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Chem 310 – Chapter 2

Measurement and Significant Figure

Tools Needed to Deal with Numbers

Types of numbers

Accuracy and precision

Uncertainty

Scientific notation

Making measurements

Rounding numbers

Significant figures

Significant figures with calculations

Three Types of Numbers

Counted numbers:

I have 11 fingers

there are 12 dogs

Defined numbers:

1 foot =

1 kilometer =

Have NO uncertainty

Will NEVER affect “significant figures”

Measured numbers:

I live 6 and a half miles away

I drink 1.5 liters of water a day

This bench is…

ALWAYS have some uncertainty to them

Will ALWAYS affect “significant figures”

Can be discussed in terms of accuracy and precision

Accuracy and Precision

Accuracy of measurements

How close a measurement is to the “actual” measurement

Determined from a single measurement or from the average of more than one measurement versus a “known” or “accepted” value.

Gaged by percent error

Measured mass: 9.9 grams

“Known” mass: 10.0 grams

Percent error: –1 %

Pretty accurate!

Accuracy and Precision

Accuracy of measurements

How close a measurement is to the “actual” measurement

Determined from a single measurement or from the average of more than one measurement versus a “known” or “accepted” value.

Gaged by percent error

Average of measured volumes: 49.031 liters

“Known” volume: 27.241 liters

Percent error: 79.990 %

Not very accurate!

Accuracy and Precision

Accuracy of measurements

How close a measurement is to the “actual” measurement

Determined from a single measurement or from the average of more than one measurement versus a “known” or “accepted” value.

Gaged by percent error

Average of measured lengths: 202.796 meters

“Known” volume: 197.850 meters

Percent error: 2.500 %

Pretty accurate!

Error Calculations

absolute error: difference between a measurement (or average) and the “known” value

average of measured masses: 11.819 grams

“known” mass: 12.039 grams

absolute error =

absolute error = 11.819 g

absolute error = 11.819 g – 12.039 g

absolute error = 11.819 g – 12.039 g = – 0.220 g

percent error: What is the size of the absolute error relative to the “known” value?

Error Calculations

average of measured masses: 11.819 grams

“known” mass: 12.039 grams

percent error =

percent error = =

percent error = = = – 1.81 %

1.81 %

– 1.81 %

Calculate the percent error given the following data obtained by a student:

Error Calculations

actual: 450.0 oC

trial 1: 545.6 oC

trial 2: 197.2 oC

trial 3: 207.5 oC

trial 4: 476.1 oC

1) Calculate the average:

2) Calculate the percent error:

Accuracy and Precision

Precision of measurements

How close measurements are to each other

Must have two or more measurements

Do NOT need “known” value

Gaged by standard deviation or relative standard deviation

Measurements: 107.5 meters

106.9 meters

109.3 meters

107.1 meters

106.1 meters

Relative standard deviation: 1.1%

Pretty precise!

Accuracy and Precision

Precision

How close measurements are to each other

Must have two or more measurements

Do NOT need “known” value

Gaged by standard deviation or relative standard deviation

Measurements: 215.9367 grams

141.0352 grams

192.4929 grams

117.5333 grams

166.0023 grams

Relative standard deviation: 23.57%

Not very precise!

Standard Deviation Calculations

standard deviation is a way of measuring the “spread” of a set of data

x1, x2, x3, … , xn = individual measurements

= mean (average)

n = number of measurements

Standard Deviation Calculations

relative standard deviation (RSD) : standard deviation divided by the mean

= mean (average)

Calculate the standard deviation and RSD given the following data obtained by a student:

Standard Deviation Calculations

trial 1: 167.8 oC

trial 2: 324.7 oC

trial 3: 121.3 oC

trial 4: 404.6 oC

mean: 254.6 oC

trial 1: 167.8 oC

trial 2: 324.7 oC

trial 3: 121.3 oC

trial 4: 404.6 oC

mean: 254.6 oC

trial 1: 167.8 oC

trial 2: 324.7 oC

trial 3: 121.3 oC

trial 4: 404.6 oC

mean: 254.6 oC

trial 1: 167.8 oC

trial 2: 324.7 oC

trial 3: 121.3 oC

trial 4: 404.6 oC

mean: 254.6 oC

Calculate the standard deviation and RSD given the following data obtained by a student:

