Uncertainty in numbersfaculty.sdmiramar.edu/faculty/sdccd/fgarces/zCourse/All_Year/Ch15… · II Accuracy, Precision & Significant Figures In general, the uncertainty of a measurement

1 05.20151.03 Significant Figures, Precisions and Accuracy

2.02 Accuracy, Precision and Significant Figures

Story: Taxi driver (14 years experience) points to a pyramid "...this here pyramid is exactly 4514 years old". After a quick calculation, the tourist asked, how can you be so certain that this particular pyramid was built in 2500 BC ?

Uncertainty in numbers

–I had a friend who when to tour Egypt in 2014. Once at Egypt, there was a taxi driver with 14 year experience, who offered to drive him around and take him on a personal tour for a nominal fee. My friend took up the taxi driver on his offer. Toward the end of their tour the cab driver pointed to a set of pyramid and said that the age of the pyramid were exactly 4514 years old. My friend was fascinated because he did not realize that the cab driver was very knowledgeable on the exact date of the grand opening ceremony. After quizzing the cab driver further, my friend discovered that the cab driver had been working with the cab company for six years and six years ago, he was told by the company that this particular pyramid were 4500 years old. 2014

20000 AD

2500 BC 4514


II Accuracy, Precision & Significant Figures

It is important to realize that any measurement will always contain some degree of uncertainty.

The uncertainty of the measurement is determine by the scale of the measuring device.

Uncertainty in Measurements



Precision: Indication of how close individual measurements agree with each other.

Accuracy: How close individual measurements agree with the true value.In general, experimental measurements are taken

numerous times to improve precision;

In generalmore precise e more accurate.

….. but not necessarily true.

Precision vs. Accuracy



Accuracy and precision. Each dot represents one

attempt at measuring a person’s height. a) shows

high precision and great accuracy; the dots are

tightly clustered by the true value. b) Shows

high precision (tightly clustered dots) but poor

accuracy (large error); perhaps the meter stick

was not made correctly. c) Shows poor precision

(dots are not tightly clustered) but, by accident,

high accuracy; the average of the measurements

would be close to the true value. d) Here there is

neither precision and nor accuracya) Precise Accurate

(laudable) -Promote the

analyst

b) Precise Inaccurate

(avoidable) -Repair the instrument

c) Imprecise Accurate (by

accident) -Train the analyst

d) Imprecise Inaccurate

(lamentable) -Hire a new

analyst

Measurements in the lab



High Accuracy - by accident marks are averaged around the target.

Low Precision - grouping is scattered.

Low PrecisionHigh Accuracy



High precision- grouping is tight.

Low Accuracy - but the marks miss the target.

High PrecisionLow Accuracy



Low Precision - grouping is scattered.

Low Accuracy - marks miss the target.

Low Precision Low Accuracy



High Accuracy- marks are averaged around the target.

High precision- grouping is tight.

High PrecisionHigh Accuracy



In general, the uncertainty of a measurement is determined by the precision of the measuring device. A 10-mL pipet with a graduation (hash mark, Δ) of 0.1mL with give an uncertainty of + 0.01 i.e., 1.7X + 0.01 In this example the volume is calibrated to the tenth of a place and the measurement is uncertain to + 0.01 mL. In other words, if a measurement is recorded at 1.75 ml, then the actual volume might be assumed to be 1.74mL or 1.76 mL in which the 4, 5 or 6 is the uncertain digit. The +0.01 mL is arrived at by placing a 1 in the position of the last certain digit (the tenth place) and dividing by 10 (0.1 mL / 10 = 0.01 mL).

A 100-mL graduated cylinder with 1-mL graduation (hash mark, Δ) will have an uncertainty of + 0.1mL. That is a measurement of 53.4 mL using this graduated cylinder could be read 53.3 or 53.4mL.

For calibration of a 50-mL graduated cylinder with a graduation (hash mark, Δ) of 0.2 mL, the uncertainty is +0.05mL.

