Measurement & uncertainty pp presentation

MEASUREMENT & UNCERTAINTY

Estimating Uncertainties In Experimental Results

All experimental scientists need to know how well they can trust their results.

The results of any experiment are only as valid as the degree of error in those results.

A lot of time, effort , and money has been spent by scientists developing more “accurate” machines to measure events more precisely.

This unit is all about making and keeping tracks of errors during experimental measurements.


Examine the image show below:

What is the diameter of the tennis ball in cm? (answer: ~ 6.4

cm)


Does this mean it is exactly 6.4 cm? Could the diameter be 6.3 or 6.5 or

even 6.44 cm?

Look again…


All measured values must be accompanied by an estimate of the error or uncertainty associated with the measured value.

The tennis ball has a diameter of

6.4 + 0.1 cm.

Measurement value

Estimated error value


Let’s look at some other possible ways of trying to report this value:

6.4 + 0.15 cm What is inconsistent here?

6 + 0.1 cm

6.42 + 1 cm

What is inconsistent here?

What is inconsistent here?


So what does 6.4 + 0.1 cm really mean?

The real or actual diameter of the tennis ball lies between a maximum and a minimum value.

Maximum value:

Minimum value:

6.5 cm

6.3 cm We can not be any more precise than this!

The actual value lies somewhere in between these two values!

Estimating Uncertainties In Experimental ResultsTypes of

Errors:Measurement errors fall into two main types:

Systematic errors:

These errors consistently influence a set of measurements in a particular direction , either too high or too low.

These errors are associated with the precision of the measuring device (eg. not calibrated correctly), or errors in experimental procedures.


Random errors:

These errors arise due to fluctuations in the experimental conditions or in the judgment of the experimenter.

These errors are random, some being too high while others being too low and tend to average out if the experimenter repeats the experiment often enough.

After you have identified the factors that may influence your results in the collection of experimental results, it is important to design strategies to minimize both of these two types of errors.


Think: Drop a tennis ball from some height allowing it

to hit the ground and measure the height to which it rebounds to.

1)Think and discuss all of the factors that could affect the outcome.

2)Think and discuss all of the possible error sources including both Systematic and Random.


Dealing with errors: Adding and Subtracting Measured Values:

A student measures the mass of a beaker to be :

123.4 + 0.1 g

A student measures the mass of a beaker + copper to be :

113.8 + 0.1 g

Mass of Copper is: 9.6 + ? g

But what about the uncertainty? What happens to it? Does it stay at 0.1? Or does change to a higher or lower number?


The rule is: When adding or subtracting numbers the numerical uncertainty is simply added!

Mass of Copper is: 9.6 + 0.2 g

In order to determine the mass of copper the student subtracted two measured values: therefore simply add the numerical error!

Numerical error


Now try these:

4.5 + 0.2 m + 2.3 + 0.1m + 6.3 + 0.3 m =

67.9 + 0.2 g - 45.7 + 0.2 g =

(34.5 + 0.2 cm) + (12.3 + 0.3 cm) - (14.3 + 0.2 cm) =

(1.5 + 0.5 m) - (4.3 + 0.5 m) + (8.8 + 0.3 m) =

13.1 + 0.6 m

22.2 + 0.4 g

32.5 + 0.7 cm

6 + 1 m


Multiplying or Dividing Measured Values:

This becomes a little more complicated. The rule is:When measured values are multiplied or divided the percentage errors are added.

What is a percentage error?

Answer: a numerical error changed to be represented as a percentage of the measured value


How is this done?Easily:

Mass of Copper is: 9.6 + 0.2 g

Remember the copper:

Percent error =

0.2

9.6 X 100 = 2%

Mass of Copper is:9.6 + 2 % g


Formula for finding Percentage Error:

Percentage error =

Numerical Error

Measured Value X 100


Now try These: Change numerical to percentage error:

13.1 + 0.6 m

22.2 + 0.4 g

32.5 + 0.7 cm

6 + 1 m

13.1 + 5 % m

22.2 + 2 % g

32.5 + 2 % cm

6 + 17 % m

Estimating Uncertainties In Experimental ResultsNow try these:

Remember when measured values are multiplied or divided, add the percentage errors!

1) 22.2 cm + 2 % x 45.2 cm + 5% = 2) 2.31 g + 2 % ÷ 0.76 mL + 3% = 3) 45 + 1 m x 342 + 3 m =

4) {(2.2 cm + 2 % x 5.4 cm + 5%) + 14 + 0.3 cm2} = Careful on this last one!

1000 cm2 + 7 %

3.0 g/mL + 5 %

15400 m2 + 3 %

26 + 1 cm2

Estimating Uncertainties In Experimental ResultsHow to determine the numerical error?

1) Reading a scale: • Use ½ of the smallest division

2) Fluctuating scale:• Look at the range of fluctuations and

divide by 2 • 1/2(maximum value – minimum error)


Volume = 12.3 + 0.3 mL

10

11

12

13

Fill water up to this point

Graduated Cylinder

Measurement & uncertainty pp presentation

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