Types of Numbers (Data)

Exact numbers (data) obtained from counting some conversions (e.g. 2.54 cm = 1 inch, exactly)

Inexact most measured data

Significant figures apply to inexact numbers!

Uncertainty in Measured Data

Measured data is written to convey two (2) things!• the magnitude of the measurement• the extent of its reliability

The more significant figures a measurement has, the more certain it is.

Worker #1 reports a mass of 12 g

Worker #2 reports a mass of 12.0142 g

12 g means 12 ± 1 g

12.0142 g means 12.0142 ±0.0001 g

12 g has 2 significant figures.

12.0142 g has 6 significant figures. 12.0142 g is the more certain (reliable) number.

Measured Values: Accuracy vs. Precision

• Accuracy is how close your measured value is to the right value (can be shown by % error).

• Precision is how well you can reproduce your measurement (can be shown by standard deviation).

accurate and

precise

precise but not accurate

not accurate not precise

Recording Data to the Correct Number of Significant Figures

23°C

22°C

21°C

recorded value = 21.6°C

23°C

22°C

21°C

recorded value = 21.68°C

The number of SFs in a measured value is equal to the number of known digits plus one uncertain digit.

Making Measurements in the Lab: Recording Volumetric Data to the Correct Number of

Significant Figures - Glassware with Graduations

20 mL

30 mL

30-mL beaker: the volume you write in your lab report should

be 13 mL

0 mL

1 mL

2 mL

Buret marked in 0.1 mL: you record volume as

0.67 mL

1. If the glassware is marked every 10 mLs, the volume you record should be in mLs. (Example A)

2. If the glassware is marked every 1 mL, the volume you record should be in tenths of mLs.

3. If the glassware is marked every 0.1 mL, the volume you record should be in hundredths of mLs. (Example B)

10 mL

Example A

Example B

Making Measurements in the Lab: Recording Volumetric Data to the Correct Number of

Significant Figures - Volumetric Glassware

On this volumetric flask is written 500mL ± 0.2 mL. You would

record the volume of the liquid in this flask as 500.0 mL.

Look on the glassware for written indication of the precision of the volumetric flask or pipet.

Making Measurements in the Lab: Recording Masses to the Correct Number of Significant

Figures

This one is easy: record EVERY number (especially zeros) that appears on the

display of the electronic balance.

Trailing zeros MUST be recorded.

How to Count Significant Figures

• All nonzero digits are significant (1.23 has 3 SFs).

• All zeros between nonzero digits are significant (1.003 has 4 SFs).

• Leading zeros are NEVER significant (0.01 has 1 SF).

• Trailing zeros WHEN A DECIMAL POINT IS PRESENT are significant (0.0780 has 3 SFs and 180. has 3 SFs.)

• Trailing zeros WHEN NO DECIMAL POINT IS SHOWN are not significant. (180 has 2 SFs.)

Scientific Notation• An unambiguous way to show the number of

significant figures (SFs) in your data• Numbers are written as the product of a number

greater than or equal to 1 and less than 10 and a power of 10.

Measurement in scientific notation #SFs

186282 mi/s0.0051900 m512.1 x 101 g

1.86282 x 105 mi/s5.1900 x 10-3 m5.121 x 103 g

654

Maintain the Correct Number of SFs When Performing Calculations Involving Measured Data

• Multiplication/Division: The answer contains the same number of SFs as the measurement with the fewest SFs.

25.2 x 6.1 = 153.72 (but only 2 SFs are allowed) = 1.5 x 102 (correct answer)

25.2 x 6.1 = 44.604747 (on my calculator)3.44627

25.2 = 7.3122535 (on my calculator)3.44627 = 7.31 (correct answer)

= 45 (correct answer)

Addition/Subtraction: The answer contains the same number of digits to the right of the decimal as that of the measurement with the fewest number of decimal places.

3.14159+ 25.2 28.34159

28.3 (correct answer)

3 SFs

33.14159 - 33.04 0.10159

0.10 (correct answer)

2 SFs

Calculators do NOT know these rules. It’s up to you to apply them!

Maintain the Correct Number of SFs When Performing Calculations Involving Measured Data

Addition/Subtraction: Dealing with numbers with no decimal places. Convert both numbers to exponential notation with the same power of ten, and then use the decimal place rule.

286.4 x 105 - 8.1 x 103 = ?

286.4 x 105

- 0.081 x 105

286.319 x 105

286.3 x 105 (correct answer)

4 SFs

Maintain the Correct Number of SFs When Performing Calculations Involving Measured Data

Addition/Subtraction: Dealing with numbers with no decimal places. Write out the numbers and underline uncertain digit.

286.4 x 105 - 8.1 x 103 = ?

28,640,000 (uncertain in 10,000 place) - 8,100 (uncertain in 100 place)

28,631,900 (take the uncertain digit farthest to the left)

28,630,000 or 2.863 x 107

4 SFs

Maintain the Correct Number of SFs When Performing Calculations Involving Measured Data

Combined operations: Do the add/subtract first, carrying all digits, then do the multiply/divide. The only time you round is at the very end of the calculation.

Maintain the Correct Number of SFs When Performing Calculations Involving Measured Data

% difference = 100 x (your value - accepted value)

accepted value

This is an exact number. It does not affect SFs!

Maintain the Correct Number of SFs When Performing Calculations Involving Measured Data

% difference = 100 x (3.015 - 3.025)

3.025

Find the percent difference between 3.015 and an accepted value of 3.025.

First, subtract 3.025 from 3.015: 3.015

- 3.025

result has 2 SFs - 0.010

Second, multiply by 100 and divide by 3.025 (4 SFs)

(in either order): - 0.3305785

Finally, round to 2 SFs: - 0.33 %

Maintaining the Correct Number of SFs When Working with Common Logarithms

• Log x = y or 10y = x The number of decimal places in y is the number of SFs in x.

• Calculators do NOT know these rules. It’s up to you to apply them!

Example 1.

Log x = 2.33

(2 decimal places)

x = 213.796209

x = 2.1 x 102 (2 SFs)

Example 2.

x = 561.3 (4 SFs)

Log 561.3 = 2.749195042

Log 561.3 = 2.7492 (4 decimal places)