1. Review of Significant Figures
Dr. C. Yau
Spring 2015
(Loosely based on Sec 1.5)
from Jespersen 7th edition)

2. Accuracy vs. Precision
Accuracy is how close a measurement is to the correct value (commonly accepted value).
Precision is how close several measurements are to each other.
When calibrated instruments are used properly, the greater the number of significant figures, the greater is the degree of precision for a given measurement.

3. Uncertainty in Measurement
When making a measurement, we record all the digits that we are certain of and then estimate one more digit. mL
What is the volume of the liquid in the graduated cylinder to the left?
to the right?
Ans mL? mL?
Ans mL? mL?
There is always uncertainty in the last digit we record.

4. Uncertainty in Measurement
Another way of looking at this is to say that we record to 1/10 th of the smallest increment of the measuring device.
Smallest increment of the graduated cylinder on the right is _____ mL.
1/10 th of that would be ___ mL.
Its reading should therefore be recorded to what decimal place?
Ans. ...to 2 decimal places (21.32 mL) 4

5. Uncertainty in Measurement
Smallest increment of the graduated cylinder on the left is _____ mL.
1/10 th of that would be ___ mL.
Its reading should therefore be recorded to what decimal place?
Ans. ...to 1 decimal place (21.3 mL) 5

6. (Graduated cylinder on the left)
21.3 mL has 3 significant figures (sig. fig.) and 1 decimal place.
(Graduated cylinder on the right)
21.32 mL has 4 sig.fig. and
2 decimal places.
The number of significant digits in a measurement may be increased by using a more precise instrument
Know the difference between the number of sig. fig. and decimal places!

7. Errors Arise From A Number Of Sources Including:
Errors - inherent error due to the equipment or procedure (instrumental or method errors)
Changing volume due to thermal expansion or contraction (temperature changes)
Improperly calibrated equipment
procedural design allows variable measurements
Mistakes-blunders that you know that you have made. Do not use these data
Spillage
Incomplete procedures
Reading scales incorrectly
Using the measuring device incorrectly

8. Reducing Error:
Errors can often be detected by making repeated measurements
Error can be reduced by calibrating equipment
The average or mean reduces data variations: it helps find a central value

9. How many sig.fig. are in a number?
All digits are significant EXCEPT:
Zeros to the left of the first nonzero digit (leading zeros) are never counted as significant.
has only 2 sig. fig.
Zeros at the end of a number without a decimal point are assumed not to be significant (Trailing without a decimal place)
5900 is assumed to have 2 sig. fig.
We don’t really know for sure.
The 2 zeroes are ambiguous, and the number should be expressed in scientific notation.
30.0 on the other hand has 3 sig.fig.
The zeroes are not ambiguous. Why not?

10. Learning Check: How Many Significant Figures Are There In the Following?
2.33 3
500.0 4
Ambiguous, assumed to be 1
1000 3
.0500
How many sig. figs. are there in the number ?
5 sig. fig.

11. Numbers with trailing zeroes & without a decimal are said to be ambiguous in terms of sig. fig. and must be expressed in scientific notation:
3400 in 3 sig. fig. ______________
1530 in 2 sig. fig. ______________
1400 x 103 in 3 sig. fig. _______________
3000 x 10-2 in 2 sig. fig._______________

12. Exact Numbers
Numbers that come from definitions are exact and have no uncertainty.
They can be assumed to contain an infinite number of significant figures.
Example:
25 mph means 25 miles/hour
25 has 2 sig. fig.
Per hour means exactly 1 hour
They do not affect the sig. fig. in a calculation.
( …. hour)

13. Measurements Limit The Precision Of Calculated Results
Rules for combining measurements depend on the type of operation performed:
Multiplication and division
The # of sig. fig. in the answer should be the same as the # of sig. figs of the least precise measurement.
Chem FAQ: How do I count significant digits in the product of two or more measurements?
How do I count significant digits in the result when one measurement is divided by another?
How are measurements rounded to the correct number of significant digits?

14. How many sig. figs. result from the following: 12.33 x 0.00002?
Ans. ONLY ONE!
= = 2 x 10-4

15. Addition and Subtraction
The answer should have the same # of decimal places as the number that is least precise, with the fewest # of decimal places.
← 3 decimal places
← 2 decimal places
← 1 decimal place
← ans rounded to 1 decimal place
Ans
How many sig. figs. result from the following: ?
Tricky! Do this carefully before blurting out the answer!
Ans. Four!
= 10.30

16. Addition and Subtraction
What if the numbers do not have a decimal points?
e.g = ?
Mathematically, the answer is 7332, but should we keep all 4 digits?
7200 is precise only to the hundred place. It will limit the precision of the answer to the hundred place. NO
7332
Ambiguous zeroes
= 7300
= 7.3x103

17. Think Carefully! How many sig. figs. result from the following? 2 3 45
none of these
10.0 x = 108.8
108.8 – 12.2 = 96.6
= 44
Ans. 2 sig. fig.

18. Let’s pinpoint exactly what is meant by “scientific notation.”
It is to express a number so that the decimal is behind the first nonzero digit.
e.g x 104 = =
1302 x =
4.271 x 106
5.1 x 10-3
1.302 x 10-2
Expressing a number in sci. notation should not change the # of sig. fig.

19. Exponential vs. Scientific Notation
How is “exponential notation” different from “scientific notation”?
Technically, “exponential notation” means expressing a number to include x10n without the decimal being behind the first nonzero digit necessarily.
0.25 in exponential notation could be written as 25x10-2 or 2.5x10-1 or x102.
0.25 in scientific notation MUST be written as 2.5x10-1.

20. Do NOT use scientific notation indiscriminately!
Scientific notation is needed ONLY....
1) ...if the number has ambiguous zeroes.
e.g x 2.0 =
2) ...if the number is very small (smaller than 0.01)
e.g =
3) ...if the number is already in exponential notation (but not scientific notation)
e.g x 1056 =
300
= 3.0 x 102
3 x 10-3
x 1057

21. What is wrong with using scientific notation when it is not necessary?
It shows your ignorance on the use of scientific notation.
Would it make sense for someone to tell you to measure out 3.5x101 mL, when all he/she needs to say is ?
Would it make sense for someone to tell you to weigh out 4.5 x 10-1 g when all he/she needs to say is ?
35 mL
0.45 g

22. Significant Figures When Calculating Averages
What is the average of 35.7 and 35.8?
You probably can come up with an answer without a calculator or doing mental arithmetic.
In math, it would be the midpoint,
In science, however, if these were measured numbers you might consider the operation:
What is wrong with this answer?
The average cannot have more precision than the numbers from which it is derived!
Ans. 35.8

23. NO!! One last question on sig. fig......
Think before you blurt out the answer!
What is the answer to the correct sig. fig. for this calculation?
3.42 x x 10-38
Ans x not 3.4 x 10-32
3.42 x 10-32
x 10-38
5.52 x 10-??
NO!!
3.42 x 10-32
x 10-32
3 sig. fig.!
More practice problems are available on my HomePage: Click here.