Solved Examples: Measurements and Errors in Measurements

Samuel Dominic Chukwuemeka (SamDom For Peace)

Significant Digits
Random Error:
Random errors are inherently unpredictable, so by taking a large number of measurements and averaging them, any errors are minimized.
To minimize the effects of random errors, make many measurements and average them.

Systematic Error:
A systematic error occurs when there is a problem in the measurement system that affects all measurements, so by identifying it, all of the measurements can be adjusted to eliminate it.
To account for the effect of a systematic error, adjust all measurements accordingly.

Absolute Error:
The absolute error describes how far a measured (or claimed) value lies from the true value.

$ Absolute\;\;error = |Measured\;\;value - True\;\;value| \\[3ex] $

Relative Error:
The relative error compares the size of the absolute error to the true value and is often expressed as a percentage.

$ Relative\;\;error = \dfrac{|Measured\;\;value - True\;\;value|}{True\;\;value} * 100 \\[5ex] = \dfrac{Absolute\;\;error}{True\;\;value} * 100 \\[5ex] $ Accuracy:
Accuracy describes how closely a measurement approximates a true value.
An accurate measurement has a small relative error.

Precision:
Precision describes the amount of detail in a measurement.
It can be misleading to give measurements with more precision than is justified by the measurement process because the measurement would be perceived as having a greater amount of detail than it actually has.
The implied precision of a number is the lowest place in the number that contains a significant digit.

Rounding Rules for Adding and Subtracting measured numbers
Round your answer to the same precision as the least precise number in the problem.
For example: 4.2 − 2 = 2

Rounding Rules for Multiplying and Dividing measured numbers
Round your answer to the same number of significant digits as the measurement with the fewest significant digits.
For example: 4.2 × 2 = 8

For ACT Students
The ACT is a timed exam...60 questions for 60 minutes
This implies that you have to solve each question in one minute.
Some questions will typically take less than a minute a solve.
Some questions will typically take more than a minute to solve.
The goal is to maximize your time. You use the time saved on those questions you solved in less than a minute, to solve the questions that will take more than a minute.
So, you should try to solve each question correctly and timely.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no negative penalty for any wrong answer.

For JAMB and CMAT Students
Calculators are not allowed. So, the questions are solved in a way that does not require a calculator.

Solve all questions.
Use at least two methods as applicable.
State the measurement.
Show all work.

NOTE: Unless specified otherwise:
(1.) Use only the tables provided for you.
(2.) Please do not approximate intermediate calculations.
(3.) Please do not approximate final calculations. Leave your final answer as is.

(1.) Discuss possible sources of error in the following measurement.
Then state whether you think the measurement is believable, given the precision with which it is stated.

(I.) The U.S. Census Bureau reported the 2015 U.S. population to be 321,418,820.

(II.) The most visited theme park in the world is Magic Kingdom Park at Walt Disney World Resort in Florida, which had 20,960,000 visitors in 2019.

(III.) The most common last name in the U.S. population is Smith, which is the last name of 0.881% of the population.

(IV.) In the fourth quarter of 2019, General Motors sold 735,909 vehicles.
During the same period, Ford sold 601,682 vehicles.

(V.) Asia has a land area of 30,875,906 square kilometers.

(VI.) According to UNAIDS (Joint United Nations Programme on HIV/AIDS), 38 million people were living with HIV/AIDS (Human Immunodeficiency Virus/Acquired Immunodeficiency Syndrome) worldwide in 2019.

(VII.) The New York Public Library has over 50 million printed items.

(I.) Random or Systematic errors could be present with the given measurement.
The measurement is not believable with the given precision.

(II.) Random or Systematic errors could be present with the given measurement.
The given value has a reasonable level of precision.

(III.) Random or Systematic errors could be present with the given measurement.
The given value has a reasonable level of precision.

(IV.) Random or Systematic errors could be present with the given measurement.
The given value has too much precision to be believable.

(V.) Random or Systematic errors could be present with the given measurement.
The measurement is not believable with the given precision.

(VI.) Random or Systematic errors could be present with the given measurement.
The given value has a reasonable level of precision.

(VII.) Random or Systematic errors could be present with the given measurement.
The given value has a reasonable level of precision.

(2.) Give examples of the following:
(a.) The absolute error is small but the relative error is large
(b.) The absolute error is large but the relative error is small.
(c.) A measurement that is precise but inaccurate.
(d.) A measurement that is accurate but imprecise.