Standard Deviation Calculations

trial 1: 167.8 oC

trial 2: 324.7 oC

trial 3: 121.3 oC

trial 4: 404.6 oC

mean: 254.6 oC

s = 132.6

Accuracy versus Precision

accurate and precise

actual: 250.0 oC

trial 1: 250.2 oC

trial 2: 249.7 oC

trial 3: 249.2 oC

trial 4: 249.6 oC

mean: 249.7 oC

% error: – 0.130 %

RSD: 0.165 %

not accurate or precise

actual: 250.0 oC

trial 1: 545.6 oC

trial 2: 197.2 oC

trial 3: 207.5 oC

trial 4: 476.1 oC

mean: 356.6 oC

% error: 42.64 %

RSD: 50.59 %

precise not accurate

actual: 250.0 oC

trial 1: 362.5 oC

trial 2: 361.9 oC

trial 3: 362.0 oC

trial 4: 362.8 oC

mean: 362.3 oC

% error: 44.92 %

RSD: 0.117 %

accurate not precise

actual: 250.0 oC

trial 1: 167.8 oC

trial 2: 324.7 oC

trial 3: 121.3 oC

trial 4: 404.6 oC

mean: 254.6 oC

% error: 1.84 %

RSD: 52.07 %

Accuracy versus Precision

accurate and precise

actual: 250.0 oC

trial 1: 250.2 oC

trial 2: 249.7 oC

trial 3: 249.2 oC

trial 4: 249.6 oC

mean: 249.675 oC

% error: – 0.130 %

RSD: 0.1647 %

accurate and precise

actual: 250.0 oC

trial 1: 250.17 oC

trial 2: 249.68 oC

trial 3: 249.19 oC

trial 4: 249.64 oC

mean: 249.670 oC

% error: – 0.132 %

RSD: 0.1604 %

accurate and precise

actual: 250.0 oC

trial 1: 250.169 oC

trial 2: 249.676 oC

trial 3: 249.194 oC

trial 4: 249.638 oC

mean: 249.669 oC

% error: – 0.132 %

RSD: 0.1596 %

around 12 cm

around 11.8 cm

All about uncertainty

ALL measured numbers have some uncertainty

Uncertainty

around 11.84 cm

around 11.834 cm

Uncertainty and Measurements

Three things to know when taking a measurement:

Units: What is the instrument measuring (mass, volume, time, length, etc.)

Graduations: What is the value of the lines on the instrument

Not always numbered

1 cm

0.2 cm

0.1 cm

0.01 cm

Uncertainty and Measurements

Three things to know when taking a measurement:

Units: What is the instrument measuring (mass, volume, time, length, etc.)

Graduations: What is the value of the lines on the instrument

Not always numbered

Uncertainty: How precise is the instrument?

Is either given or assumed to be one graduation divided by 10

1 cm

0.2 cm

0.01 cm

± 0.1 cm

± 0.02 cm

± 0.001 cm

Precision in Measurements

Read to the “ones” place, uncertainty in the tenths (±0.1)  11.8 cm

Read to the “tenths” place (0.1), uncertainty in the hundredths (±0.01)  11.82 cm

Read to the “hundredths” place (0.01), uncertainty in the thousandths (±0.001)  11.837 cm

11.8 cm

Determine the size of the graduations and the degree of uncertainty

State the digits you can be sure of!

Guess the next digit! (and only the next digit)

uncertainty: ± 0.1 cm

Making Measurements

I KNOW it is between 11 cm and 12 cm

measurement:

uncertainty:

uncertainty: divide graduations by 10 (unless it is given)

graduations (marks): 1 cm

graduations (marks):

11.8

11.84 cm

Determine the size of the graduations and the degree of uncertainty

State the digits you can be sure of!

