1.2 Uncertainties and errors Random/systematic uncertainties Absolute/fractional uncertainties...

1.2 Uncertainties and errors

• Random/systematic uncertainties• Absolute/fractional uncertainties• Propagating uncertainties• Uncertainty in gradients and intercepts

Let’s do some measuring!

1.2 Measuring practicalDo the

measurements yourselves, but leave space in your table of results to record

the measurements of 4 other people from

the group

Errors/Uncertainties

Errors/Uncertainties

In EVERY measurement (as opposed to simply counting) there is an uncertainty in the measurement.

This is sometimes determined by the apparatus you're using, sometimes by the nature of the measurement itself.

Estimating uncertainty

As Physicists we need to have an idea

of the size of the uncertainty in each

measurement

The intelligent ones are

always the cutest.

Individual measurements

When using an analogue scale, the uncertainty is plus or minus half the smallest scale division. (in a best case scenario!)

4.20 ± 0.05 cm

Individual measurements

When using an analogue scale, the uncertainty is plus or minus half the smallest scale division. (in a best case scenario!) 22.0 ± 0.5 V

Individual measurements

When using a digital scale, the uncertainty is plus or minus the smallest unit shown.

19.16 ± 0.01 V

Significant figures

• Note that the uncertainty is given to one significant figure (after all it is itself an estimate) and it agrees with the number of decimal places given in the measurement.

• 19.16 ± 0.01

• (NOT 19.160 or 19.2)

Repeated measurements

When we take repeated measurements and find an average, we can find the uncertainty by finding the difference between the highest and lowest measurement and divide by two.

Repeated measurements - Example

Pascal measured the length of 5 supposedly identical tables. He got the following results; 1560 mm, 1565 mm, 1558 mm, 1567 mm , 1558 mm

Average value = 1563 mm

Uncertainty = (1567 – 1558)/2 = 4.5 mm

Length of table = 1563 ± 5 mm

This means the actual length is anywhere between 1558 and 1568 mm

Average of the differences

• We can do a slightly more sophisticated estimate of the uncertainty by finding the average of the differences between the average and each individual measurement. Imagine you got the following results for resistance (in Ohms)

• 13.2, 14.2, 12.3, 15.2, 13.1, 12.2.

Precision and Accuracy

The same thing?

Precision

A man’s height was measured several times using a laser device. All the measurements were very similar and the height was found to be

184.34 ± 0.01 cm

This is a precise result (high number of significant figures, small range of measurements)

AccuracyHeight of man = 184.34 ± 0.01cm

This is a precise result, but not accurate (near the “real value”) because the man still had his shoes on.

Accuracy

The man then took his shoes off and his height was measured using a ruler to the nearest centimetre.

Height = 182 ± 1 cm

This is accurate (near the real value) but not precise (only 3 significant figures)

Precise and accurate

The man’s height was then measured without his socks on using the laser device.

Height = 182.23 ± 0.01 cm

This is precise (high number of significant figures) AND accurate (near the real value)

Precision and Accuracy

• Precise – High number of significent figures. Repeated measurements are similar

• Accurate – Near to the “real” value

Random errors/uncertainties

Some measurements do vary randomly. Some are bigger than the actual/real value, some are smaller. This is called a random uncertainty. Finding an average can produce a more reliable result in this case.

Systematic/zero errors

Sometimes all measurements are bigger or smaller than they should be by the same amount. This is called a systematic error/uncertainty.

(An error which is identical for each reading )

Systematic/zero errors

This is normally caused by not measuring from zero. For example when you all measured Mr Porter’s height without taking his shoes off!

For this reason they are also known as zero errors/uncertainties. Finding an average doesn’t help.

Systematic/zero errors

Systematic errors are sometimes hard to identify and eradicate.

UncertaintiesIn the example with the table, we found the length of the table to be 1563 ± 5 mm

We say the absolute uncertainty is 5 mm

The fractional uncertainty is 5/1563 = 0.003

The percentage uncertainty is 5/1563 x 100 = 0.3%

UncertaintiesIf the average height of students at BSW is 1.23 ± 0.01 m

We say the absolute uncertainty is 0.01 m

The fractional uncertainty is 0.01/1.23 = 0.008

The percentage uncertainty is 0.01/1.23 x 100 = 0.8%

Let’s try some questions.