(a.) A chemist has 2.9 mg of substance, but a scale measures 2.1 mg.

$ Abolute\;\;error = |Measured\;\;value - True\;\;value| \\[3ex] = |2.1 - 2.9| \\[3ex] = |-0.8| \\[3ex] = 0.8\;mg \\[5ex] Relative\;\;error = \dfrac{Abolute\;\;error}{True\;\;value} * 100 \\[5ex] = \dfrac{0.8}{2.9} * 100 \\[5ex] = 27.5862069\% \\[3ex] $ The absolute error is only 0.8 mg, but the relative error is approximately 28%.

(b.) A company projects sales of $7.30 billion and true sales turn out to be $7.32 billion.

$ Abolute\;\;error = |Measured\;\;value - True\;\;value| \\[3ex] = |7.3\;billion - 7.32\;billion| \\[3ex] = |-0.02\;billion| \\[3ex] = 0.02\;billion \\[5ex] Relative\;\;error = \dfrac{Abolute\;\;error}{True\;\;value} * 100 \\[5ex] = \dfrac{0.02\;billion}{7.32\;billion} * 100 \\[5ex] = 0.273224044\% \\[3ex] $ The relative error is less than 1% even though the absolute error of $0.02 billion represents $20 million.

(c.) A measurement that is precise but inaccurate is a laptop that weighs 12.28 tons.
In this case, a laptop could not possible weigh 12.28 tons, so the measurement is inaccurate.
However, the number 12.28 is given two several decimal places, so it is a very precise number though it is inaccurate.

(d.) A measurement that is accurate but imprecise is a piece of paper being 1 foot long.

(4.) You are trying to measure the outside temperature at a particular time.
If you use three thermometers and place all three in direct sunlight, the sunlight is likely to cause your measurements to suffer from what kind of error?

A. Your measurements will suffer from a decrease in precision because at the same time of day tomorrow the amount of sunlight might not be the same.
B. Your measurements will suffer from a decrease in precision because the thermometer only uses whole numbers.
C. Your measurements will suffer from random errors because they will be expressed as a percentage.
D. Your measurements will suffer from systematic errors because they will never quite match the true temperature.
E. Your measurements will suffer from random errors because of the randomness of cloud movements. Sometimes the clouds block sunlight.
F. Your measurements will suffer from systematic errors because the sunlight affects all measurements in the same way by making them too high by the same amount.

F. Your measurements will suffer from systematic errors because the sunlight affects all measurements in the same way by making them too high by the same amount.

(5.) In 2015 and 2016, a mosquito-borne disease called Zika fever spread through much of the Americas and a few other areas of the world.
Researchers studying the progression of the Zika epidemic faced at least two major challenges in determining how many people had been infected:
(1st.) Some people who were suffering from Zika infection were misdiagnosed as having other diseases, and vice versa, and
(2nd.) many cases of Zika infection occurred in poor or remote areas where medical diagnosis was not available.
Decide whether each problem involved random or systematic errors.

Problem (1st.) is a random error.
Problem (2nd.) is a systematic error.

(6.) A testing service makes an error that causes all the SAT scores for several thousand students to be low by 57 points.
Which statement is true?

A. All the scores had the same absolute error, but the relative errors varied.
The relative error for any given score will be that score divided by 57.

B. Both the absolute and relative errors were the same in all cases because the same error is made for each score.

C. All the scores had the same absolute error, but the relative errors varied.
If all of the scores have the same absolute error, then they will be that many points off from the true value of the exam score.

D. All the scores had the same relative error, but the absolute errors varied.
The percentage was the same, but the difference between the measurement and the true value varied.

E. All the scores had the same relative error, but the absolute errors varied.
If all of the scores have the same relative error, then they will be the same amount of points less than the actual score.

C. All the scores had the same absolute error, but the relative errors varied.
If all of the scores have the same absolute error, then they will be that many points off from the true value of the exam score.

(7.) Decide whether these statements make sense (or is clearly true) or does not make sense (or is clearly false).
Explain your reasoning.

(I.) Next year's federal deficit will be $675.734 billion.

(II.) Someone weighs 111.3627 pounds.

(III.) In many developing nations, official estimates of the population may be off by 10% or more.

(IV.) A $1 million error may sound like a lot, but when compared to our company's revenue it represents a relative error of only 0.1%.

