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Calculations involving margin of error

by Frank
(Ottawa, Ontario, Canada)










































How do I compute the ± margin of error in calculations? For example (6.00 ± 0.01) divided by (2.00 ± 0.01)?

Comments for Calculations involving margin of error

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Nov 20, 2011
Calculations
by: Staff


Question:

by Frank
(Ottawa, Ontario, Canada)


How do I compute the ± margin of error in calculations? For example (6.00 ± 0.01) divided by (2.00 ± 0.01)?


Answer:

I’m not quite sure what your two numbers represent, but they look like they represent an engineering tolerance rather than a margin of error.

An engineering tolerance is not the same thing as a margin of error. In general, an engineering tolerance is a maximum deviation from the specified value such as 6.00 ± .01 mm.


The margin of error is a sampling statistic.

The margin or error quantifies the random sampling error in survey results.

The margin of error is reported like this:
6.00 ± 0.01 with a 95% confidence level.

Without specifying the confidence level (is it 90%, 95%, 99% ?) the any number purporting to be the margin or error is meaningless.

The Margin of Error can be calculated as follows:

Margin of Error formula = (Z value)*(Standard Deviation)/( square root of sample size)



The Z value is determined by the confidence level you are trying to achieve

Confidence level, z* value

(ref: http://people.richland.edu/james/lecture/m170/ch08-int.html)

90% 1.645
95% 1.96
98% 2.33
99% 2.58


So where does that leave the answer to your original question?

The best thing I can suggest would be to use the number of significant digits as a basis for the division.

For example (6.00 ± 0.01) divided by (2.00 ± 0.01)?

(6.00 ± 0.01) / (2.00 ± 0.01)

Round each number to the nearest 1/10

= (6.0) / (2.0)

= 3.0 (not 3.00, but 3.0)


On this basis (significant digits), the answer to your question = 3.0






Thanks for writing.

Staff
www.solving-math-problems.com


Nov 20, 2011
Response to staff's comment
by: Frank


Thanks for your input.

The numbers were meant to be general examples of numbers with margins of error, not engineering tolerances.

Both the numerator and the denominator have a margin of error of ± 0.01.

How is this margin of error reflected in the quotient? Does it remain ± 0.01? Or are the individual margins of error added, giving an answer of 3.00 ± 0.02? Or something else? Clearly the quotient must contain a margin of error.

I don’t think that consideration of significant figures comes into play.


Nov 21, 2011
Margin of Error in Calculations
by: Staff



First, let’s summarize:


When imprecise numbers are used in calculations, there will always be a “propagation or error” (or, propagation of uncertainty) in the result.

We know that (6.00 ± 0.01) ÷ (2.00 ± 0.01) will definitely contain an error.


(1) The easy way to compensate for the error in the quotient

As far as I know, the most straightforward (albeit crude) way to deal with the “propagation of error” is rounding to significant figures (significance arithmetic).


(2) Improved Calculation using Relative Uncertainty

You can improve upon the rounding method by actually calculating the error (not just compensating for it) in the quotient of (6.00 ± 0.01) ÷ (2.00 ± 0.01). Use the relative uncertainty of both numbers as a starting point.


(there are some assumptions involved which I am not going to list)

Relative uncertainty = the absolute uncertainty of a measurement divided by the best value for that estimate.

Relative uncertainty of 1st value = (1st margin of error)/(1st value)

Relative uncertainty of 1st value = (0.01)/(6.00)


Relative uncertainty of 2nd value = (2nd margin of error)/(2nd value)

Relative uncertainty of 2nd value = (0.01)/(2.00)



Relative uncertainty for division = sqrt{[(1st margin of error)/(1st value)]² + [(2nd margin of error)/(2nd value)]²}

Relative uncertainty of (6.00 ± 0.01) ÷ (2.00 ± 0.01)

= sqrt{[(0.01)/(6.00)]² + [(0.01)/(2.00)]²}

= sqrt(10)/600

= 0.0052704627669

Remember, we are using relative values. We will convert the answer shown above into a percent.

Relative uncertainty of (6.00 ± 0.01) ÷ (2.00 ± 0.01)

= 0.52704627669 %


The division of (6.00 ± 0.01) ÷ (2.00 ± 0.01)

= 6.00 ÷ 2.00

= 3.00 ± Relative uncertainty of (6.00 ± 0.01) ÷ (2.00 ± 0.01)


= 3.00 ± 0.52704627669 %

= 3.00 ± 0.0158114

= 3.00 ± 0.02


The division of (6.00 ± 0.01) ÷ (2.00 ± 0.01) = 3.00 ± 0.02


(3) An alternative Calculation using margin of error

1st value = 6.00 ± 0.01

5.99 ≤ 1st value ≤ 6.01



2nd value = 2.00 ± 0.01

1.99 ≤ 2nd value ≤ 2.01


Quotient of (6.00 ± 0.01) ÷ (2.00 ± 0.01)

2.9801 ≤ Quotient ≤ 3.0201


Averaging the extremes


Quotient = [(2.9801 + 3.0201)/ 2] ± [(3.0201 - 2.9801) / 2]


= 3.0001 ± 0.02

= 3.00 ± 0.02


The division of (6.00 ± 0.01) ÷ (2.00 ± 0.01) = 3.00 ± 0.02


Thanks for writing.

Staff
www.solving-math-problems.com


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