Advanced Higher Physics Unit 1

Uncertainties

Types of uncertainties

There are four ways in which uncertainty from a measurement can arise:

1. Systematic

2. Calibration

3. Scale

4. Random

Systematic

Arising from the system being used.For example, a slow running stopclock.These affect every reading and cannot quantified.

They need to be considered in an evaluation for an experiment.

Calibration

Arising from the accuracy of the instruments used-how well have they been made?

Instrument Calibration uncertainties

Metres Stick (wood) 0.5 mm

Ruler made of steel 0.1 mm

Digital Metre 0.5% of reading + 1 in the last digit

Calibration-examples

0:01 35 s

Stopclock

For this time measurement, the calibration uncertainty will be:

(0.5% of 1.35s) + 0.01 =0.01 + 0.01 =± 0.02 s

Scale

Arising from the accuracy of how the instruments are read.

Type of Meter Scale uncertainty

digital ±1 in the least significant digit

analogue ±half the smallest unit

Scale-example

0:01 35 s

Stopclock

For this time measurement, the scaleuncertainty is:

Δt = ±0.01s

Random

Arising from fluctuations in repeated measurements:

valuesofnumber

valuevalue .min.max

This formula can be found in the Higher part of the data booklet.

Random uncertainty

Random-example

n

ttt MINMAX

5

19.123.1 t

Five time intervals have been measured:

1.23s, 1.21s, 1.19s, 1.20s, 1.21s

The random uncertainty is given by:

st 01.0

Overall Uncertainty in a reading

Generally:

222 randomscalencalibratioX

This formula is NOT included in the data booklet.

However, if one of these is three times the others, it dominates. It will be the percentage uncertainty in the final result.

Overall uncertainty-example

222 randomscalencalibratiot

222 01.001.002.0 t

For the earlier time measurements:•Calibration uncertainty = ±0.02s•Scale uncertainty = ±0.01s•Random uncertainty = ±0.01s

The overall uncertainty is:

st 001.0

Uncertainty in a calculation

First, calculate the percentage uncertainty in all the measurements being used.

If one of these is three times the others, it dominates.It will be the percentage uncertainty in the final result.

If this does not happen then we need to combine the uncertaintiesin the measurements.

Addition and subtraction

ZYX If

then 22 ZYX

(This formula can be found in the data booklet)

Multiplication and division

Z

YXorYZX If

then22

Z

Z

Y

Y

X

X

(This formula can be found in the data booklet)

Powers

nYX If

then YnX %%

(This formula cannot be found in the data booklet)

Graphical Uncertainties

1. Plot points with error bars

X

Y

0

2. Centroid is plotted: mean of all the X and Y coordinates

X

Y

0

3. Best fit line is drawn through centroid

X

Y

0

4. A top parallel line is drawn through point furthest above it.

X

Y

0

5. A similar bottom parallel line is drawn below the best fit line.

X

Y

0

6. Draw vertical lines through 1st and last plotted points so that it construct a parallelogram round best fit line.

X

Y

0

7. Draw the diagonals of this parallelogram

X

Y

0

8. Calculate the gradients m1 and m2 of the diagonals.

X

Y

0

9. The uncertainty in the gradient is given by:

2221

n

mmm

n is the number of data points plotted

(This formula can be found in the data booklet)

10. Find y intercepts c1 and c2.

X

Y

0

c1

c2

11. The uncertainty in the intercept is given by:

2221

n

ccc

n is the number of data points plotted

(This formula can be found in the data booklet)

Summary

Measurements including: Scale Calibration Random

2. Overall Uncertainty

222 randomscalencalibratioX

3. Calculations Start with additions, subtractions and power

Then do formula22 ZYX YnX %%

22

Z

Z

Y

Y

X

X

Activity

1. Measure the density of a metal cube

2. Measure the specific heat capacity of block

3. Measure the average speed of a trolley

Think:

What formula are you going to use?

What quantities will you need to measure?