(V.) Someone used a yard stick to measure the length of her kitchen to the nearest micrometer.

(I.) This statement does not make sense because there are too many significant digits.

(II.) The statement does not makes sense because no information about the precision of the measurement process is given, and it is highly unlikely that a person knows their weight to the nearest ten-thousandth of a pound.

(III.) The statement makes sense because it is difficult to estimate populations.

(IV.) The statement makes sense because the relative error is low.
Thus, the $1 million dollar error is not very big when compared with the actual revenue.

(V.) The statement does not make sense because a yard stick measures length in yards not micrometers.
Therefore, the measurement will not be precise to the nearest micrometer.

(8.) At a particular moment, the U.S. National Debt Clock says that the federal debt is $11,946,495,375.68.
What is a good description of this reading?

A. Very precise, but not necessarily accurate.
It is quite precise because it seems to show the exact amount, but is not necessarily accurate because of a large relative error.

B. Very accurate, but not necessarily precise.
It is quite accurate due to a small relative error, but is not necessarily precise because it's difficult to calculate the exact amount.

C. Very accurate, but not necessarily precise.
This projection was made by the same economists who predicted the 2001 surplus, so it is not very precise.

D. Very precise, but not necessarily accurate.
This projection was made by the same economists who predicted the 2001 surplus, so it is not accurate.

E. Both very precise and very accurate.
This projection was calculated by qualified and experienced economists.

F. Both very precise and very accurate.
This projection is calculated by a computer, so it is very precise and accurate.

A. Very precise, but not necessarily accurate.
It is quite precise because it seems to show the exact amount, but is not necessarily accurate because of a large relative error.

(11.) The 2020 census claimed a U.S. population of 331,449,281 on April 1, 2020.
According to the U.S. Geological Survey, the land area of the United States is 3,531,005 square miles (mi²).
Dividing the population by the area, we find that the population density of the country is 93.86825592 people/mi²

(a.) Discuss the possible sources of error in this calculation.
(b.) State the population with the precision you think are justified.
(c.) State the land area with the precision you think are justified.
(d.) Give an estimate of the population density that you think is reasonable.

(a.) Both the population and area are subject to random and systematic errors and are stated with too much precision.
(b.) A reasonable precision for the population is 331,400,000 people.
(c.) A reasonable precision for the land area is 3,531,000 mi²
(d.) A reasonable estimate of the population density is 93.85 people/mi²

(12.) Before taking off, a pilot is supposed to set the aircraft altimeter to the elevation of the airport.
A pilot leaves Denver (altitude 5280 ft) with her altimeter set to 2667 feet.
Explain how this affects the altimeter readings throughout the flight.

A. This is a random error.
The pilot made an unintentional and unpredictable mistake.

B. This is a random error.
Sometimes altimeters set to random altitudes if not properly calibrated.

C. This is a systematic error.
All altitude readings will be about 2613 feet too low.

D. This is a systematic error.
The plane will be 7947 feet too high to land in Denver.

$ Absolute\;\;error = |Measured\;\;value - True\;\;value| \\[3ex] = 5280 - 2667 \\[3ex] = 2613\;feet \\[3ex] $ C. This is a systematic error.
All altitude readings will be about 2613 feet too low.

(15.) Describe possible sources of random and systematic errors in these measurements.

(I.) The annual incomes of 200 people obtained from their tax returns.
(II.) Speeds of cars recorded by a police officer using a radar gun.
(III.) A count of every different meadowlark that visits a three-acre region over a 2-hour period.
(IV.) Lap times in the Indianapolis 500 auto race.

(I.) Random errors could occur when taxpayers make honest mistakes or when the income accounts are recorded incorrectly.
Systematic errors could occur when dishonest taxpayers report income amounts that are lower than their true amounts.

(II.) (A.) Possible sources of random errors are:
(a.) Errors made by the police officer in reading the radar gun.
(b.) Errors made by the police officer in recording the speeds.
(c.) Errors made when calibrating the radar gun.

(II.) (B.) Possible sources of systematic errors are:
(a.) Errors made by the police force in using a radar gun that is known to be malfunctioning.
(b.) Errors made by an officer known to consistently read the radar gun output higher than the actual value displayed.

(III.) Random errors could occur due to not counting some birds and double counting other birds.

(IV.) (A.) Possible sources of random errors are:
(a.) Electrical noise in the lap clock.
(b.) Jitter in the lap clock.
(c.) Electrical noise in the car detector.

