Science and Measurement
Chapter 1.4
The Foundations of Physical
Science

Working With Measurements
 All measurements involve a degree of uncertainty
 How much error or uncertainty is acceptable?
 In the real world, it is IMPOSSIBLE to make measurement of the exact true value of anything (except counting)

Significant Digits
 Significant Digits or Significant Figures (or Sig
Figs for short) are the meaningful digits in a measured quantity
 If you measure a paperclip and it is between
2.6 and 2.7. We usually say 2.65 cm
 But to a scientist 2.65 means between 2.62 and 2.67
 The final digit in 2.65 is the 5, it represents the smallest amount and is always considered to be rounded up or down

Rules for Significant Digits
 Digits that are ALWAYS significant
 1. Non-zero numbers
 Examples: 54982 = 5 2365149898= 10
 2. Zeros between two significant digits
 Examples: 93100008= 8 1001= 4
 3. All final zeros to the right of a decimal point
 Examples: 4451.630= 7 5012677.26090= 12

Rules for Significant Digits
 Digits that are NEVER significant
 1. Leading zeros to the right of a decimal point
 Examples: 0.002= 1 0.000456= 3
 2. Final zeros in a number that does not have a decimal point
 Examples: 9900000= 2 8765210= 6

WANT A SHORT CUT???
 This method depends on a question .
When you are attempting to determine the number of sig fig’s in a number, ask yourself:
 Is the decimal point PRESENT or
ABSENT ?

Is the decimal point PRESENT or ABSENT?
 If the decimal point is NOT WRITTEN in the number, start on the RIGHT side of the number, go through any zero until you get to the first nonzero digit, underline it and all other numbers.
 The number of underlines is the number of sig fig’s!
300 = __ 1 ___ 2100 = __ 2 ___ 7890900 = __ 5 ___

Is the decimal point
PRESENT or ABSENT?
 If the decimal point IS WRITTEN in the number, start on the LEFT side of the number, go through any zero until you get to the first nonzero digit, underline it and all other numbers. The number of underlines is the number of sig fig’s!
 0.0033 = __ 2 ___ 0.000 000 4040 = __ 4 ___
 2.0000 = ___ 5 __

Significant Figures
Addition/Subtraction Rule
 For addition or subtraction, your answer should be rounded off to the LEAST number of decimal places in the problem.
 325.471
 + 77.210
 402.681  Using the rule, this answer should be reported
 as 402 .68

 1207.2
- 756.739
450.461  Using the rule, this answer should be reported
 as 450.5

Significant Figures
Multiplication/Division Rule
 For multiplication and division problems, your answer should have the same number of sig. figs. as the LEAST number of sig. figs. in the problem.
 How many sig. figs.? 3  2 = 2 .
2.51
 61 = __ 153.11
_____ Final Answer _ 150 ___
 How many sig. figs.? 2 / 3 = 2 .
450/32.1 = 14.0186915
Final Answer ___ 14 ___

Accuracy, Precision, and
Resolution
 Accuracy- how close a measurement is to an accepted or true value
 Precision- describes how close together or reproducible reproducible repeated measurements are
 Resolution- refers to the smallest interval that can be measured

Significant Differences
 In everyday conversation, “ same ” means two numbers that are the same exactly, like 2.56 and 2.56.
 When comparing scientific results “same” means “ not significantly different ”.
 Significant differences are differences that are MUCH larger than the estimated error in the results

Error and Significance
 How can you tell if two results are the same when both contain
(uncertainty)?
error
 When we estimate error in a data set, we will assume the average is the exact value .
 If the difference in the averages is at least three times larger than the average error , we say the difference is “ significant” .

Error
 How you can you tell if two results are the same when both contain error.
 Calculate error
 Average error
 Compare average error