Devices that measure the volume of liquids. Several devices are used to measure the volume of liquids. Considerations include ease of use as well as accuracy and precision. From left to right; the blue liquid is in a 10-mL graduated pipet that can disperse liquids with an accuracy of +0.06 mL; the reddish liquid is in a volumetric flask that will contain 100.00+0.08 mL when filled to the mark; the yellow liquid is in a 100mL graduated cylinder that measures volumes to +0.1 mL; the green liquid is in a 10-mL graduated cylinder that measures volumes to +0.01mL; the red liquid at the right is in a 10-mL transfer pipet that delivers the liquid with an accuracy of +0.04 mL

Making a measurement



32.33°C 32.3°C

–Reading a temperature measurement The number of significant figures in a measurement depends on the measuring device. Two thermometers measuring the temperature of the same object are shown with expanded views. The thermometer on the right is graduated in 1°C and reads 32.3°C; the one on the left is graduated to 0.1°C and reads 32.33°C. Therefore, a reading with more precision (more certainty) and therefore with more significant figures can be made with the thermometer on the left which is the more precise thermometer



The more accurate the measurement the smaller the errorAccurate number e small errors

The more precise the measurement, the lower the uncertainty. Precise number e small uncertainty

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

D = 1cm error is tenth 1/104.4 cm + .1

When reading a measurement of length, remember that in general-



The more accurate the measurement the smaller the errorAccurate number e small errors

The more precise the measurement, the lower the uncertainty. Precise number e small uncertainty

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

D = 1cm error is tenth 1/104.4 cm + .1

D = 0.1cm error is tenth 1/104.42 cm + .01More precise measurement

When reading a measurement of length remember that in general



SF is an alternative method of measuring uncertainty.

Scientist have found that it is useful to tell the degree of

certainty of a measured number using significant figures (SF).

Merely writing down all the digits that are certain and not writing

down any that are not certain expresses the error in the

measurement.

So why are significant figures (SF) so important?


Significant Figures vs. Precision

Rules of Significant Figures:Nonzero integers - always count as significant figures.Zeros –

Leading zeros - are zeros that precede all of the nonzero digits. They never count as significant figures.

Captive zeros - are zeros that fall between nonzero digits. They all count as significant figures.

Trailing zeros - are zeros at the right end of the number. They are significant only if the number contains a decimal point.

Exact numbers - such as tallies or conversion factors have unlimited number of significant figures.

–How many significant figures do these numbers contain ?

Forget memorizing these rules

because there is a simpler

method

–2?12.000 0.1005 .0005020 2500

–Significant

–5–Figures –4–3 ???

–Chp1: Measurements


•To determine the number of Significant FiguresPacific-Atlantic Rule

Decimal Present vs. Decimal AbsentPlace the number in the center of the map.

Depending if the number has a decimal point or no decimal point determines if the analysis starts at the Pacific or the Atlantic part of US.

Decimal Present

For numbers with a

decimal point present,

draw a line starting from

the Pacific to the first

non-zero number, all digits

shown including the non-

zero number are

significant. i.e., 0.040050

Decimal Absent

For numbers with a decimal

point absent, draw a line

starting from the Atlantic

(right) to the first non-zero

number, all digits shown

including the non-zero

number are significant.

i.e., 30500

– Place Number here

0.040050 30500

–5 Significant Figures –3 Significant Figures–Chp1: Measurements

P A

AP

305000.040050P – Place Number here

AP

0.04005030500


Significant Figures example-

12.000 five significant figures

0.1005 four significant figures

.0005020 four significant figures

2500 two significant figures

Now determine the number of significant figures for the examples we had earlier.

2?12.000 0.1005 .0005020 2500

Significant

5Figures 43 ???

Chp1: Measurements


II Accuracy, Precision & Significant FiguresMore on significant digits and the

different type of uncertainties

i12.000

i0.1005

i0.0005020

i2500

Number of significant figures

5 4 4 2

Implied uncertainty

+ 0.001 + 0.000 1 + 0.000 0001 + 100

Relative uncertainty

+ 0.00112.000

+ 0.000 10.1005

+ 0.000 000 10.000 502 0

+ 1002500

% relative uncertainty

+ 0.008 % + 0. 100 %

+ 0.020 % + 4.0 %

Implied uncertainty is based on the right most significant digit.

i i i i



Manipulation of Significant Figures:

(add and subtract)

Wrong Answer20.4 + 1.32 + 83 + 0.6 = 105.32Wrong Answer


Math operation: Addition-subtraction

Example - keep exponent the same in addition and subtraction operation and round off answer to least precise of the data.

Consider the following example: Three individual who each gave their loose change to the Salvation Army collection pan. The first, had two dollar bills, two quarters and two pennies. The second, donated three dollar bills, five nickels and one penny. The third, gave two quarters

and a few pennies. What is the total amount that was collected?

$2.52 + $3.26 + ~ $0.80 =



What is the difference between significant figures and precision. For

example in the example shown, which number has the fewest

significant figures and which has the least precise ?