(IV.) (B.) Possible sources of systematic errors are:
(a.) Frequency drift in the lap clock.
(b.) Misalignment of the car detector.

(19.) Suppose you want to cut 20 identical boards of length 4 feet.
The procedure is to measure and cut the first board, then use the first board to measure and cut the second board, then use the second board to measure and cut the third board, and so on.

(a.) What are the possible lengths of the 20th board if, each time you cut a board, there is a maximum error of $\pm \dfrac{1}{4}$inch?

(b.) What are the possible lengths of the 20th board if, each time you cut a board, there is a maximum error of ±0.5%?

(a.) Let us represent this information in a table for easier understanding.
Please NOTE: Each time you cut a board, there is a maximum error of $\pm \dfrac{1}{4}$inch

4 feet = 4(12) inches = 48 inches
Minimum error = $-\dfrac{1}{4}$
Maximum error = $\dfrac{1}{4}$
Position	Minimum Length	Maximum Length
1st	$48 - \dfrac{1}{4}$	$48 + \dfrac{1}{4}$
2nd	$48 - 2\left(\dfrac{1}{4}\right)$	$48 + 2\left(\dfrac{1}{4}\right)$
3rd	$48 - 3\left(\dfrac{1}{4}\right)$	$48 + 3\left(\dfrac{1}{4}\right)$
20th	$ 48 - 20\left(\dfrac{1}{4}\right) \\[5ex] 48 - 5 \\[3ex] 43 $	$ 48 + 20\left(\dfrac{1}{4}\right) \\[5ex] 48 + 5 \\[3ex] 53 $

The board could be as short as 43 inches or as long as 53 inches.

(b.) Please NOTE: Each time you cut a board, there is a maximum error of ±0.5%

$ 0.5\% = \dfrac{0.5}{100} = 0.005 \\[5ex] $

4 feet = 4(12) inches = 48 inches
Minimum error = $-0.005(length)$
Maximum error = $0.005(length)$
Position	Minimum Length	Maximum Length
1st	$ 48 - 0.005(48) \\[3ex] 48(1 - 0.005) \\[3ex] 48(0.995) \\[3ex] 47.76 $	$ 48 + 0.005(48) \\[3ex] 48(1 + 0.005) \\[3ex] 48(1.005) \\[3ex] 47.76 $
2nd	$ 47.76 - 0.005(47.76) \\[3ex] 47.76(1 - 0.005) \\[3ex] 47.76(0.995) \\[3ex] Substitute\;\;47.76\;\;for\;\;48(0.995) \\[3ex] 48(0.995)(0.995) \\[3ex] 48(0.995)^2 $	$ 47.76 + 0.005(47.76) \\[3ex] 47.76(1 + 0.005) \\[3ex] 47.76(1.005) \\[3ex] Substitute\;\;47.76\;\;for\;\;48(1.005) \\[3ex] 48(1.005)(1.005) \\[3ex] 48(1.005)^2 $
3rd	$ 48(0.995)^{3} $	$ 48(1.005)^3 $
20th	$ 48(0.995)^{20} \\[3ex] 43.42130305 $	$ 48(1.005)^{20} \\[3ex] 53.0349877 $

The board could be as short as 43.42130305 inches or as long as 53.0349877 inches.

(20.) A tax auditor reviewing a tax return looks for several kinds of problems, including (1.) mistakes made in entering or calculating numbers on the tax return and (2.) places where the taxpayer reported income dishonestly.
Discuss whether each problem involves random or systematic errors.

(I.) Does mistakes made in entering or calculating numbers on the tax return involve random or systematic error?
Explain.

A. This involves a random error because most taxpayers enter random numbers to hide their real income.
B. This involves a systematic error because the mistake makes the taxes too high by the same amount.
C. This involves a systematic error because the tax return form is part of the tax system.
D. This involves a random error because unintentional mistakes are unpredictable and random.

(II.) Does places where the taxpayer reported income dishonestly involve random or systematic error?
Explain.

A. This involves a random error because there is no way to predict dishonesty before it happens.
B. This involves a systematic error due to underreporting of income.
C. This involves a systematic error due to failure to report hourly pay.
D. This involves a random error because most people are honest about their income.

(I.) D. This involves a random error because unintentional mistakes are unpredictable and random.

(II.) B. This involves a systematic error due to underreporting of income.

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