Uncertainties+in+calculated+results

As we saw earlier there is some extent of uncertainty when measuring a value. For example as measuring cylinder might measure values to 0.1 cm 3 . Thus if you take a reading of say 10.0 cm 3  it can be plus or minus by 0.1 cm 3 , so you state it as follows 10.0±0.1 cm 3 . Remember that 10.0 cm 3  has onc d.p and 0.1 also has 1 d.p—you can’t say 10.000± 0.1 cm 3  since the in this case the value has more decimal places (is more accurate) than the uncertainty range, which is not possible. (±0.1) This is called absolute uncertainty which shows the actual value of the uncertainty.

There is also percentage uncertainty. This is calculated as follows:



Percentage uncertainty shows what percentage of the value is likely to be an error. Making use of the previous example: 10.0 ± 0.1 cm 3 . The absolute uncertainty is 0.1 cm 3  while the percentage uncertainty is

So the percentage uncertainty is ±1%.

//Calculating uncertainty is the answer: // You might have to add/subtract, multiply/divide or find the root of values such as (2.5 ±0.1) and (3.0±0.5), what do you do?? How can you know the uncertainty in the answer??? Once again there are simple rules to follow:

<span style="font-family: Tahoma,sans-serif; font-size: 14pt;">When you add (2.5 ± 0.1) + (3.0 ± 0.5) you get 5.5 ± ???. <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">To get the uncertainty in the answer add the uncertainty of the values, so add 0.1 + 0.5= 0.6. <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">Therefore the answer is 5.5 ± 0.6
 * <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">Whenever you add/subtract <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> numbers, the absolute uncertainty in the answer is the sum of the uncertainties of the values in the calculation. Eg:

<span style="font-family: Tahoma,sans-serif; font-size: 14pt;">So, (2.5 ± 0.1) X (3.0 ± 0.5)= 7.5 ± ???. what do you do? <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">Find the percentage uncertainty in each value:
 * <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">Whenever you multiply/ divide <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> values, the uncertainty in the answer is the sum of the percentage uncertainties of the values in the calculation.





<span style="font-family: Tahoma,sans-serif; font-size: 14pt;">Add the two 4+16.7=20.7 %. <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">20.7% is the percentage uncertainty in the answer. To find the absolute uncertainty:



<span style="font-family: Tahoma,sans-serif; font-size: 14pt;">So the absolute uncertainty in the answer is ±2. (remember that an uncertainty should always be given to one significant figure). __//<span style="font-family: Tahoma,sans-serif; font-size: 14pt;">The same procedure is to be followed when dividing numbers. //__

<span style="font-family: Tahoma,sans-serif; font-size: 14pt;">So, if you have: (2.5± 0.1) 2 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">. 2.5 2 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> = 6.3. <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">First find the percentage uncertainty <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">Multiply it by the power (2) 4% X 2 = 8% <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">Therefore 8% is the percentage uncertainty in the answer. <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">Find the absolute uncertainty in the answer: <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">So the final answer is 6.3 ± 0.5
 * <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">If you raise a number to the power ‘n’, multiply the percentage uncertainty of the number by ‘n’ to get the percentage uncertainty of the answer.
 * <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">Similarly, if you find the ‘nth’ root of a number, divide the percentage uncertainty of the number by ‘n’ to get the percentage uncertainty of the answer.

<span style="font-family: Tahoma,sans-serif; font-size: 14pt;">//Likewise if you find the nth root you divide the percentage uncertainty by n.// Eg: <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">The percentage uncertainty in the value is <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">Divide by the power (3) you get 4/3 <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">Find the absolute uncertainty in the answer = <span style="font-family: Tahoma,sans-serif; font-size: 14pt;"> <span style="font-family: Tahoma,sans-serif; font-size: 14pt;">So the final answer is 1.4 ± 0.02

NEXT UP- GRAPHICAL TECHNIQUES
 * By the end of this lesson you should be able to: **
 * ** State uncertainties as absolute and percentage uncertainties **
 * ** Determine the uncertainties in you answer when making calculations. **
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