Error in Measurement - PowerPoint Presentation

Slide1

Error in Measurement

PrecisionAccuracyError TypesSignificant DigitsError Propagation

Error in MeasurementSlide2

"Accuracy" means truth.

"Precision" means detail.

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Accuracy

Accurate: measured value very close to actual value.What affects accuracy?How can you improve accuracy?Systematic (correctable) errorsCalibration of instrument, e.g. zero and gain Slide4

Precision

Precise: Multiple measurements close together-What affects precision?- How can you improve precision?Random errors (not correctable, need trials)

Instrument capabilitiesSlide5
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Significant Digits in Measurement

Conveys precisionRule of thumb: estimate 1 digitDigital: half of the last digitSlide8
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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. Slide10

Sig Fig in calculations

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, 2

100, 36

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Pendulum Lab M

easurementsSlide12

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 errorSlide13

HistogramsSlide14

PracticeSlide15

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)

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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.Slide16

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

PracticeSlide17
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Error Propagation

Spirit Travel Example: Length = 4825 +/- 5 m Stop at 3260 +/- 10 mHow far to go? 1565 m, +/- ??

Can calculate extremes: 1580 and 1550 m

Or add uncertainty

Either 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. Slide19

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:

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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