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B. UNCERTAINTIES & SIGNIFICANT FIGURES - Coggle Diagram
B. UNCERTAINTIES & SIGNIFICANT FIGURES
Random error
These lead to fluctuations around
the true value because of difficulty
taking measurements
Effect: Some measurements are too small and others too large. Unbiased.
Detectability and correction: Spread about best-fit line. Best-fit line smooths out error.
Repeating measurements: Reduces error by averaging
accuracy & precision : Results may not be precise, but can be accurate
Instruments
Analog instrument – the uncertainty is ± half of the smallest width of the graduations on the instrument.
Digital instrument – the uncertainty is ± 1 the last digit the instrument can read.
Source: Observer
Systematic error
These are instrumental, methodological or human errors causing “lopsided” data which consistently deviates in one direction from the true value.
Effect: Biases measurements in the same direction i.e. either too large or too small
Detectability & correction: Hard to detect. Best-fit line will not pass through origin, but with a straight line.
Accuracy & precision: Not accurate, but can be precise.
Repeating measurements: repeating measurements has no effect
Source: Observer & instrument
Precision vs Accuracy
Accuracy:
The closeness of a measured value to a standard / known / real or true value. Therefore a measurement of the ability to get the true value
Precision:
The closeness of two or more measurements to each other. Therefore a measure of the reproducibility of a set of measurements.
Significant figures
The number of digits used to express a number that carries information about how precisely the number is known
Rules for identification
Any zeros between two significant digits are significant.
All trailing zeros after the decimal comma are significant.
Non-zero digits are always significant.
Rules for calculation
Adding and subtracting: The number of decimal digits in the answer must be equal to the least number of decimal places in the numbers added or subtracted.
Therefore, you are ONLY looking at the decimal portion to determine the number of significant figures
. Calculations only requiring addition and subtraction are only rounded at the end, i.e. no intermediate rounding is done.
Multiplication and division: The answer must have as many significant figures as the least precisely known number entering the calculations.
Therefore, you are looking at the entire number, not just the decimal portion to determine the number of significant figures.
Calculations only requiring multiplication or division are only rounded at the end, i.e. no intermediate rounding is done.
Combination calculations: Calculations involving both multiplication or division and addition or subtraction are rounded in between steps, i.e. you must round between a multiplication or division step and an addition or subtraction step.
An answer is no more precise than the least precise number used to get the answer
Rules for rounding off numbers when calculating
If the digit to be dropped is greater than 5, the last retained digit is rounded up. e.g., 15,6 is rounded to 16.
If the digit to be dropped is less than 5, the last remaining digit is left as it is. e.g., 15,4 is rounded to 15.
If the digit to be dropped is 5, and if ANY digit following it is not zero, the last remaining digit is rounded up. e.g., 15,51 is rounded to 16.
Odd-even rounding rule
When rounding when the last digit to be dropped is 5 and it is followed ONLY BY ZEROES or NOT followed by any other digit we do not
round up if the digit before the 5 is even, but we do round up when the digit before the 5 is odd.
Uncertainty
Random uncertainties occur when an experiment is repeated, and slight variations occur in the measurements
Expression
Percentage uncertainty: The fractional uncertainty expressed as a percentage; i.e. ∆x/x ×100.
Percentage uncertainty= ∆A/A×100
Fractional uncertainty: The proportion of the measured value that is its uncertainty; usually given as ∆x/x. (∆L = ±0,1/11,2 =±1/112 cm)
Absolute uncertainty: The actual uncertainty associated with any measured value; this may be the smallest increment on a measuring instrument. (∆L = ± 0,1 cm)
Causes
Scale reading uncertainty is a measure of how well an instrument scale can be read. This depends on the instrument
Systematic uncertainties occur when readings taken are either all too small or all too large. This depends on flawed measurement techniques or experimental design flaws.
Calculations
Power and roots: When a quantity is raised to the power n, the total percentage uncertainty is n multiplied by the percentage uncertainty.
Addition and subtraction: When quantities are added or subtracted, the uncertainty is the sum of the absolute uncertainties.
Multiplication and division: When quantities are multiplied or divided, the total percentage uncertainty is the sum of the percentage uncertainties.