2. Uncertainty Quantified doubt of a measure

3. A few basics An uncertainty value can only have one significant figure * Recall that leading zeros are never significant The measured value must end at the same decimal place as the uncertainty value If the uncertainty value is in the tenths place, then the measured value can not end in the thousandths place

4. Most often The uncertainty value is ½ the value of the most precise division of the measuring tool – If the smallest division is 1 mm, then the uncertainty value would be + / - 0.5 mm – Example: Ι I I I I I I I I I Ι 1 mm intervals   1.3 + or -.5 mm

5. Another method: average deviation from the mean Multiple measures (of the same thing) can result in slightly different results In this case, calculate the mean measure Then sum all the absolute deviations from that mean Divide by the number of measures – Example: 1.50g, 1.60g, 1.70g mean = 1.60g deviations =.10,.00,.10 sum =.20 / by 3 =.07 Therefore: 1.60 +/-.07g

6. Addition of uncertainty Sum the uncertainty Values 4.05 +/- 0.02g + 3.06 +/- 0.03g = (4.05 + 3.06) +/- (0.02 + 0.03) = 7.11 +/- 0.05g Wow it really is that easy!!!!!!

7. Subtraction example 4.05 +/- 0.02g - 3.06 +/- 0.03g = (4.05 – 3.06) +/- (0.02 + 0.03) = 0.99 +/- 0.05g Just take care to add the uncertainty values! If your uncertainty value is larger than your measure – then your measure is not all that useful!

8. Multiplication/division Sum the % of the uncertainty values 4.05 +/- 0.02g x 3.06 +/- 0.03g = (4.05)(3.06) +/- (0.02/4.05 x 100 + 0.03/3.06 x 100) = 12.39 +/- (.49% +.98%) = 12.39 +/- 1.47% = now chg. The % to a measure 12.39 +/-.18 = 12.4g +/- 0.2g Note the uncertainty value only has one sig fig Note the decimal place of the measure is the same as the decimal place of the uncertainty

9. Other calculations: If multiplying a measure and uncertainty value by a coefficient: Keep the uncert. % the same for the multiplied value. Multiply the % of the uncertainty value: (.5)(8 +/- 2) =.5(8) +/- (25%) = 4 +/- 25% = 4 +/- 1 If 7.5 +/- 2 Limit measure to same decimal place as uncertainty value = 8 +/- 2 If uncertainty value is greater than the measure then the measure tells us nothing!