Uncertainty in Measurement Professor Bob Kaplan University Department of Science

1

Limitations of the Instrument

Individual Skill

Random Conditions

( not under control )

2

Numbers reported should have:

1)  All Digits Known

2)  One Estimated Digit

3

All known digits

andEstimated digit

( uncertain digit )

4

5

6

7

Smallest unit of measurement

on glassware, ruler, scale, etc.

Place value of increment or unit

in the reported number.

8

Degree of uncertainty generally depends on the smallest division or increment of the measuring instrument used (e.g. ruler, graduated cylinder, etc.).

But it also depends on:

Skill of the individual

One person may feel comfortable splitting the

Division in half ( +/- 0.5 unit )

Another person may feel confident splitting the

Division in tenths ( +/- 0.1 unit )

9

10

Reported number: 34.746 meters

Uncertainty level: +/- 0.001 meters

= +/-- 1 mm

Reported number: 34.73579 meters

Uncertainty level: +/- 0.00001 meters

= +/-- 0.01 mm

11

In measured numbers, the “sig figs” include:

All the reported numbers including

the estimated digit.

When we do calculations, we will need to count the significant figures in each of the numbers used .

In each individual number,

all non-zero numbers are counted as “sig figs”.

Zeros may or may not be significant,

depending on their position in the number.

 

12

 

Zeros between integers

Always significant  [ e.

g. 1004 ] 13

Zeros that precede

integers in decimals

Never

significant  [ e.g.  0.0001234

]  

14

Zeros that follow integers -

End of a number

Never

significant  [ e.g.

1,004,000 ]     

15

The purpose of significant figures is to tell

you where to round off the number.

If the first digit to be dropped is 4 or less,

it and all the following digits

should be dropped.

If the first digit to be dropped is 5 or greater,

the last retained digit of the number is

increased by 1.

16

Answer can be no more precise

than the least precise quantity.

Example: Solution of nitric acid that is precise to

+/-0.0001 molarity (moles / liter).

Mix that with another solution that was

not measured precisely at all.

Is the precision of my original solution retained ?

Of course not !!!!

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Result should contain the same number of sig figs

as the measurement that has the least number of

sig figs.18

3 * 1, 465, 876 = 4, 000,000

3 * 1.465876 = 4.0

32 * 550 = 17,600 = 18,000

32 * 560 = 18,920 = 19,000

32 * 568 = 18,176 = 18,000

32 * 575 = 18,400 = 18,000

32 * 580 = 18,560 = 19,000

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“Limiting term” :

Term with fewest

decimal places.

The result is rounded off the

same as number with

fewest decimal places.

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57898.32

+ 33.34567

_____________

57931.66

57931.67

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Consider the number: 564.32

What is the place value of the 5 ?

Hundreds is correct

In a number like 564.32, what single thing determines the uncertainty ???

Last number (or digit) is correct !!

22

What about the last digit is important ???

Place Value !

What is the place value of the last digit

in the number 546.32 ?

Hundredths

So what is the level of uncertainty ????

+/- .01

23

Precision :

Measures repeatability

Accuracy :

Distance from true value 24

True value:

32.146

Accurate measurements:

32.132 , 32.150 , 32.161

Precise measurements:

36.456 , 36.468 , 36.345

25

Systematic Error

Instrumentation

Calibration

Standards of

Measurement 26

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See you all there !27