1. Accuracy, Precision, Percent Error, Significant Figures and Rounding
Uncertainty in Data
Accuracy, Precision, Percent Error, Significant Figures and Rounding

2. Numbers in Chemistry are either Exact or Inexact
EXACT NUMBERS
An exact number results from counting objects or is part of a definition
There are 7 skittles in the bag (counting)
There are 12 eggs in a dozen (definition)
There is NO UNCERTAINTY in an exact number

3. Numbers in Chemistry are either Exact or Inexact
INEXACT NUMBERS
An exact number results from a measurement and contains some uncertainty

4. Practice Classify each of the following as exact or measured numbers
A measuring tool is required MEASURED
A defined relationship EXACT
A measuring tool is used to determine length MEASURED
The number of hats is obtained by counting EXACT
The volume of soda is measured MEASURED

5. Accuracy and Precision are terms used with measured values
Accuracy refers to how close a measured value is to an accepted value
Precision refers to how close a series of measurements are to one another

6. Center of Target is Accepted Value

7. Practice
Two students massed the same sample on two different laboratory balances. The results were as follows:
Balance A g g Balance B g g
Which balance is more precise?
  Balance B
If the mass of the sample is actually 12.1 g, which is more accurate?
Balance A

8. Significant Figures
Precision of measured numbers is often limited by the tools available
Can record time to the nearest minute
Can record time to the nearest tenth of a second

9. Significant Figures
The precision of a measurement (how good the measurement is) is indicated by the number of digits reported
Better measuring devices give better measurements with more digits reported
Reported digits are called Significant Figures (sig figs) or Significant Digits
Sig figs include all known digits plus one estimated digit (one digit past the smallest hash mark on the measuring device)

10. Practice
What is the correct measurement for the same piece of wood using each of the rulers below: (remember to record one digit past the smallest hash mark on the measuring device) A
4.5 cm B
4.56 cm
Which measuring instrument allowed us to record a measurement with more digits?
Which is a better measurement?
(For Discussion Only) In which situations would each instrument be appropriate?

11. Rules for Recognizing Significant Figures
Rule 1: Nonzero numbers are always significant
Rule 2: Embedded zeros are always significant
Rule 3: Leading zeros are never significant
Rule 4: Trailing zeros are only significant if there is a decimal point

12. Examples: Circle the significant figures
(rule 1)
62, (rule 2)
(rule 4)
(rule 3)
(rule 4)
(rule 4)
(rule 4)
(rule 3)
405, (rule 4)

13. Rules for Rounding Significant Figures
When rounding calculator answers, look at the digit after the one you can keep:
Greater than or equal to 5, ROUND UP
rounded to 2 sig figs is 2.5
Less than 5, KEEP THE SAME
rounded to 3 sig figs is 45.8
Sometimes, a calculated answer requires more sig figs. In this case, one or more zeros are ADDED.
Calculated answer = 4
Zeros added to give 3 sig figs = 4.00

14. Examples: Round to 3 significant figures
761.50
14.334
10.44
0.08
10789 35
762
14.3
10.4
(added 2 zeros)
10800
8020
204
35.0 (added zero)

15. Percent Error
Percent error can be used to describe the accuracy of results of experimental data if the ACTUAL result is known.
Ex: The known density of water is 1, but when we calculated this in class, some groups got different values
Percent error is always positive (take the absolute value)

16. Percent Error Example:
A student measures the density of an unknown substance (sucrose) and comes up with a measurement of g/mL
The known density of sucrose is 1.59 g/mL
What is the student’s percent error?
1.59 g/mL – 1.54 g/mL x = 3%
1.59 g/mL

17. End here

18. Rounding Numbers
There are TWO main instances when you’ll need to round numbers
When multiplying and dividing numbers
When adding and subtracting numbers
In these two instances, different rules apply

19. Rounding Numbers Multiplying and Dividing
When multiplying and dividing numbers, the answer should have the same number of sig figs as the original data with the fewest number of sig figs
Example:
22.44 / 14.2 =
To round this answer to 3 sig figs, look at the number after the 3rd sig fig
If it’s 0-4, leave the third number as it is
If it’s 5-9, round the third number up to the next number
So, the answer would be 1.58
Fewest # of sig figs (three)

20. Rounding Numbers Multiplying and Dividing
When multiplying and dividing numbers, the answer should have the same number of sig figs as the original data with the fewest number of sig figs
Example:
57 x 1000 = 57000
To round this answer to 1 sig fig, look at the number after the 1st sig fig
If it’s 0-4, leave the first number as it is
If it’s 5-9, round the first number up to the next number
So, the answer would be (not 6!!!!!!)
Fewest # of sig figs (one)

21. Rounding Numbers Adding and Subtracting
When adding and subtracting measurements, the answer must have the same number of decimal places as the original value having the fewest number of decimal places
Example =
To round the answer the ONE decimal place, look at the number right after the 2
If it’s 0-4, leave the 2 as is
If it’s 5-9, round the 2 up to 3
So, the answer would be 77.2
Fewest # of decimal places (one)

22. Rounding Numbers Adding and Subtracting
When adding and subtracting measurements, the answer must have the same number of decimal places as the original value having the fewest number of decimal places
Example =
To round the answer the NO decimal places, look at the number after the 8
If it’s 0-4, leave the 8 as is
If it’s 5-9, round the 8 up to 9
So, the answer would be 769
Fewest # of decimal places (NONE)