In the manufacture of a cylindrical part for a high-precision engine, the radius is required to be $r = 2.50$ in and the height $h = 8.00$ in. If the machinery has a relative error bound of $1\%$ in any linear measurement, show algebraically whether the maximum relative error in the calculation of the volume of the cylinder ($V = \pi r^2 h$) will be $1\%$, $2\%$ or $3\%$. Assume that the relative errors are small enough to add their linear approximations.
A student must calculate the value of $y = \sqrt{x}$. He knows that the exact value of $x$ belongs to the interval $[15.8, 16.2]$. If he decides to use $x = 16$ to…
A student must calculate the value of $y = \sqrt{x}$. He knows that the exact value of $x$ belongs to the interval $[15.8, 16.2]$. If he decides to use $x = 16$ to simplify, obtaining $y = 4$, determine the upper bound of the absolute error he is making in the variable $y$.
When calculating the compound interest of an initial capital of $\$10\,000$ at $5\%$ per year for $10…
When calculating the compound interest of an initial capital of $\$10\,000$ at $5\%$ per year for $10$ years, an accountant decides to truncate the annual growth rate from $1.05$ to $1.0$ in his preliminary calculations. Calculate the absolute error and the relative error he makes in the calculation of the final capital. What conclusion can you draw about the propagation of errors in operations with high exponents?
Back to the start: your app was off by $0.03$ mi on a $2.43$-mi walk, so $E_r = \tfrac{0.03}{2.43} \approx 1.2\%$. That is a decent estimate for a walk. The same $0.03$ mi miss on a $3{,}000$-mile cross-country flight would be $E_r = 0.001\%$, which is excellent. The same absolute error can be good or bad depending on the size of the quantity, and that is what relative error tells you.