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Uncertainty & Significant Digits

Uncertainty and Significant Digits

Uncertainty in Measurement

  • Depends on the precision of the measuring device
    • For example a measurement of 1.682956 grams is more precise measurement than 1.7 grams
  • Reliability in Measurements
    • Accuracy: the closeness to the actual scientific value
    • Precision: getting repeated measurements in repeated trials
  • Type of Errors
    • Random: error in measurement has equal probability of being high or low
    • Systematic: errors all occur in the same direction

General Rules for Determining if a Number is Significant

  1. Draw a box around all nonzero digits beginning with the leftmost nonzero digit and ending with the rightmost nonzero digit in the number.
  2. If a decimal is present, draw a box around and trailing zeros
  3. Consider any all boxed digits significant
    • Example 1: 20406 = 5 significant digit
    • Example 2: 0.0045 = 2 significant digit
    • Example 3: 4000 = 1 significant digit
    • Example 4: 4000. = 4 significant digit
    • Example 5: 0.002500 = 4 significant digit
    • Example 6: 3.00 = 3 significant digit

Addition of Subtraction using Significant Figures

  • The answer can only be as precise as your least precise measurement

Add / Subtract Example:

1
2
3
4
5
    2.87     (precise 2 places to the right of the decimal)
    3.5673   (precise 4 places to the right of the decimal)
+ 301.2      (precise 1 place to the right of the decimal)
------------
  307.6373   (rounds off to 307.6 because answer can only have 1 significant digit after the decimal)

Multiplication of Division Using Significant Figures

  • The answer can have no more significant figures than there are in the measurement with the fewest number of significant figures.

Multiply / Divide Example

1
2
3
4
   12.257    (5 total significant digits)
X   1.162    (4 total significant digits)
------------
  14.2426    (rounds off to 14.25 because answer can only have 4 significant digits)
RULE of THUMB

As a rule, when performing a series of calculation, wait until the very end to round off to the proper number of significant figures instead of rounding off each intermediate result.