Guess the next digit! (and only the next digit)

uncertainty: ± 0.01 cm

I KNOW it is between 11.8 cm and 11.9 cm

measurement:

11.8

uncertainty:

uncertainty: divide graduations by 10 (unless it is given)

graduations (marks): 0.1 cm

graduations (marks):

11.84

Making Measurements

520

520 cm

between 500 cm and 600 cm

measurement:

graduations (marks):

uncertainty:

graduations (marks): 100 cm

uncertainty: ± 10 cm

Determine the size of the graduations and the degree of uncertainty

State the digits you can be sure of!

Guess the next digit! (and only the next digit)

Making Measurements

The final digit in a measurement must ALWAYS be in the SAME PLACE AS, and be a MULTIPLE OF the uncertainty!

If your instrument has an uncertainty of ± 0.01 mL, your measurement will end with .00 .01 .02 .03 .04 .05 .06 .07 .08 or .09

If your instrument has an uncertainty of ± 0.2 g, your measurement will end with .0 .2 .4 .6, or .8

If your instrument has an uncertainty of ± 0.005 cm, your measurement will end with .000 or .005

Uncertainty and Measurements

Relationship between uncertainty of an instrument and its measurements:

The final digit in a measurement must ALWAYS be in the SAME PLACE AS (have the same precision as) the uncertainty!

If uncertainty is ± 0.1, then all measurements must have one decimal place (no more and no less)

If uncertainty is ± 0.0001, then all measurements must have four decimal places (no more and no less)

If uncertainty is ± 10, then all measurements must be to the tens place (no more and no less)

Uncertainty and Measurements

29.3 mL, 234.0 g, 0.2 s

0.5787 g, 8.0001 m, 17.3650 s

870 km, 20 m, 10 s

Relationship between uncertainty of an instrument and its measurements:

The final digit in a measurement must ALWAYS be a multiple of the uncertainty!

If uncertainty ends with a 2, (i.e. ± 0.0002, ± 0.02, etc.) then all measurements’ final digit must be a multiple of 2

If uncertainty ends with a 5, (i.e. ± 0.005, ± 0.5, etc.) then all measurements’ final digit must be a multiple of 5

If uncertainty ends with a 1, (i.e. ± 0.01, ± 1, etc.) then all measurements’ final digit must be a multiple of 1

Uncertainty and Measurements

1.04 mL, 234.0 g, 27.06 s

0.0235 g, 8.000 m, 17.5 s

89 kg, 2.4 m, 1.00 s

Which of the following would be correct if measured on this ruler?

Uncertainty and Measurements

too many decimals

(more precise than ruler)

too few decimals

(less precise than ruler)

too many decimals

(more precise than ruler)

assume ± 0.1 cm uncertainty

a) 7.0 cm

b) 2.50 cm

c) 12 cm

d) 0.6 cm

e) 4.00 cm

Which of the following would be correct if measured on this ruler?

Uncertainty and Measurements

too many decimals

(more precise than ruler)

too few decimals

(less precise than ruler)

too few decimals

(less precise than ruler)

assume ± 0.02 cm uncertainty

a) 11.5 cm

b) 7.50 cm

c) 1.004 cm

d) 7.5 cm

e) 12.68 cm

For the graduated cylinder to the right, provide the following information (Given: the uncertainty is ± 0.5 mL)

What is the size of the graduations?

1 mL

What is the volume?

67.6 mL ± 0.5 mL

Uncertainty and Measurements

Significant Figures

Significant figures tell you about the uncertainty in a measurement.

Significant figures include all CERTAIN (known) digits, and ONE UNCERTAIN (guessed) digit

11.84 cm

4 significant figures

The guessed digit is always the last significant figure in a value

Significant Figures

Significant figures include all CERTAIN (known) digits, and ONE UNCERTAIN (guessed) digit

520 cm

2 significant figures

This is called a “place-holder” zero. These are NEVER significant (though they are important)

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60. mL

70. mL

50. mL