Error in Measurement

PrecisionAccuracyError TypesSignificant DigitsError Propagation

Error in Measurement

"Accuracy" means truth."Precision" means detail.

Accuracy

• Accurate: measured value very close to actual value.– What affects accuracy?– How can you improve accuracy?

– Systematic (correctable) errors– Calibration of instrument, e.g. zero and gain

Precision

Precise: Multiple measurements close together-What affects precision?- How can you improve precision?

- Random errors (not correctable, need trials)- Instrument capabilities

Significant Digits in Measurement

• Conveys precision• Rule of thumb: estimate 1 digit• Digital: half of the last digit

Sig Fig Rules•

Examples – 7000.00 kg has six: there is a decimal point, so all the digits are significant. – 0.00040 kg has two: there is a decimal point, so the 4 and last 0 count. The

other zeros are all "leading". – 208,000 g has three: the 2, the leftmost 0, and the 8. The trailing zeros do not

count, since there is no decimal point. – 4440700 km has five: the 4s, the sandwiched 0, and the 7. The trailing

zeros do not count, since there is no decimal point. • In scientific notation, significant digits count in the first part of the

number.

Examples – 5.8 x 10^3 has two significant digits. – 2.00 x 10^-8 has three significant figures.

Sig Fig in calculations

For multiplication and division, your answer should match the smallest number of significant digits in the question.

For addition and subtraction, your answer should match the smallest number of decimal places in the question.

1. If I ran 10.0 m in 2.0 s, how should my speed be rounded off?2. What if I ran it in 2.01 s? What about 5 sec?3. What value of force causes a 25.0 kg ball to have an acceleration of 4 m/s^2?4. If I raise my 36 m flagpole by 20.0 cm, how tall is it?

5.0, 4.98, 2100, 36

Pendulum Lab Measurements

Which were accurate?

Which were precise?

How do you know?

How do you compensate?

What kind of errors were there?

How many sig fig did you use?

Reporting error

• Error: disagreement between measurement and true/accepted value. We don’t always know this.

• Uncertainty: refers to measured value, it is the interval within which repeated measurements are expected to lie.

• Examples:5.0 +/- 0.2 cm (absolute)Mean +/- 1 std dev (68%, or about 2/3)9.8 +/- 2% (relative)

Reporting error

Histograms

A metal rod about 4 inches long has been passed around to several groups of students. Each group is asked to measure the length of the rod. Each group has five students and each student independently measures the rod and records his or her result.

Student Group

Student 1 Student 2 Student 3 Student 4 Student 5

Group A 10.1 10.4 9.6 9.9 10.8

Group B 10.135 10.227 10.201 10.011 10.155

Group C 12.14 12.17 12.15 12.14 12.18

Group D 10.05 10.82 8.01 11.5 10.77

Group E 10 11 10 10 10

1) Which group has the most accurate measurement? 2) Which group has the most precise measurement? 3) Which group has the greatest error? 4) Which group has the greatest uncertainty?

We now receive a report from the machine shop where the rod was manufactured. This very reputable firm certifies the rod to be 4 inches long to the nearest thousandths of an inch. Answer the questions below given this new information. Note that the questions are slightly different. (4.000 inches = 10.160 cm) 5) Which group has the least accurate measurement? 6) Which group has the least precise measurement? 7) Which group has the smallest error? 8) Which group has the smallest uncertainty?

Practice

Answers

• Answers:• 1) Unknown. Accuracy can only be decided when the true or accepted value is known. Every value

above is as likely as any other value.• 2) Group C: The measurements of this group are very close to each other (repeatable). Individual

measurements in Group B have greater precision but they do not agree with each other as closely as Group C.

• 3) Unknown. Error can only be decided when the true/accepted value is know. We may suspect error for measurements such as 8.01 cm or 12.18 cm but unless we observe a clear mistake in measurement method we must accept all values here on the same basis.

• 4) Group D: The average variation in measurements from this group is nearly 1 cm, almost 3 times more than any other group. (read about average deviation)

• ==============• 5) Group C: While they are the most consistent set presented, they are consistently far away from the

accepted value.• 6) Group D: Precision refers to the data set itself, not to the comparison between the data and the

accepted value. The values for Group D do not agree with each other very well.• 7) Group A: This data shows a pretty wide scatter of values but the average is closer to the accepted

value than for any other group.• 8) Group C: Uncertainty is not changed by knowledge of the accepted value. This group has the least

variation so the least uncertainty. Here you see a clear difference between uncertainty and error.

In a ballistic pendulum experiment, suppose the digital timer shows 0.02 s for the time of flight of the projectile.

The manufacturer information about the precision of the timer is nowhere to be found. What error would you quote

on your measurement?

0.002 s

0.005 s

0.01 s

In a projectile motion experiment suppose you have the following series of measurements of the distance x

traveled by the projectile: 30 cm, 32 cm, 29 cm, 28 cm, 31 cm. The mean value is therefore 30 cm. What is the

error (standard deviation) of the result?

0.8 cm

1 cm

1.4 cm

2 cm

Practice

Suppose you counted the number of people in line each time you came to your favorite coffee shop last month. Here are the results of your

investigation.

Number of people 0 1 2 3 4 5 6 7 8 9 10

Occurrences 1 3 3 4 5 8 5 4 3 2 1

What is the error determined by the 2/3 rule?

1.7

2

2.5

3

3.5

Answers: In a ballistic pendulum experiment suppose the digital timer shows 0.02 s for the time of flight of the projectile. The manufacturer information about the

precision of the timer is nowhere to be found. What error would you quote on your measurement? .005 s

In a projectile motion experiment suppose you have the following series of measurements of the distance x traveled by the projectile: 30 cm, 32 cm, 29

cm, 28 cm, 31 cm. The mean value is therefore 30 cm. What is the error (standard deviation) of the result? 1.4 cm

Suppose you counted the number of people in line each time you came to your favorite coffee shop last month. Here are the results of your

investigation. 2.5

Error PropagationSpirit Travel Example:

Length = 4825 +/- 5 mStop at 3260 +/- 10 m

How far to go? 1565 m, +/- ??

Can calculate extremes: 1580 and 1550 mOr add uncertaintyEither way, get 1565 +/- 15 m

General rules For sum and diff: add absolute errors

For Product/Quotient:add relative errors

For Powers:relative error of the original quantity times the power.

Practice

You measure the following quantities:A = 1.0 m ± 0.2 m, B = 2.0 m ± 0.2 m, C = 2.5 m/s ± 0.5 m/s, D = 0.10 s ± 0.01 s.Choose the correct answers for the following expressions:A + B = 3.0 m ± 0.0 m 3.0 m ± 0.2 m 3.0 m ± 0.4 m

B − A = 1.0 m ± 0.0 m 1.0 m ± 0.2 m 1.0 m ± 0.4 m

C × D = 0.25 m ± 0.05 m 0.25 m ± 0.08 m 0.25 m ± 0.51 m

B / D = 20. m/s ± 4 m/s 20. m/s ± 10 m/s 20. m/s ± 20 m/s

3 × A = 3.0 m ± 0.2 m 3.0 m ± 0.3 m 3.0 m ± 0.6 m

The square root of (A × B) = 1.4 m ± 0.1 m 1.4 m ± 0.2 m 1.4 m ± 0.4 m

Answers:________________________________________A + B = 3.0 m ± 0.4 m B − A = 1.0 m ± 0.4 m C × D =0.25 m ± 0.08 m B / D = 20. m/s ± 4 m/s

3 × A = 3.0 m ± 0.6 m The square root of (A × B) = 1.4 m ± 0.2 m