Working with Uncertainties

Sci 10 PIB

Uncertainties and errorsUncertainties and errors

When measuring physical quantities 3 types of errors may arise

Types of measurement errorsTypes of measurement errors

RandomSystematicReading

Random errorsRandom errors

Almost always due to the observerShows up as fluctuating measurements

about some central valueCan be reduced by averaging over

repeated mesurements

Systematic errorsSystematic errors

Have to do with the system i.e. the equipment and the procedure

Attributable to both the observer and the measuring instrument.

Do not result in fluctuating valuesCannot be reduced by repeated

measurements

Reading errorsReading errors

Relates to the difficulty in reading the instrument with absolute precision

Cannot be improved upon by repeated measurements.

E.g. the reading error for a metrestick is ± 0.05 cm. When a metrestick is read, the best precision we can obtain is to the nearest 0.05 cm on either end of the measurement i.e a total of ± 0.1 cm (add uncertainties when subtracting)

Random errors for repeated Random errors for repeated measurementsmeasurements

For repeated measurements, it is reasonable to expect that half the time, the values will be above the mean and the other half will be below the mean. Therefore, we calculate the uncertainty in the mean (average) as

∆ Mean = ± (Max Value-Min Value)/2

Systematic errorsSystematic errors

Most common source is incorrectly calibrated instrument e.g. if an electronic scale is off by 1 g, then, all the measurements will be off by 1 g

Zero errors also give rise to systematic errors. E.g. a rounded metrestick may yield measurements understated by a few mm. An analog ammeter (measures electric current) whose needle starts at 0.1 Amp will have all the current values overstated by 0.1 Amp

Systematic errors Systematic errors

Systematic errors also arise as a result of the experimenter not being properly aligned with the measuring instrument when reading the instrument. The reading will be either overstated or understated depending on where the experimenter is positioned. This is also known as “human parallax” error

Repeated measurements

For a number of repeated values, first find the average or mean. The uncertainty in the average is plus or minus one-half of the range between the maximum and the minimum value. e.g.

L1 = 140. m, L2 = 136 m , L3 = 142 m Lmean = (140. m +136 m +142 m)/3 =139.33m ∆Lmean =Lmax-Lmin = 142m – 136m = ±3 m L ± ∆L = (139 ± 3) m

Reading ErrorsReading Errors

Instrument Reading error

Metrestick ± 0.5 mm

Vernier calipers ± 0.05 mm

Micrometer ± 0.005 mm

Volumetric (measuring) cylinder ± 0.5 mL

Electronic weighing scale ± 0.1 g

Stopwatch ± 0.01 s

Uncertainties with addition

L = r + w r ±∆r =(6.1±0.1)cm ; w±∆w=(12.6±0.2)cm L=6.1cm + 12.6 cm=18.7 cm ∆L = ∆r + ∆w = 0.1 cm + 0.2 cm = 0.3 cm L±∆L = (18.7±0.3) cm

Uncertainties with subtraction

L = w - r r ±∆r =(6.1±0.1)cm ; w±∆w=(12.6±0.2)cm L= 12.6 - 6.1 cm=6.5 cm ∆L = ∆r + ∆w = 0.1 cm + 0.2 cm = 0.3 cm L±∆L = (6.5 ±0.3) cm

Uncertainties with Multiplication

Area = Length x Width A = L x W L = (24.3 ± 0.1) cm W = (11.8 ± 0.1) cm A = 24.3 cm x 11.8 cm = 286.74 cm² Note ΔA % = ΔL % + ΔW % ΔA % = [(0.1/24.3)x100] + [(0.1/11.8)x100] ΔA % = 0.412% + 0.847% = 1.259% ≈1% A ± ΔA = 286.74 cm² ± 2.8674 cm² A ± ΔA = (287 ± 3) cm²

Uncertainties with Division

Speed = Distance/Time v = s/t ; s = (12.4 ± 0.2) m t = (5.43 ± 0.01) s v = 12.4/5.43 = 2.2836 ms-1

Δv% = Δs% + Δt% Δv% = [(0.2/12.4)x100] + [(0.01/5.43)x100] Δv% = 1.6129% + 0.1842% = 1.7971%≈ 2% v ± Δv = 2.2836 ms-1 ± 0.045672 ms-1

v ± Δv = (2.28 ± 0.05) ms-1

Line of best fit Line of best fit Graph the data with error barsGraph the data with error bars

Extension x/cm Tension Force T/N (± 10)

0.1 16

0.2 36

0.3 56

0.4 84

0.5 100

0.6 116