1. Math Toolkit
ACCURACY, PRECISION
& ERROR

2. Uncertainty in Measurements
A digit that must be estimated is called uncertain.
A measurement always has some degree of uncertainty.
Measurements are performed with instruments.
No instrument can read to an infinite number of decimal places.
to an infinite number of
decimal places

3. Accuracy & Precision
Accuracy refers to the agreement of a particular value with the true value.
Precision refers to the degree of agreement among several measurements made in the same manner.
Precise? Accurate? Neither? …or Both?
Neither accurate nor precise
Precise but not accurate
Precise AND accurate

4. Types of Error
Random Error (Indeterminate Error) - measurement has an equal probability of being high or low
Systematic Error (Determinate Error) - occurs in the same direction each time (high or low), often resulting from poor technique or incorrect calibration.
This can result in measurements that are precise but not accurate.
This can result in measurements that are precise, but not accurate.

5. Significant
Figures

6. Sig Figs
As mentioned, the measurements we make in lab have a limited degree of precision. The number we report for the measurement must indicate how precisely our measurement was made.
The digits included in the number we report are referred to as significant figures (or SIG-FIGS). A digit is therefore“significant” if it has been measured.
For every value there are significant figures which are certain digits and there is one uncertain (or estimated) digit. The uncertain digit results from estimating between the divisions on the measuring device’s scale.

7. Counting
Sig Figs

8. Rules for Counting Sig Figs
Nonzero integers always count as Sig Figs.
3456 has 4 sig figs.
Zeros
Leading zeros do not count as Sig Figs.
has 3 sig figs
Captive zeros always count as Sig Figs.
16.07 has 4 sig figs
Trailing zeros are significant only if the number contains a decimal point.
9.300 has 4 sig figs
9300 only has 2 sig figs
Exact numbers have an infinite number of sig figs.
1 inch = cm, exactly

9. Sig Fig Exercise #1 1.0070 m  5 sig figs 17.10 kg  4 sig figs
Count/Identify the SIG-FIGS in each of the following.
m 
5 sig figs
17.10 kg 
4 sig figs
L 
5 sig figs
3.29 x 103 s 
3 sig figs
cm 
2 sig figs
g 
2 sig figs

10. Calculations
with
Sig Figs

11. Rules for Calculations with Sig Figs
Multiplication and Division: # sig figs in the result equals the number in the least precise measurement used in the calculation.
6.38 x 2.0 =
ANSWER: 13 (2 sig figs) 11

12. Sig Fig Exercise #2 Calculation Calculator says: Answer 3.24 m x 7.0 m
100.0 g ÷ 23.7 cm3
g/cm3
4.22 g/cm3
0.02 cm x cm
cm2
0.05 cm2
710 m ÷ 3.0 s
m/s
240 m/s
lb x 3.23 ft
lb·ft
5870 lb·ft
1.030 g ÷ 2.87 mL
g/mL
2.96 g/mL

13. Rules for Calculations with Sig Figs
Addition and Subtraction: The number of decimal places in the result equals the number of decimal places in the least precise measurement. =
ANSWER: 18.7 (1 dec place/3 sig figs) 13

14. Sig Fig Exercise #3 Calculation Calculator says: Answer 3.24 m + 7.0 m
100.0 g g
76.27 g
76.3 g
0.02 cm cm
2.391 cm
2.39 cm
713.1 L L L
709.2 L
lb lb lb lb
2.030 mL mL
0.16 mL
0.160 mL

15. Rules for Rounding Numbers
If the digit immediately to the right of the last significant figure you want to retain is:
Greater than 5, ROUND UP !
Increase the last digit by 1 and turn all digits following to 0.
Less than 5, ROUND DOWN !
Do not change the last digit and turn all digits following to 0.

16. Rules for Rounding Numbers
If the digit following the place to which the number is to be rounded is equal to 5, use the EVEN/ODD RULE.
*** If the number before the 5 is even, leave it as it is.
*** If it is odd, raise it by 1 to make it even.
e.g., 2.5 (rounded to 1 SF) = 2 (EVEN RULE)
3.5 (rounded to 1 SF) = 4 (ODD RULE)