UNCERTAINTIES IN MEASUREMENTS

PROPERLY PROCESSING DATA

Every measurement has a doubtful digit.

MEASUREMENTS WITH UNCERTAINTIES

This ruler measures to the mm (0.1 cm)

Every measurement made with a ruler like this has an uncertainty of ± 0.05cm. This is half the smallest division.

This measurement is 1.02 ± 0.05 cm.

It lies somewhere between 0.97 cm and 1.07 cm.

Notice that the measurement and the uncertainty have the same level of precision.

Every measurement has a doubtful digit.

MEASUREMENTS WITH UNCERTAINTIES

This stopwatch measures to 0.01 seconds.

This measurement could be 3min and 53.170 seconds or3 min and 53.179 seconds.

This means the uncertainty is ±0.01 seconds and the measurement is 237.17 ± 0.01s.

For IB labs, each measurement needs to be repeated at least 3 times.

Measurement, measurement

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

Raw data should be presented in an easy to understand data table. The headings should state the name of the quantity, its symbol, the units it is measured in and its uncertainty.

Recording the data

Lengthl/cmΔl = ±0.1 cm

29.4

38.5

48.7

60.5

Of course, you will be measuring at least two variables. Here’s an example of using repeated values.

Recording the data

Lengthl/cm

Δl =±0.1 cm

Time for 10 Periods

10T/sΔ10T = ±0.01s

60.516.77

15.42

15.79

Of course, you will be measuring at least two variables. Here’s an example of using repeated values.

Recording the data

Lengthl/cm

Δl =±0.1 cm

Time for 10 Periods

10T/sΔ10T = ±0.01s

60.516.77

15.42

15.79

Average Time for 10 Periods

16.0

Of course, you will be measuring at least two variables. Here’s an example of using repeated values.

Recording the data

Lengthl/cm

Δl =±0.1 cm

Time for 10 Periods

10T/sΔ10T = ±0.01s

60.516.77

15.42

15.79

Average Time for 10 Periods

16.0

Average Period

1.60

When dividing a measurement by a pure number, divide its uncertainty as well.

Basic Rules

This is called the absolute or rawuncertainty

Basic Rules

cmcmcmwrL

cmcmcmL

rwL

3.02.01.0

5.61.66.12

Basic Rules

%100%100)()()(B

B

A

ABABBAA

tionMultiplica

This is called the fractional or relative uncertainty A

A

The relative uncertainty multiplied by 100% is called the percentage uncertainty.

Basic Rules

Basic Rules

When adding or subtracting uncertainties, you ADD the ABSOLUTE uncertainties.

Summary of the Basics

When multiplying or dividing uncertainties, you ADD the PERCENTAGE uncertainties.

When stating uncertainties the uncertainty must have 1 sig fig and must have the same level of precision of the measurement itself.

To straighten some curves, we use the reciprocal values. The uncertainties MUST be processed properly.

Reciprocal calculation

Some graphs need a variable raised to an exponent to be linearized. Again, the uncertainties MUST be properly processed.

Power Function

Calculate the volume of a sphere whose radius is measured to be

You Try

Gradient Uncertainties

Error bars must be shown on the graph for the variable with the most significant uncertainty.

Data point

Min value the data point could have

Max value the data point could have

Sample Graph

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Series1

This data has the responding variable with a fixed absolute value of uncertainty of ±2 cm

Sample Graph

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Series1

Here the uncertainty in the responding variable is a fixed percentage of 5%

Sample Graph

y = 23.226x - 0.1833

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Linear (Series1)

Here is the best straight line. The variables are proportional since the best straight line passes through the origin (accounting for error) and through each point.

But the slope measurement must have uncertainties associated with it as well.

Sample Graph

To get the uncertainty in the slope value, we will look at the maximum slope and the minimum slope then calculate half the range.

y = 23.226x - 0.1833

y = 25.357x - 2

y = 19.643x + 2

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max best fit

min best fit

Linear (Series1)

Linear (max best fit)

Linear (min best fit)

Sample Graph

The max slope is 25.357. The min slope is 19.643. The best slope is 23.226.

y = 23.226x - 0.1833

y = 25.357x - 2

y = 19.643x + 2

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max best fit

min best fit

Linear (Series1)

Linear (max best fit)

Linear (min best fit)We would record the slope as 23.226 ±2.857.

With sig figs, it is 23 ±3.