Error in Measurement Precision Accuracy Error Types Significant Digits Error Propagation

"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 – kg has six: there is a decimal point, so all the digits are significant. – 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. – 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 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

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

Histograms

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 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? s 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

Error Propagation Spirit Travel Example: Length = /- 5 m Stop at /- 10 m How far to go? 1565 m, +/- ?? Can calculate extremes: 1580 and 1550 m Or add uncertainty Either way, get /- 15 m General rules For sum and diff: absolute add absolute errors For Product/Quotient: relative add relative errors For Powers: relative 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 m0.25 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 mB − A = 1.0 m ± 0.4 mC × D =0.25 m ± 0.08 mB / D = 20. m/s ± 4 m/s 3 × A = 3.0 m ± 0.6 mThe square root of (A × B) = 1.4 m ± 0.2 m