. Fewest Significant figures Least precise

a) 123 vs 1.2

b) 1.23 vs 1.200

c) 0.123 vs 0.00012

d) 30 vs 3600

The number with the fewest significant figures is the number which contains the lower amount of digits (significant digits). The least precise number, on the other hand, is the number with the largest uncertainty (error). For example 0.1 (or 1/10) has more uncertainty (larges error or less precision) than 0.01 (or 1/100) (smallest error or more precise).

Difference between Significant figures Precision

1.2 123

1.23 1.230.1230.00012

360030



Addition and Subtraction:Uncertainty of answer (Significant

figures of answer) is limited to the

value with the least precise value

(number with fewest digit after

decimal place - the number 83 in

our example).

Least precise

Manipulation of Significant Figures:

(add and subtract)

Correct Answer



Try this at your desk:

Correct Answer (c)

1056.30 - 23 + 456.500 + 30.2 =

1056.30- 23+ 456.500+ 30.21520.

a) 1520.000b) 1520.00c) 1520.d) 1520e) No right answer


Fewest number of significant figures


Multiplication and division:Uncertainty of answer (Significant.

Figure) is limited to the value with

the fewest significant figures. In

our example, 0.17 limits the value.

7 . 548

4 4 . 40 . 1 7

3 S. F.

2 S. F.

2 S. F.

7 .5Answer with 2 s.f.

Manipulation of Significant Figures:(mult and divide)



Try this at your desk:

Correct Answer (d)

(1056.30 * 23) / (456.500 * 30.2) =

(1056.30 * 23)(456.500 *30.2)

a) 1.76225b) 1.76c) 2d) 1.8e) No right answer =


€

0.0742 × 6.01512( ) + 0.9258 × 0.190100( ) =

.446 .01760 = 0.622

Arithmetic Examples:Combinations using addition-subtraction / multiplication - division

Carry out addition / subtraction operation first before the multiplication or division operation.

€

48.3335.2 - 29.0( )

= 7.7952 = 7.81*

2*

3

€

48.35 - 35.18( ) ∗ 0.12

33.792 - 31.426( ) =

Workingit out

Lower layer

0.742 ∗ 6.01512( ) + 9.9258 ∗ 0.0090( ) =

4.464.46321904

0.0890.0893322

= 4.554.5493322

4*


0.742 ∗ 6.01512( ) + 9.9258 ∗ 0.0090( ) =

4.464.46321904

0.0890.0893322

= 4.55 4.5493322

Arithmetic Examples:Combinations using addition / subtractionand multiplication / division

Carry out addition / subtraction operation first before the multiplication or division operation.

€

48.35 - 35.18( ) ∗ 0.12

33.792 - 31.426( ) = 0.668 = 0.67

€

48.3335.2 - 29.0( )

= 7.7952 = 7.81

2

4


• Exact # or Tallies, i.e., 5 fingers, 176 students.

• *Definition of numbers (quantities).

i.e., Exactly 1 m = 100 cm, or 1 hr = 3600 sec

• Integral numbers that is part of the equation.

circumference of square = length x 4, (4 is an exact number)

*Define conversion versus measured conversion. AWH-atom = 1.008 g/mol


These type of numbers contain unlimited significant figures (do not influence the number of significant figures in the final answer).

Exception to significant figure rules:


One last important note:

Conversion factor comes in two forms, the first are conversion factors from definitions. 60 min = 1 hr, 100cm = 1m, 5280 ft = 1 mile, 1 gal = 3.785 L, 100 pennies = $1.00.

Other conversion factors are based on measured values. 65 mph (65 mi = 1 hr) , $10/hr (10 $ = 1 hr), 0.76Euro/$ (0.76 euro = 1.00$)

Measured conversion factors do have fix amount of significant figures unlike defined conversion factors that have infinite (unlimited) number of significant figures.

Thus in the problem; If a runner ran for 139.9 minutes at 11 mph, how far will the runner travel in miles? The answer is rounded to two significant figures.

Conversion factors: Measured versus defined

139.9%min%⊗ % 1)hr60)min

%⊗ % 11%mi1)hr

%=%25.6483%mi%=%26%mi


II Summary: Significant FiguresWritten digits of results must have right number of significant figures.

Can determine number of significant figures for any number using U.S. map analogy.

• Addition/Subtraction

Least precise number in the data determines number of significant figures in the final answer. (number of SF may change in the operation)

• Multiplication/Division

data with fewest significant figures determines number of significant figures in the answer.

* Remember the exception to the rules for significant figures.

Documents

Uncertainty in numbersfaculty.sdmiramar.edu/faculty/sdccd/fgarces/zCourse/All_Year/Ch15… · II Accuracy, Precision & Significant Figures In general, the uncertainty of a measurement