• 1.2 Uncertainty questions

Let’s read!Pages 7 to 10 of Hamper/Ord ‘SL

Physics’

Homework

Complete “1.2 Measuring Practical”• Taking one measurement;

i. Decide whether it is precise and/or accurate. Explain your answer.

ii. Are there liable to be systematic or random uncertainties? (Explain)

iii.How could a better measurement be obtained?

DUE Friday 12th September

Homework due today

• On your tables can you compare your answers to the questions

• Did you all agree?!

Propagating uncertainties

When we find the volume of a block, we have to multiply the length by the width by the height.

Because each measurement has an uncertainty, the uncertainty increases when we multiply the measurements together.

Propagating uncertainties

When multiplying (or dividing) quantities, to find the resultant uncertainty we have to add the percentage (or fractional) uncertainties of the quantities we are multiplying.

Propagating uncertainties

• Data book reference• If y = ab/c• Δy/y = Δa/a + Δb/b + Δc/c

• If y = an

• Δy/y = nΔa/a

Propagating uncertainties

Example: A block has a length of 10.0 ± 0.1 cm, width 5.0 ± 0.1 cm and height 6.0 ± 0.1 cm.

Volume = 10.0 x 5.0 x 6.0 = 300 cm3

% uncertainty in length = 0.1/10 x 100 = 1%% uncertainty in width = 0.1/5 x 100 = 2 %% uncertainty in height = 0.1/6 x 100 = 1.7 %

Uncertainty in volume = 1% + 2% + 1.7% = 4.7%

(4.7% of 300 = 14)

Volume = 300 ± 10 cm3

This means the actual volume could be anywhere between 286 and 314 cm3

Propagating uncertainties

When adding (or subtracting) quantities, to find the resultant uncertainty we have to add the absolute uncertainties of the quantities we are multiplying.

Propagating uncertainties

• Data book reference• If y = a ± b• Δy = Δa + Δb

Propagating uncertainties

One basketball player has a height of 196 ± 1 cm and the other has a height of 152 ± 1 cm. What is the difference in their heights?

Difference = 44 ± 2 cm

Who’s going to win?

New York TimesLatest opinion poll

Bush 48%

Gore 52%

Gore will win!

Uncertainty = ± 5%

Who’s going to win?

New York TimesLatest opinion poll

Bush 48%

Gore 52%

Gore will win!

Uncertainty = ± 5%

Who’s going to win?

New York TimesLatest opinion poll

Bush 48%

Gore 52%

Gore will win!

Uncertainty = ± 5%

Uncertainty = ± 5%

Who’s going to win

Bush = 48 ± 5 % = between 43 and 53 %

Gore = 52 ± 5 % = between 47 and 57 %

We can’t say!

(If the uncertainty is greater than the difference)

Let’s try some more questions!

1.2 Propagating uncertainties

1.2 Graphing uncertaintities practical

Error bars/lines of best fitMass of dog/kg

Time it takes dog to burn/s

Minimum gradientMass of dog/kg

Time it takes dog to burn/s

Minimum gradientMass of dog/kg

Time it takes dog to burn/s

Maximum gradientMass of dog/kg

Time it takes dog to burn/s

Error bars/line of best fits

Error bars/line of best fits

Some Maths!

B α L

Proportional?

If B α L then

B = kL

Proportional = straight line through origin

B = kLBoredom/B

Length of time in class/s

k = ΔB/ΔL

B = kLBoredom/B

Length of time in class/s

ΔL

ΔB

Inversely proportional?

Inversely proportional?

U α 1/WUniform conformity/U

Number of weeks of school/W

Inversely proportional?

U = k/W

UW = kUniform conformity/U

Number of weeks of school/W

U1W1 = U2W2

UW = kUniform conformity/U

Number of weeks of school/W

U1

U2

W1 W2

y = mx + c

y

x

y = mx + c

y

x

c

c Δx

Δy m = Δy/Δx

E = ½mv2

E = ½mv2

Energy/J

v2/m2/s-2

½m

R = aTb

R = aTb

lnR = lna +blnT

lnR = lna + blnT

lnR

lnT

lna

b

Gradient to a curve

Gradient to a curve

Let’s try an IB question!

• Paper 3 – Question 1 is always a ‘data response’ question to do with error bars, lines of best fit, gradients etc.

1.2 Period of a pendulum practical

HOMEWORK

• Complete “Pendulum investigation (DO what it says on the sheet!)

• Due NEXT FRIDAY 19th September